ES raga i in Raw AMERICAN MATHEMATICAL SOCIETY COLLOQUIUM PUBLICATIONS, VOLUME VIII NON-RIEMANNIAN GEOMETRY BY LUTHER PFAHLER EISENHART PROFESSOR OF MATHEMATICS IN PRINCETON UNIVERSITY NEW YORK PUBLISHED BY THE AMERICAN MATHEMATICAL SOCIETY 501 WEST 116TH STREET 1927 v “ v v CEE + o> « el = CERRY vo ves Rem =n - - bees “ serv wows . tev het; v > ES -" PEI « ® - sev ves vi vee «®» - « wena Fy» “a, ere wT . ,e > . - «v ol. ‘ee LUTCKE & WULFF, HAMBURG, GERMANY PREFACE The use of the differential geometry of a Riemannian space in the mathematical formulation of recent physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi- Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Kddington, Einstein and Weyl, in their efforts to formulate a combined theory of eravitation and electromagnetism, proposed a simultaneous generalization of this process and of the definition of paral- lelism. This generalization consisted in using general functions of the coordinates in the formulas of covariant differentiation in place of the Christoffel symbols formed with respect to the fundamental tensor of a Riemannian space. This has been the line of approach adopted also by Cartan, Schouten and others. When such a set of functions is assigned to a space it is said to be affinely connected. From the affine point of view the geodesics of a Riemannian space are the straight lines, in the sense that the tangents to a geodesic are parallel with respect to the curve. In any affinely connected space there are straight lines, which we call the paths. A path is uniquely determined by a point and a direction or by two points within a sufficiently restricted region. Conversely, a system of curves possessing this property may be taken as the straight lines of a space and an affine connection deduced therefrom. This method of departure was adopted by Veblen and the writer in their papers dealing with the geometry of paths, the equations of the paths being a generalization of those of geodesics by the process described in the first paragraph. 111 710173 iv PREFACE In presenting the development of these ideas we begin with a definition of covariant differentiation which involves functions Lj: of the coordinates, the law connecting the corresponding functions in any two coordinate systems being fundamental. Upon this foundation a general tensor calculus is built and a theory of parallelism. Much of the literature on this subject deals with the case where the connection is symmetric, that is Lj — Lf; When the paths are taken as fundamental, this is the type of connection which is derived. This restriction is not made in the first chapter, which deals accordingly with asymmetric connections. Vectors parallel with respect to a curve for an asymmetric connection retain this property for certain changes of the connection. This is not true of symmetric connections. How- ever, it is possible to change a symmetric connection with- out changing the equations of the paths of the manifold. Accordingly when the paths are taken as fundamental, the affine connection is not uniquely defined, and we have a group of affine connections with the same paths. a situation analogous to that in the projective geometry of straight lines. Accordingly there is a projective geometry of paths dealing with that theory which applies to all affine connec- tions with the same paths. In the second chapter we develop the affine theory of symmetric connections and in the third chapter the projective theory. For a sub-space of a Riemannian space there is in general an induced metric and consequently an induced law of parallelism. There is not a unique induced affine connection in a sub-space of an affinely connected space. If the latter is of order m and the sub-space of order =, each choice at points of the latter of m—mn independent directions in the enveloping space but not in the sub-space leads to an induced affine connection, and to a geometry of the sub- space in many ways analogous to that for Riemannian geometry. Under certain conditions there are preferred choices of these directions, which are analogous to the normals to PREFACE v the sub-space. The fourth chapter of the book deals with the geometry of sub-spaces. A generalization of Riemannian spaces other than those presented in this book consists in assigning to the space a metric based upon an integral whose integrand is homogeneous of the first degree in the differentials. Developments of this theory have been made by Finsler, Berwald, Synge and J. H. Taylor. In this geometry the paths are the shortest lines, and in that sense are a generalization of geodesics. Affine properties of these spaces are obtained from a natural generalization of the definition of Levi-Civita for Riemannian spaces. Berwald has also obtained generalizations of the geometry of paths by taking for the paths the integral curves of a certain type of ditferential equations, and Douglas showed that these are the most general geometries of paths; he also developed their projective theory. References to the works of these authors are to be found in the Biblio- graphy at the end of the book. This book contains, with subsequent developments, the material presented in my lectures at the Ithaca Colloquium. in September 1925, under the title The New Differential (Geometry. I have given the book a more definitive title. In the preparation of the manuscript I have had the benefit of suggestions and criticisms: by Dr. Harry Levy, Dr. J. M. Thomas and Mr. M. S. Knebelman, the latter of whom has also read the proof. September, 1927. s L.uTHER PFAHLER KISENHART. Section 1. pe RN = 11, 12. NN DN eed CONTENTS CHAPTER 1 ASYMMETRIC CONNECTIONS Transformation of cobrdinates . ........ «ove Coctlicionts of conneelion 0. ual Covariant differentiation with respect to the L’s.. Generalized identities of Rice... ....0. 0... Other fundamental tensors ....... aed Covariant differentiation with respect to the 7's. . Parallelism. . Pathe 0... i... 0.00... A theorem on partial differential equations. ...... Fields of parallel contravariant veetors.......... Parallel displacement of a contravariant vector around an infinitesimal eiveliit.. Jo. 0. oes Pseudo-orthogonal contravariant and covariant vec- tors. Parallelism of covariant vectors........... Changes of connection which preserve parallelism . Tensors independent of the choice of ¥;........- Semi-Symmetric CONNECHONS vv noi vvicv wivis vidas Transversals of parallelism of a given vector-field and assoclate'vector-flelds -...«. . 0. ii. in Associate divectlonS oo. io aan a ae Determination of a tensor by an ennuple of vectors and Invariants... co 0 a ar a The invariants y," of an emnuple............... Geometric properties expressed in terms of the invariants Fo aie th Lee a ee Ld de CHAPTER 11 SYMMETRIC CONNECTIONS Geodesic cotrdinates a i The curvature tensor and other fundamental tensors vi 36 50 53 5d CONTENTS Section . 99. Bquations. of the paths. ..........00. oii, 93. Normal eobrdinates. . ... iv hus niin alt dus 94, Curvature of a eurve. ..... .. ie. ai cons 25. Extension of the theorem of Fermi to symmetric CONNECHIONR: 0... va ainsi penis Bl 906. Normal tensors... i. simi ya 27. Fxtensions of a 1O0K0F .. 20s nnn, 28. The equivalence of symmetric connections. ....... 29." Riemannian spaces. Flat spaces. .........v...., 30. Symmetric connections of Weyl................. 31. Homogeneous first integrals of the equations of the PRthS. a ee CHAPTER III PROJECTIVE GEOMETRY OF PATHS 32. Projective change of affine connection. The Weyl fensor in Ll a Se a eae 33. Affine normal coordinates under a projective change of connection... so. iil. 0 wasn an a, aa 34. Projectively flat spaces ......... 0.0 hia 0s 35. Coefficients of a projeetive connection ........... 36. The equivalence of projective connections........ %.. Normal affine connection =. iw iin sen cia 38, : Projective parameters of a path .....c.aui nn. 39. Coefficients of a projective connection as tensors. . 40. Projective cotrdinates . . ......0 0 oh ial nk 41. Projective normal codrdinates .................. 42. Significance of a projective change of affine connection 43. Homogeneous first integrals under a projective change 44. Spaces for which the equations of the paths admit n(n-+1)/2 independent homogeneous linear first in- fagvals 0. em a ei 45. Transformations of the equations of the paths. ... 46. Collineations in an affinely connected space . ..... 47, Conditions for the existence of infinitesimal col- HReations «oi an a ag a 64 68 2 4 8 81 83 91 94 98 100 104 106 107 110 111 116 117 119 122 124 129 viii CONTENTS Section ; Page 48. Continuous groups of collineations............... 133 49, , Collineations ina Riemannian space............. 134 CHAPTER 1V THE GEOMETRY OF SUB-SPACES 50. Covariant pseudonormal to a hypersurface. The vector=fleld »% =... 2 0 ea ne en 137 51. Transversals of a hypersurface which are paths of cL the enveloping 8paes. ... 0... ooo indivi ini 141 52. Mensors in a hypersurface derived from tensors in the enveloping space 0... .............. cu. 143 53. Symmetric connection induced in a hypersurface .. 148 54. Fundamental derived tensors in a hypersurface ... 150 Db. The generalized equations of Gauss and Codazzi.. 152 56. Contravariant psendonormal . ................ ... 154 57. Fundamental equations when the determinant o is NOL BPA Se ia a 158 58. Parallelism and associate directions in a hypersurface 160 59. Curvature of a curve in a hypersurface.......... 162 60. Asymptotic lines, conjugate directions and lines of enrvature of a -hypersupface ........o.0. 000, 163 61. Projectively flat spaces for which 3; is symmetric 166 62. Covariant pseudonormals to a sub-space ......... 170 63. Derived tensors in a sub-space. Induced affine GOMNECHION: on. i a 171 64. Fundamental derived tensors in a sub-space...... 172 65. (Generalized equations of Gauss and Codazzi...... 3 66. Parallelism in a sub-space. Curvature of a curve Mea subspaces: io: nL a aa a 177 67. Projective change of induced connection ......... 178 BIGLIGREADPHY oo ys a ae 181 CHAPTER 1 ASYMMETRIC CONNECTIONS 1. Transformation of codrdinates. Any ordered set of » independent real variables 2%, where 7 takes the values 1, ---, n, may be thought of as coordinates of points in an n-dimensional space V, in the sense that each set of values of the x's defines a point of V;,. The terms manifold and A oi are Synonymous with space as here defined. If ¢?(x!, ..., 2") for i = 1, oi are real functions, whose jacobian is net identically zero, So equations (1.1) 2 Sh gq" Ct, sive 79 (7 = 1, “vr n) define a transformation of codrdinates in the space V. If 22 and 2" are functions of the z's and z’’s such that ; a OF (1.2) Al — J ox in consequence of (1.1), 4’ and 2" are the components in the respective coordinate systems of a contravariant vector. In (1.2) we make use of the convention that when the same index appears as a subscript and superscript in a term this term stands for the sum of the terms obtained by giving the index each of its n values; this convention will be used throughout the book. From (1.2) we have by differentiation ol ax" on oe Pal Bf = i RE 2 1:5) 0x’ 28 8% dad 8x" 0x8. Ba) It is assumed that the reader is familiar with relations con- necting the components of a tensor in two codrdinate systems.* * (Of. 1926, 1, pp. 1-12. References are to the Bibliography at the end of the text. 1 a2. I. ASYMMETRIC CONNECTIONS He will observe that, because of the presence of the last term in the right-hand member of (1.3), the derivatives of 2? and 2'° are not the components of a tensor. Consider further a symmetric covariant tensor of the second order whose components in the two codrdinate systems are g;; and Jus such that the determinant (1.4) 9 = |g] is different from zero. From the equations ie bal dat Yep oa 9s Br” 5B we get by differentiation / . : n 0p ay 0 ij dak. Pot: Dak ak ee ox’? 0x" p4® 84f ox’ rnd AZ ne) J Nn, i Spl on na om Bx? Bus Rr Bx ay A similar observation applies to these equations. However. there are n(n +1)/2 of these equations, and they can be 9 3 ia oF x2 2 : solved for the n*(n-+1)/2 quantities IS We obtain x xX 52,4 = pl nak a intl a.) 2 bo | 1 0a fi 3 | 0 E ox’ oa'd ~ Vkl pa” ba’ \eBl pu where I are the Christoffel symbols of the second kind, that is, 0gjn Filo. apnoy tania ld [22 tg (1.6) Viel 7 is, Lh 2 2 \ank 2 oad pal)’ where ¢” are defined by Dg gn =4d, G=1or0asi=joritj 20 c sity 0* x 3 acti When we eliminate Te from (1.3) by means of (1.5). a ont we obtain 21926, 1, ». 19. 92. COEFFICIENTS OF CONNECTION 3 Ng /{ nrc 0 x* 0 xX # ; RS LET (1.8) == ln At Bud? where we a0 yt re 02" 1? 0 2 sm ah) i 2 B= = 8 A 22 (1 )) J Bs +7 \hj if 548 of We From (1.8) we see that A’; and 4'“ 3 are the components of a tensor in the two coordinate systems. Thus we have formed a tensor by suitable combinations of the first derivatives of the components of a vector and a tensor. It g; is the fundamental tensor of a Riemannian space, then A’; is the covariant derivative of 2’. However, the theory of covariant differentiation in a Riemannian space has nothing to do with the fact that the tensor g¢; is used to define a metric. Consequently this theory can be applied to any space, if we make use of any tensor ¢;; such that g + 0.* 2. Coefficients of connection. We have just seen that when a symmetric tensor g; is specified for a space we have an algorithm for obtaining tensors from other tensors: by differentiation. But this process is a special case of a much more general one. In fact, the fundamental element in the ; ; ; aa , 92 xt former consisted in the elimination of — re ox (1.3) by means of (1.5). From this it is evn that if Lj and L'%s are functions of the x's and z's satisfying the equations ; from equations . Bt [nad Ban 5 8a (2.1) Wry diy, Sn pes dlisely dx" 0x ox" 8x” ox” the quantities 2°; and 4'“;, where 0 At 7 i (99 steal 2b 7 pO Ol Ar (2.2) 4; id Li, 2 gol a gi rd are in the is (2.5) Pym 2% dat oa'P Ea 3 : ox’ and consequently are components of a tensor. * Of. 1926, 1, pp. 26-30. 4 I. ASYMMETRIC CONNECTIONS Conversely, if equations (2.3) are to hold for any vector, it follows from (1.3) that we must have 02 xt af ; 0k dat auf, TE ea which are equivalent to (2.1), since pa’? oa 8 dad ba" ; RE ET TE TY If we take any set of functions La of the z's, equations (2.1) determine the corresponding functions in any other coordinate system x’° such that equations (2.2) define the components of the same tensor in the two coordinate systems. The articular form of the functions Lj: in (1.5) arose from a tensor g;j, and there are other ways (cf. § 18) in which ° we get functions Lj and LZ; ¢g in two coordinate systems satisfying (2.1). Whenever in any way such a set of functions is assigned to a manifold we say that the latter is connected and that the L's are the coefficients of the Sa From (1.6) it is seen that the symbols I ” in 7 and k. We remark that from the fore of (2.1) it follows that, if the Z's are symmetric in the subscripts in one coor- dinate system, the corresponding coefficients in any coordinate system are symmetric. We do not make the restriction that they be symmetric, and for the present consider the more general case where the connection is asymmetric. Cartan® uses the terms with torsion and without torsion for the asymmetric and symmetric connections respectively. When we express the conditions of integrability of equa- tions (2.1), making use of (2.1) in the reduction, we obtain are symmetric 2 i . Aa" dxl 0xk dat 9 (2.5) Loa— = 22 = Lah —— dx'® dxf bal £1923. Bb, np. 325, 326. 3. COVARIANT DIFFERENTIATION BH where al 0 y @6) Lu = 5 — 70 +L Lie — Li Lia, and similarly for Lij,. From the form of (2.5) it follows that Lj; and Les, are the components of a tensor. Also if in (2.6) he functions Lk are replaced by the Christoffel symbols formed with respect to g¢;, the tensor Li J becomes te Riemannian curvature tensor of a Riemannian space with the fundamental tensor g;.* Accordingly we call Lj the curvature tensor of the space.t 3. Covariant differentiation with respect to the L’s. Since (2.2) are a generalization of (1.9), we call the tensor of components A’; the first covariant derivative of A* with respect to the given connection, or briefly, with respect to the L's. If 2; are the components of a covariant vector-field, it is readily shown by means of (2.1) that the quantities 4; j, given by 3.1) Thy we l Ba — A Li, are the components of a tensor of the second order. It is the first covariant derivative of the vector A; with respect to the L's. In general it can be shown that, if SRM, are the com- Pp ponents of a tensor, the quantities ga Von = a TE nit (3.2) gy Ni % iy 5 Blo eyes Te are the components of a tensor of order m +p +1, the first covariant derivative of the given tensor. As a consequence of this definition we have £1926, 1, p. 19. T Cf. Schouten, 1924, 1, p. 83. 6 I. ASYMMETRIC CONNECTIONS The first covariant derivative with respect to the L's of the tensor 0; is zero. If in (3.2) the L’s are replaced by the corresponding Christoffel symbols (1.6) of the second kind, we obtain the formulas for covariant differentiation with respect to the fundamental tensor of a Riemannian geometry.* As in the latter we can establish the theorem: Covariant differentiation of the sum, difference, outer and immer product of tensors obeys the same rules as ordinary differentiation. In order that this theorem may hold for the case of an invariant obtained by the inner multiplication of a contra- variant and a covariant vector, it is necessary that we define the first covariant derivative of an invariant (or scalar) to be its ordinary derivative. : Since the Christoffel symbols i: are symmetric in j and k, equations (3.2) are not the only generalization of covariant differentiation in Riemannian geometry. Thus if in (3.2) we replace I and Hi by I and Th. we again obtain com- ponents of a tensor, as follows from the following considerations. When we put (3.3) TL = 35, we have from (2.1) that 92 are the components of a tensor. Consequently the differences between the quantities defined by (3.2) and those obtained by the change described above are the components of a tensor. Still other definitions of covariant differentiation are possible. Thus recently KinsteinT was led to the consideration of the equations : oy — an Liy— ap; Lis = 0. From (3.2) and (3.3) it follows that the left-hand members of these equations are the components of a tensor. However, ©1926, 1, p. 28. +1925, 11. 4. GENERALIZED IDENTITIES OF RICCI T it ij = Ai pj, the above theorem as regards products does ‘not hold,* if we define the left-hand member to be the covariant li 'derivative of aij. When dealing with asymmetric connections, we shall ad- here to (3.2) as the formula for covariant differentiation. Several subscripts preceded by a solidus (|) indicate repeated covariant differentiation with respect to the Z's. 4. Generalized identities of Ricci. If ¢ is an invariant, its second covariant derivative is given by 0% 6 0; = at Bal — LE 9 fs From this expression it follows that : k (4.1) By— 0 = —205 25, where or are defined by (3.3); we recall also that they are the components of a tensor. Proceeding in like manner with a contravariant vector AZ, a covariant vector Z; and a covariant tensor a;, we obtain respectively ni i Al 7 ; 4.2) 2 jn — Kj — — 4 2 Li —2 Zu 22 43) Afp—day == Ay Lin—20 2. (44) aym— gm = apy Lig + an L a Bn 2. And in general we have / ” % Ton Aa rr ! Tn oy: Tm rh (4.5) Qs vea8, he Ds, <8, |Uk = a, “Sq MSg yy a, Ys or ce, mM In Py Ty Trg y+ Pe 24 vey 7 = Adria, wl re 8,| (17.40 The foregoing identities are generalizations of the Ricci identities of Riemannian geometry t+ When covariant differen- tiation is used, it is advantageous to use (4.5) in place of the * Cf. J. M. Thomas, 1926, 13, p. 189. T1926, 1, p. 30; cf. Schouten, 1924, 1, p. 85. 8 I. ASYMMETRIC CONNECTIONS customary conditions of integrability of ordinary differentiation, namely ? | 2 ) me 2 he dal \ Bo dat \ dx! the derivation of the generalized identities of Ricci. 5. Other fundamental tensors. If we put J which are used in fact in (5.1) Th = 5 (T+ Th, it follows from these equations and (3.3) that (5.2) Ihe = Th +h. Thus rh and 2 are the symmetric and skew-symmetric parts respectively of Lj. Substituting these expressions in (2.6), we obtain a i 5 i (5.3) Lia = Biju + Lj, where i 0 ry , TE i = i oy J J (5.4) Bj mm "Y ho e ri. — 7 Jl ri and Lie i i oh (35) Lu = Qp— DL + & Lh 0 2 — Djs & 2 —32 24 7A £5. £ L i From equations (2.1) and (5.2) we have ia a 2 0a dak a Bal ee — "2 en = #2 = (3.5) pf ou +7 earl da” Since these equations are of the form (2.1), it follows that Bj, are the components of a tensor. This is evident also from (5.3), since Lj are the components of a tensor. From (5.3), (5. 4) and (5.5) it follows that the tensors La, Blu and Qj; are skew-symmetric in the indices % and /. If Bj; denotes the contracted tensor Bj; and bj and Bj ‘denote respectively the symmetric and skew-symmetric parts of Bj, we have from (5.4) 0 I'ty J = L Ta + + ji Chie— Lj Tha, Li] Loar 0a’ 0 xh 1 (3.7) bp = I 5. OTHER FUNDAMENTAL TENSORS 9 0 xk 0 xt We shall show that gj; is the curl of a vector. In fact," let ¢;; be any symmetric tensor of the second order and form the Christoffel symbols of the second kind; if we put ee 1 9 rl Ir (5.8) fn = 1 ¢ ii 21h) (5.9) he = J lel -+ a, it follows from (1.5) and (5.6) that af are the components of a tensor symmetric in the indices ; and k. Since {2 I: 2 logVyg ila ox’ 2 it follows from (5.8) and (5.9) that 10m Ew, ; Leh (5.10) Bj ee 9 Sh ou), % = Sa 2 gt If in place of taking the tensor g; we had taken any other tensor g;, the function g/g is a scalar, and consequently a; in (5.10) would have been replaced by a; plus a gradient. From (5.4) and (5.8) we obtain (5.11) Sy = By = 24, If we put 2 (5.12) G = 0; — — 0, we have from (5.5) by contraction for 7 and / and for / and j respectively A i I i h b. 13) & Qj; ri —_— —(Q Qi ly + Q, 25 he 25 Op and ! ; 0 £2 0 8, op ot 7 k (5.14) Dy = yg — — oak dal As a consequence of (5.3), (5.10), (5.11) and (5.14) we have: The skew-symmetric tensor Lb, is the curl of the vector — (ai + L;), where a; is determined to within an additive arbitrary gradient. * 1926, 1, p. 18. 10 I. ASYMMETRIC CONNECTIONS When the expressions (5.9) are substituted in (5.4) and we denote by B, , the components of the Riemannian curvature tensor for the tensor ¢;, we have = i a P i alt (5.15) By = Boyt Uke f. + a aly a — Aj; Wy» where a semi-colon followed by an index indicates covariant differentiation with respect to the ¢’s. Contracting for 7 and / and for ¢ and 5, we have 5 : a TAT i I oh (5.16) By, = 5 +a, —a iT a ols Wi. a, and (5.10), in consequence of (5.11). From (5.16) it follows that the symmetric part of Bj is a = 1 (5.17) by = EB, + 9 (a, + 0 wh. + a; ahr — Ch, ty If the symmetric tensor 0;, defined by (5.7), satisfies the condition that the determinant |b; is not identically zero, it may be made to play a role for the manifold analogous in some respects to that of the fundamental tensor in Riemannian geometry. It is the tensor which would naturally be used for the tensor g; in the above equations to give determinateness to these equations.* From (5.8), (5.10) and (5.11) we have a nw eh q 0 Ly; 0 Ip; 0 a; 0a; A = — a — — = Y oat ox’ 0 xt ox’’ from which it follows that a function ¢ is defined by the equations 0 log Vg h (5.18) I Liq —i. From the relation (5.6) it follows that between gy and the corresponding function ¢’ in another coordinate system = we have the relation Yi Vo y g | 0x’ ® CL. Fisenhart, 1923, 4, p. 373. 6. COVARIANT DIFFERENTIATION 11 Consequently we have the invariant integral 1 = | Vy dot: do? = { Vi dat. da’, If in (5.18) we replace a; by a; + pi arbitrary scalar, then Vg is replaced by Vg 6. Hence for a given symmetric connection there is no uniquely defined | fundamental integral like the volume integral of a Riemannian space. If, however, the tensor Sj; is zero for the connection, the function ¢ defined by (5.18) with a; — 0 is uniquely defined and thus we have a volume integral for the space® which is analogous to that of a Riemannian space.t 6. Covariant differentiation with respect to the 7s. Since the I's satisfy (5.6), which are of the form (2.1), it follows from (3.2) that the quantities , where # is an rote oat Vg 1,..-,m ; Poe ein Sf Tg IO ga a a mo — = ; he > a fi 1/ a1 mJ oo < « P J pH! da’ x Pp JK yy a 1 m / oid ; sp 5 CIERRL) 1KSpy-s, (6.1) are the components of a tensor. This may be seen also by substituting the expressions (5.2) in (3.2) and observing that the differences between the resulting expressions and (6.1) are components of a tensor. The process defined by (6.1) we call covariant differentiation with respect to the I's and use a comma followed by indices to denote this type of covariant differentiation. In terms of covariant differentiation with respect to the I's equations (5.5) become ; i i i Bel Lo (6.2) Qa = Lie — Lj,r + Ln Le — Lp La _* Cf. Veblen, 1923, 8; Eisenhart, 1923, 9. 71926, p. 18. CE 1996. 1, p. 23, 12 I. ASYMMETRIC CONNECTIONS If # is an invariant, we have (6.3) bu—0, = 0. Also we have the following generalized identities of Ricei: pricey) 7, ea Pe ao 7 a $9, or 7, CA >i Agen FL Wes og ll TE > Wiis. Ts , Ds akl 4 ; » « 1 a—1"%a1;" (6.4) an Wk Tyo Pp oy hlpey: Ty 73 = Ds: 2S By . 7. Parallelism. Paths. In a general manifold there is no a prior: basis for the comparison of vectors at different points. For a Riemannian manifold parallelism of vectors, as defined by Levi-Civita,* serves as a basis for such com- parison. This definition may be generalized for a connected manifold. We say that a curve is the locus of points for which the coordinates xz? are functions of a parameter f. Let C' be any curve and consider the system of differential equations (1.1) 2 47h pl where the x's in the L’s are replaced by the functions of ¢ for C'. A solution of these equations, that is a set of functions A, ..., A" satistying them, is determined by arbitrary values of the 4’s for a given value of #, in accordance with the theory of differential equations. Consider such a solution. Since the 4's are functions of ¢ and likewise the x's, the 4’s are expressible, in many ways, as functions of the x's. Assume that the 2's considered as functions of the x's are substituted in (7.1) and that the . Je == (J, resulting equations are multiplied by and 7 is summed, 2’ being the coordinates of any other system for the space. By means of equations obtained from (2.1) by interchanging the x's and »'’s, the resulting equation is reducible to di’ 10 3 dx’ mtr army EIT, 0 7. PARALLELISM. PATHS 13 where Consequently a set of functions A¢ satisfying (7.1) are for each value of 7 the components of a contravariant vector. We say that they are parallel to one another with respect to the curve, and that any one of them may be obtained from any other by a parallel displacement of the latter along the curve. From the above remarks it follows that a family of vectors exists parallel to any given vector at a point of C. Since parallelism has thus been defined in terms of the connection, we say that the connection is affine and that the L’s are the coefficients of affine connection. Two vectors at a point are said to have he same direction, if corresponding components are proportional. Accordingly, it a set of functions 4° satisfy equations (7.1), the vectors of components (7.2) 7 — gL, where ¢ is any function of ¢, should be interpreted as parallel with respect to the given curve C. From (7.1) and (7.2) we have Co dr’ da® nl, (7.3) LPI =I, where og it 2 Conversely, if we have any set of functions 4° of ¢ which satisfy (7.3), they are the components of a family of contra- variant vectors parallel with respect to C; and by means of (7.2) and (7.4) we find the vectors A¢ satisfying (7.1). From (7.3) we have, on eliminating #(¢#) and omitting the bars, as) Li ai . dak ns at J At J dil 14 I. ASYMMETRIC CONNECTIONS as the conditions of parellelism which hold for (7.2) what- ever be ¢. As a particular example of the foregoing we consider the curves whose tangents are parallel with respect to the curves. From (7.5) it follows that the equations of these curves are dad. [d? ai i dak 2 re Lg At Ten a dot z 2 | J dx" gad = 0 hE LR (7.6) ai and that. conversely, any curve defined by these equations possesses the above property. We call these curves the paths of the manifold. They are an evident generalization of the geodesics of a Riemannian manifold.™* From the form of (7.6) it is evident that all connected spaces for which the I's are the same but £2j, are arbitrary have the same paths. Later (§ 12) it will be shown that this is not a necessary condition. 8. A theorem on partial differential equations. (on- sider a system of partial differential equations (8.1) a = Yr) (ea=1,-.-, M;i=1, .... nm), 0 at where the ys are functions of the 6's and z's. The conditions of integrability of these equations are Gy a 20 oer O03 PVD Bar i ox! 068 oad Ql Sor ont’ where 8 and y are summed from 1 to JM. If these equations are satisfied identically, the system (8.1) is completely inte- grable and the general solution involves M arbitrary constants. For in this case we can obtain developments in powers of the x's, with constant coefficients, which satisfy (8.1), the coefficients being determined by the initial values of the 8's. *1926, 1, p. 50. 8S. PARTIAL DIFFERENTIAL EQUATIONS 15 If equations (8.2) are not satisfied identically, we have a set Fy, of equations, which establish conditions upon the 6's as functions of the z's. If we differentiate each of these . y : 3 JH equations with respect to the »’s and substitute for oT x from (8.1), either the resulting equations are a consequence of the set F; or we get a new set F,. Proceeding in this way we get a sequence of sets, Fy, £5, -.., of equations, which must be compatible, if equations (8.1) are to have a solution. If one of these sets is not a consequence of the preceding sets, it introduces at least one additional condition. Consequently, if the equations (8.1) are to admit a solution, there must be a positive integer NV such that the equations of the (V-41)th set are satisfied because of the equations of the preceding N sets; otherwise we should obtain more than M independent equations which would imply a relation between the 2's. Moreover, from this argument it follows that N << J. Conversely, suppose that there is a number N such that the equations of the sets (8.3) A are compatible and each set introduces one or more conditions independent of the conditions imposed by the equations of the other sets, and that all of the equations of the set (8.4) Fy are satisfied identically because of the equations of the sets (8.3). Assume that there are p (<= M) independent conditions imposed by (8.3), say G, (8, x) = 0. Since the jacobian Gy matrix | — f 00% as solved for p of the #’s in terms of the remaining 6's and the 27s, and the equations are then of the form (by suitable numbering) : | is of rank p, these equations may be regarded (8.5) 0% — ¢° (pr+1, vey oY, x) = 0 (0 == 1; rvs, P) 16 I. ASYMMETRIC CONNECTIONS From these equations we have by differentiation 20° 0g° 00" 0g" i | 3 50° Do Ba =0 v=p—+1,---, M). 06° 9 xt Replacing by means of (8.1), we have 0° i 0 ¢° 0, bh SE oO." which equations are satisfied because of the sets (8.3) and (8.4), as follows from the method of obtaining the latter. Accordingly we have by subtraction 06° Su IY dat iT er \ Ba (8.6) — wi) — 0. From these equations it follows that, if the functions r+... 6M are chosen to satisfy the equations 0 0” 0 xt (8.7) Tons yr (grt, ST AY, 7), where Wr is obtained from ¥% on replacing 6° (0 = 1, .--, p) by their expressions (8.5), then equations (8.1) for « =— 1, --., p are satisfied by the values (8.5). Since the equations of the set F are satisfied identically because of (8.5), it follows that equations (8.7) are completely integrable; for, the equations arising from expressing their conditions of inte- grability are in the set Fj. Consequently there is a solution in this case and it involves M—p arbitrary constants. When p — M, we have in place of (8.5) 9° = ¢%{(z) and in place of (8.6) that the functions 6“ satisfy (8.1). In this case there are no constants of integration. Hence we have: In order that a system of equations (8.1) admit a solution, it is mecessary and sufficient that there exist a positive integer N(L M) such that the equations of the sets Fy, ---, Fy are compatible for all values of the x's in a domain, and that 8. PARTIAL DIFFERENTIAL EQUATIONS 17 the equations of the set Fy i1 are satisfied because of the former sets; if p is the number of independent equations in the first N sets, the solution involves M— p arbitrary constants.™ It is evident from the above considerations that when an integer N exists such that the conditions of the theorem are satisfied, they are satisfied also for any integer larger than N. However, it is understood in the theorem and in the various applications of it that NN is the least integer for which the conditions are satisfied. The above theorem can be applied also to the case when there are certain functional relations between the 6’s and 2’s which must be satisfied in addition to the differential equations (8.1). In this case we denote by F, this set of conditions, and include in the set /' of the theorem also such conditions as arise from F}, by differentiation and sub- stitution from (8.1). Then the theorem proceeds as above with the understanding that the sets Fj, Fi, ..., Fy shall be compatible, and that the set Fi; shall be satisfied because of the former. In certain cases (cf. § 36) the equations of the set F} consist of two sets FY and FY’, such that, if FY and Fy’ are those which follow from Fy and FY’ respectively, then the set FY’ is a consequence of Fy. In this case equations Fy are a consequence of Fy and so on. Hence we have that all the solutions of FY, ..., Fn, satisfy the set Fi.i». Accordingly in applying the theorem we have only to consider -the sequence FY, ..., F/, -.. When the functions ¥¢ in (8.1) are linear and homogeneous in the #’s, the same is true of the equations of the sets 1, Fs, + --; moreover p is at most equal to M—1. From algebraic considerations it follows that the conditions of the problem are that there exist a positive integer N such that * This theorem for the case M = p was used by Christoffel, 1869, 1, p. 60 in the solution of a certain problem (cf. § 28) and was used for the general case in the same problem by Wright, 1908, 1, pp. 16, 17; cf. also, Bianchi, 1918, 1, pp. 9-13; Levi-Civita, 1925, 5, pp. 40-43 and Veblen and J. M. Thomas, 1926, 6, pp. 288-290. 18: i I. ASYMMETRIC CONNECTIONS the rank of the matrix of the sets F}, ..., Fyis M—p(p=>1) and that this is also the rank of the matrix of F, ---, Fxi1. When these conditions are satisfied, the solution of the problem reduces to the integration of a completely integrable set of equations (8.7), in which now the ¥’s are linear and homogeneous. Consequently any solution is expressible as a linear function with constant coefficients of p particular solutions, and such an expression with arbitrary constant coefficients is a solution. Most of the applications of this theorem which we shall make are to equations of this linear type. Moreover, these equations are of the form in which the #’s are components of a tensor and in place of their derivatives we have first covariant derivatives. 9. Fields of parallel contravariant vectors. When we have any contravariant vector-field of components 47, the vectors of the field at points of a curve C are parallel, if dh (9.1) TE (ah am Bg) = 0, as follows from (7.5). In order that these equations be satisfied for the vectors of the field along any curve of the space it is necessary that WB te— 0, = 0, from which it follows that (9.2) Al L== Al Wi where ui is a covariant vector. When gw; is not a gradient, I do” ob = depends upon the curve, so that if the vector A’ at a point P is subjected to parallel displacement around a closed circuit the resulting vector at / will depend upon the path; this is shown in § 10. This will not be the case if wi is a gradient, in which case (9.2) may be written the function uy - (9.3) Mp we po 0E 9. FIELDS OF PARALLEL VECTORS 19 A field of vectors satisfying equations (9.3) is said to be a parallel field. 1f we change the components replacing 4¢ by Aig, the new components satisfy the equations 0 Al Ee 5 TZ L Ly A — 0 . (9.4) hy == If 4% are the components of a parallel field they define a congruence of curves along any one of which it is possible dat to choose a parameter { so that A = or Then from (9.3) or (9.4) and (7.6) we have: The curves of a congruence of curves determined by a Joli of parallel vectors are paths. From (4.2) we have that the conditions of integrability of equations (9.4) are (9.5) 2h = 0. When (9.6) Li ji p= GQ. equations (9.4) are completely integrable, that is, a solution is uniquely determined by arbitrary values of A’ at a given point. Hence we have: A necessary and sufficient condition that there exist a field of contravariant vectors parallel to an arbitrary vector is that (9.6) be satisfied. From equations (9.5) we have also: A necessary and sufficient condition that a V, admit n linearly independent fields of parallel contravariant vectors is that the curvatnre tensor Lj be a zero tensor. If equations (9.6) are not satisfied, on differentiating (9.5) covariantly, we have in consequence of (9.4) 9.9) 7 2 Lom) My 0. Proceeding in like manner with these equations, we obtain the sequence of equations 90 I. ASYMMETRIC CONNECTIONS h ri 2 Lj mm, —_— 0, q \ h ri (9.8) A Lj mm, --m = 0, Equations (9.4) are of the form (8.1). Hence in consequence of the results of § 8 we have: A necessary and sufficient condition for the existence of ome or more fields of parallel contravariant vectors is that there exist a positive integer N such that the first N sets of equations (9.5), (9.7) and (9.8) admit » (C1) fundamental sets of solutions, which satisfy the (N+ 1)th set of equations; if these conditions are satisfied, there are r linearly independent fields of parallel vectors and any linear combination, with constant coefficients, of these vectors is a parallel field. ™ Having thus obtained the conditions for one or more fields of parallel contravariant vectors in invariantive form, we shall show how all such fields may be obtained by making a suit- able choice of coordinates. Suppose we have » fields of parallel vectors of com- ponents A(, where «, for « = 1, ..., », denotes the vector and ¢ the component; we use the notation that an index in paren- theses indicates an entity, one without parentheses a com- ponent. *In another coordinate system z’‘ we have 1] god ny Bu (9.9) v(ct) = Ale or . Consider the system of differential equations c 2 d0 (9.10) Xo ll) = Ale TT ha Since by hypothesis the functions es satisfy (9.4), the Poisson operator applied to equations (9.10) gives * This theorem for the case 2, — 0 was established by Veblen and TZ. Y. Thomas, 1923, 1, p. 530. 9. FIELDS OF PARALLEL VECTORS : 21 od he oy 5 j 06 9.11) (XeX;—X3Xe) 0 = 2a Xp 2 We consider first the case when ok == (), that is, when equations (9.4.) become (9.12) FE) In this case equations (9.10) form a complete system, and thus there are n— 1» independent solutions 6° (x!, ..., 2") for 6 = r-+1,...,n. If we omit any one of equations (9.10), the remaining ones form a complete system and admit in addition to the above another independent solution. In this way we get v other functions 020, ..., a") flor a = 1, ..., r, 6% being the additional solution when X.(8) = 0 is omitted. If we put (9.13) 2 = {, 0), from (9.9) and (9.10) it follows that in the coordinate system 2‘, the components A{e, are zero unless i = «. Suppose now that equations (9.12) are expressed in this coordinate system, which we call #?; then the components of the » vectors are of the form : (9.14) 2 DS J. We, « not being summed. From (9.12) we have (9.15) rh mad 8 uf Since the 7s must be symmetric in the lower indices, it is necessary that ¥. be a function at most of x% 211, ..., 27 Hence we have the theorem: The most general space with a symmetric connection admitting r fields of parallel contravariant vectors is obtained by choosing arbitrarily the coefficients I 2 Jor o and © equal Of Goursar, 1391, 1, p. 532. 29 Z I. ASYMMETRIC CONNECTIONS tor—+1, ---, m, and for the others expressions of the form (9.15). where Wy 95 a function of 2% 277, +. ., When the connection is asymmetric, the quantities (9.14) satisfy (9.4) if d a% where « is not summed. Hence we have: A space with asymmetric connection admitting r fields of parallel contravariant vectors is defined by (9.16) where Ye are any functions of the x's and the other L's are arbitrary. In particular, if » = n, it follows from the above results that the tensor Lj is zero. This is readily verified for the expressions (9.16). If in equations (2.1) we replace Ll by expressions of the form (9.16) jor o, 8,7 = 1, -.., nm, we have 0 2 2 Ii 0 x) 0 ak 0 log Ye, 0 at : Y == ht Se , 3 ~ = yn = ( - 2% 0x THE pa” px’ 2x? ax” ) o where e is not summed in the last term. Since by hypothesis Lig are zero, and Lio are zero when v, are arbitrary functions of the z's, it follows from § 2 that the above equations are completely integrable. Hence we have: When the curvature tensor of a space with asymmetric connection is zero, a coordinate system exists for which the coefficients have the form (9.16), the n functions VW. being arbitrary. If the ¥’s are constants, the coefficients must be zero, which is possible only in case of a symmetric connection. as is evident from (2.1) if we take Lj = 0. In this case we have as a corollary of the above theorem: When the curvature temsor of a space with a symmetric connection is zero, a codrdinate system exists for which all of the coefficients of the comnection are zero. 10. Parallel displacement of a contravariant vector around an infinitesimal circuit. In order to consider *® Eisenhart, 1922, 1, p. 210. 10. DISPLACEMENT OF A VECTOR 23 the parallel displacement of a vector around an infinitesimal circuit, we consider a surface, that is a manifold of two dimensions, defined by equations af = f%(u,»), where the functions / and their derivatives to the third order exist and are continuous at a point P of the surface. We consider the circuit comprising the points P (u,v), Qu + du,v), Ru—+ du, v+dv), Su, v4 dv) and P. We take a vector A¢ at P and find from (7.1) the components of the vector at () parallel to it, then in the same way the vector at RE parallel to this vector at @ and so on. The components of the resulting vectors are given by (Mr = (Mg + Lo oT : [7 oo + (10.1; nue - i, = (Hp an : (C2) ae Tp I - . Af) 2 " where the quantities such as (22) ’ [2 , and so forth \ Av] Q A) are obtained from equations of the form (7.1). When all of the above equations are added, we obtain ; on y Lt i AQ)p = (W)p— (Mp = [ie = & ) Jao ee (10.2) + [2 i Sd i dv + - ! 5 : + - 2) | du? [+ (5) Joes At P we have dA ; ee — {Li faa dA? | BJ. Po 5 Bab pat | Sar AJ i [a 7 32 L i du? an “I (2 Ji = Loall ver ce 0 24 I. ASYMMETRIC CONNECTIONS 3 Ic When the functions Lj and ow at () are replaced by their expansions about P and use is made of (7.1), we have adr nl 75] al = (Li 4 ov /p { 3 i 27 pn 0aF Bat eal a ae i fo (10.4) bi | Du Lijx Bo Lux Li on 2 Sut » da? A a] ; Daf n 82k dx (For) = v2, Lin i — Lu Ln oof + In like manner we obtain Rid — (44) dup dur 5 2a wl las [= (250 5 Pe Lin Ln du du 2 {2 7; oo 5 Bak 2] : ok (1 or Lin Lig — Tr : et . Sd ~ 2 dele Ydvls {2/0.; a2 pn Bak Bad om ARI | or Li; J i [= (Li ov Liu dv Ou lip a oak dal . 0 i 9 gh Vy ot bi La (2 a Li Th —— | dv +. d? Ai a2 Al We remark also that the expressions for ie 5. and ft 70 kid s : rd 2) differ from those for fay and ” TT) respectively, as given by (10.3) and (10.4), only in terms of the first and higher orders of the differentials. When these expressions are substituted in (10.2) we have oy fy 00 af 7 _— J 5 - v aise, 0.3) AGH 2 Lhe 22 22 3 dn dot From the considerations of § 9 it follows that A(A)p = 0 when Lj; = 0. The same is true when 2° belongs to a field 10. DISPLACEMENT OF A VECTOR 25 of parallel vectors. But in the general case when a vector undergoes parallel displacement around an infinitesimal circuit the difference between its final and original position is of the second order and depends upon the value of the com- ponents Lia at the starting point.™ Let Ale be the components of » independent contravariant vectors at P, where «, for « = 1, ..., n, indicates the vector and 4, for i = 1, -.., n, the component. If {hese veciors are displaced about an infinitesimal circuit and we denote by 7% the determinant Aly |, then from (10.5) we have ;, 0xF xt .6 / 2 = — (i 2 —— ee u se ti Sa al (10.6) AA) Li ) du dv | Hence for this variation to be of the third or higher order it is necessary that [cf. (2.6)] -7 i ; 45 Lin ge — > — re pen 0. | From these equations it follows that ¢ Blogy (10.7) 1; = ah In another coordinate system z* we have and we desire to find the relation between ¢ and ¢’. Krom (2.1) we have : ; 0 ; dod Li == 5 log A+ Lj; i rs Cs? x xz i where A is the jacobian Consequently we have, to or U within a negligible constant factor, * CL. Schouten, 1924, 1, p. 34. T Of. Schouten, 1924, 1, p. 89. 26 I. ASYMMETRIC CONNECTIONS (10.8) gp’ a 9A, that is, ¢ is a scalar density. BL denote the components of the vectors in the co- ordinates 2’° and 2’ denotes the determinant |i(,, . we have 2 A oy that is, 4 is a relative invariant of weight —1. Accordingly L9 is an invariant (or scalar). If now we take » linearly independent contravariant vectors parallel with respect to a curve ' and let ¢ be any scalar density, we have ds 0g sX dat eo Em np TL id bs vy i dt (10.9) = Consequently if the invariant 2¢ so formed with respect to every curve in space is to be constant along the curve, it is necessary and sufficient that (10.7) hold, the function ¢ being thus determined. In particular, if the connection is symmetric. we have in place of (10.7) : g 0 log ¢ (10.10) =r Then from (5.8) and (5.11) we have that (10.11) Bl = 0, By = Bj. Conversely, when conditions (10.11) are satisfied, we have (10.10), as follows from (5.8) and (5.11). Hence we have:* If for a symmetric connection the contracted tensor By is symmetric, the magnitude of the determinant A of n linearly independent contravariant vectors May is unaltered to within terms of the third and higher order, when the vectors undergo parallel displacements about an infinitesimal circuit, and conversely. * CL. Schouten, 1924, 1, p. 90. 11. PSEUDO-ORTHOGONAL VECTORS. PARALLELISM 27 11. Pseudo-6rthogonal contravariant and covariant vectors. Parallelism of covariant vectors. If yu; are the components of any covariant vector, there are evidently 2 —1 linearly independent contravariant vectors Af, such that (11.1) wiley = 0. (e=1,.--,5n~—1) We say that each of the vectors Ale is pseudo-orthogonal to wi; in Riemannian geometry (11.1) is the definition of ortho- conality, when pg; are the covariant components of a contra- variant vector p’.* Evidently any vector pseudo-érthogonal to w; is expressible in the form (11.9) MWo= q® {la=1,-.., n—1), where the «’s are invariants; here, and in similar cases later, « is supposed to be summed for its values 1, ..., n—1. Consider any curve C of the space and n—1 linearly independent families of contravariant vectors Fh parallel with respect to C. From (7.3) it follows that we have Ai dt Soh ; da (11.3) Lilly 2 = Hen fol) = 1+, n—1), a being not summed in the right-hand member. The equations (11.4) Veywi = 0 define, to within a common factor, the components wu; of a family of covariant vectors pseudo - érthogonal to the given Mw. We say that these vectors w; are parallel with respect to C. Differentiating (11.4) with respect to # and making use of (11.3), we obtain J =o. 5 [dm LE dak Ae TL Lx Modi * 1926, 1, p. 38. 28 I. ASYMMETRIC CONNECTIONS Comparing these equations with (11.4), we find . dw; = ak ; (11.5) ay 1 eT wip (1) as a necessary condition of parallelism. In order to show that it is a sufficient condition, we con- sider n—1 linearly independent covariant vectors u\”* satis- fying equations of the form (11.5), that is, Ap” dt kk j (o) dx ( ¢ — Linu) it. wi? go” (1), (11.6) where « is not summed in the right-hand member. Then the equations ad == (} determine quantities 4’, to within a common factor, which are the components of a contravariant vector pseudo-ortho- gonal to each of the n— 1 vectors uf”. Differentiating these equations, we find that 4° satisfies equations of the form (11.3) and consequently defines a family of contravariant vectors parallel with respect to C. Suppose now that we have any family of vectors um; satisfying (11.5) and we associate with it »—1 vectors satisfying (11.6) all n being linearly independent. The vector wg; and each set of n—2 of the set ui” determine a contravariant vector pseudo-ortho- gonal to w, In this way we obtain »—1 linearly inde- pendent families of parallel contravariant vectors pseudo- orthogonal to the vectors w;. Hence we have: Any family of covariant vectors whose components satisfy equations of the form (11.5) are parallel with respect to the gen curve, that is, they are pseudo-srthogonal to n—1 linearly independent families of contravariant vectors parallel with respect to the curve. * Here «, where « = 1, ..., n —1, indicates the vector and ¢ the component. 12. CHANGES OF CONNECTION 29 Incidentally we have: Any nn — 1 linearly independent families of covariant vectors parallel with respect to a curve are pseudo-orthogonal to a family of contravariant vectors parallel with respect to the curve. By processes analogous to those used in § 9 we have that when the equations (11.7) i — Lx pj = 0 admit a solution, the vector-field w; is parallel. However, we cannot say that the existence of such a field is equivalent to the existence of » — 1 linearly independent fields of parallel contravariant vectors. For on differentiating (11.4) covariantly and making use of (11.7) we have i RC which are equivalent to i Ji a ak AB) where for each value of « and 8 the a’s are components of a covariant vector. From (11.7) and (4.3) we have: A necessary and sufficient condition for the existence of n linearly independent fields of parallel covariant vectors is that the curvature tensor be zero. When the connection is symmetric, it follows from (11.7) that w; is the gradient of a function ¢. Since in this case the covariant vector wm; at any point in space is pseudo- orthogonal to every displacement in the hypersurface ¢ = const. containing the point, we call it the covariant pseudonormal to the hypersurface. Hence we have: When a space with symmetric connection admits a parallel Jield of covariant vectors, they are the covariant pseudonormals to a family of hypersurfaces. 12. Changes of connection which preserve parallel- ism. Let Lj and Lj be the coefficients of two different connections. We inquire whether it is possible that parallel directions along every curve in the space are the same for 30 I. ASYMMETRIC CONNECTIONS the two connections. To this end we make use of the equations of parallelism in the form (7.5). Subtracting these equations from the corresponding ones in the L’s, we have 7 da J (oh I 0’ ane) 2 Tk 7 where i Bey tip. = Lj — Lik. From (2.1) it is seen that aj; are the components of a tensor. Since these equations must hold for any curve and for vectors parallel to any vector with respect to this curve, we must have a" ia + J} air ee al ah —_— Jf ws ===), Contracting for /# and », we have ¢ — 948 ww Wy, 2 of 0, where vy; is the vector defined by 2ny, — o,. Conversely, if we take (12.1) Lik ES Lot 2 0) Wy, where yy is an arbitrary vector, the above conditions are satisfied. Hence we have: Lquations (12.1) in which YP; ws an arbitrary covariant vector defines the most general change of connection which preserves parallelism. From the form of equations (12.1) it is seen that both sets of coefficients cannot be symmetric in the subscripts. In § 14 we discuss the case where one set does possess this property. Hence we have: It is not possible to have two symmetric comnections with respect to which parallel directions along every curve in the space are the same for both connections. : When the condition for parallelism is written in the form (7.3), that is, * Cf. Friesecke, 1925, 1, p. 106; also J. M. Thomas, 1926, 3, p. 662. 12. CHANGES OF CONNECTION 31 ar 7A = I di = ¥ 719), the function f(#) for the connection (12.1) is given by d a O¢ 2 1. (12.2) Fee If we have a field of parallel vectors A defined by Ji 0 nt oo 27 Fx A= 0, then for the connection defined by (12.1) we have 0 A . 28 TE - + 2 Th =i 2 Uy, ox . which is discussed in § 9. ¥ Ij and Qh denote the symmetric and skew-symmetric parts of Lj, as in (3.3) and (5.1), we have (12.3) Fi = Ti+ of, + 6% wy, and : (12.4) 0 = 0, Ld und From the definition (§ 7) of the paths of a connected manifold it follows that the paths are the same for all connections related as in (12.1). This can be shown directly by means of (12.3). Conversely, if we apply to equations (7.6) the same reasoning as was applied to (7.5), we can show that expressions of the form (12.3) give the most general relation connecting the Is so that the equations (7.6) are unaltered. Hence we have: Equations (12.3) and an arbitrary choice of 2, define the most general change in connection which preserves the paths. If Lj denote the components of the curvature tensor for the L’s defined by (12.1), from (2.6) we have a ly i {dW oY 12.5) Lim = Lia +29} Sak 9 xl 2, 32 I. ASYMMETRIC CONNECTIONS In like manner we have from (5.4) and (12.3) (12.6) Bh = Boy-+ of (yy — Yi) + 07 Wj — 0 Wir, where (12.7) Ye = Yjx—Y; Yu, Yj. being the covariant derivative of 1; with respect to the I's. From equations analogous to (5.3) and from (12.5) and (12.6) we have (128) By = Oyu OF (Wy. — wig) + OL Wj — 0% Yi . From (12.5) by contraction we have (12.9) Lin= Lh = Ly+2 (50s — 2) oak xd and = i 2Y Qu I=17 = Apt 2m (22 — 2%). From this result and the theorem of § 5 we have: The vector WW; can be chosen so that for the mew linear connection Liji = 0. From (12.5) we have: When, and only when, Yr is a gradient, Lia = Lia. Contracting (12.6) for ¢ and / and 7 and j, we have respec- tively {12.11) Bir = Bjx + nj — Pi; and — 1 {12.12) Bue = Bu + 2 (Yu— Ya), in consequence of (5.11). If in accordance with (5.10) we put no 18s = (12.13) Bu: =— 9 [2 ox ]’ we have from (12.12) that (12.14) Bi = B+ n+1) Yi +a, where o; is the gradient of an arbitrary function o. 12. CHANGES OF CONNECTION 33 Again contracting (12.8) and using the notation of § 5, we have (12.15) 2 = 25 + (2—m) Yir— Wij and : : : (12.16) Oy = P+ (n—1) (Yu — Yi). From (12.4) and (5.12) we have (12.17) 2p ed Q+m—1) Ye so that (12.16) is consistent with (5.14). As an example of the second theorem of this section we consider the asymmetric connection which can be assigned to a Riemannian space so that the geodesics be the paths, that there be » independent vector-fields of parallel unit vectors and that the angle between two directions at a point and the parallel directions at any other point be equal.* In order that the first two conditions be satisfied we must have respectively \ (12.18) Lx == I! i + of Uy, + Ok Wy + where the Christoffel symbols are given by (1.6), and Li =0 (§ 9), which in consequence of (5.3) and (6.2) are (12.19) Rj + & @ I 25 1% < Qj — ok Gia == 0 where covariant differentiation is with respect to the g's and Rh are the components of the Riemannian curvature tensor. The third condition is ¢;x = 0, which are reducible by means of (12.18) to (12.20) 2g; Yu +gn + gin Yi + La + Qu = 0, where Qi = gin 2. Multiplying (12.20) by ¢¥ and summing for 4 and j, and by ¢"* and summing for 7 and %, we find that * Cf. Cartan and Schouten, 1926, 12. 34 I. ASYMMETRIC CONNECTIONS (12.21) wee, nls, and hence from (12.20) (12.29) Qin + yp. = 0. When we take the sum of the three equations obtained from (12.19) by permuting the suberipts cyclically and make use of known identities in the R’s, we have 5s i yi hn (12.28) 8h — Oh, — 9; 9 0 — 00, 0, hn — oh 0 = 0, so that (12.19) may be written in the form n (12.24) Bij + Lint, j + Lu Ray = 0. From these equations because of (12.22) and well-known identities in oe we obtain oh Jos Bon = — Le Dy B+ Quip, Ly — Ly 2) and hence from w 24) (12.25) Qu; — 5 0,0 oY — Gy ot Qa Qh). From (12.24) we have Rj — 2 9. With the aid of (12.22) and (12.25) we obtain (12.26) Ber = 0. A solution of these equations is furnished by the Einstein spaces, that is, spaces for which Bj; = cgy, where c¢ is a constant. When this condition is not satisfied, it follows from (12.26) that the spaces are a sub-class of those considered by the authori (cf. § 29). For further considerations of the preceding case see the paper by Cartan and Schouten. * 1926, 1, p. 21. + Cf. 1923, 3. 13. TENSORS INDEPENDENT OF vx 35 13. Tensors independent of the choice of y;. From (12.5), (12.9), (12.10), (12.4) and (12.17) it is seen that the following tensors are independent of the choice of the vector in (12.1); (13.1) Moy = ntl In, ; gL; (13.2) Line — = 0; Ai, n (13.3) Lj : Lj 3 (13.4) Th = 25+ - Lo (OF 2; — 0) 2). From (12.15) and (12.16) we have Ff 1 y Vir = ns = — T | 2 —_— Qi “+ UY (Dy, TT Dj.) | 1 Wg = —— =r {Dy Oy), When these expressions are substituted in (12.8), we find that the tensor of components Th = + — [ (0 $ 0, — 64 rd Dy) (13.5) ot ; (0% Mj — 0; ® ike) = (n—1) is independent of the choice of ¥;.* From (13.5) we have by contraction ppl i on (136) Tjpi = 0, Tia — 1} Ou+ (Qu ir). Other tensors independent of the choice of 1; are obtained in § 32. We close this section by establishing the following theorem: A necessary and sufficient condition that a vector Yi can be chosen so that tensor Lig be zero is that the tensor Aji be zero. * These results for (18.2), (13.4) and (13.5) are due to .J. M. Thomas, 1926, 3, pp. 667, 668. 8% 36 I. ASYMMETRIC CONNECTIONS Evidently it is a necessary condition. Conversely, if the condition is satisfied, we have (13.7) Ly + 6 Ly = 0. Contracting for 7 and j, we have, using the notation of (12.10), Ay ee n Ly mm From (5.3), (5.11), (5.14) and (12.13) we have (13.8) Ap = (B+ 2) —— (81+ 20). . yk Hence equations (13.7) become Gut Lol Dato atm] =o J n J 5 xt Pk ae j . Comparing these equations with (12.5) we see that Li z=), if we take Yi Ee a 2 (8 = 2) Ey Ki Ic? where o is any function of the x's. 14. Semi-symmetric connections. In § 12 it was shown that parallelism with respect to every curve in space cannot be the same for two symmetric connections. How- ever, if for an asymmetric connection we have (14.1) oy, = of op — 0% aj, where a; are the components of a vector, and we. take YP = — ti, then 2% = 0, as follows from (12.4). Con- versely, in order that it be possible to choose ¥; so that 2 = it is necessary that Qj be of the form (14.1). Following Schouten™ we say that the connection is semi- symmetric in this case. Hence we have: * 1924, 1, p. 69. 14. SEMI-SYMMETRIC CONNECTIONS 37 A necessary and sufficient condition that parallelism be the same with respect to every curve for two commections one of which is symmetric is that the other be semi-symmetric. From (12.4) it follows that, when a connection is semi- symmetric, the other connections with the same parallelism are semi-symmetric with the exception of a unique symmetric connection. We establish the following theorem due to J. M. Thomas:* A mecessary and sufficient condition that an asymmetric connection be semi-symmetric is that there exist a coordinate system for each point of space in terms of which any vector at the point and that arising from it by a parallel displace- ment to any nearby point are progariensl If such a codrdinate system ¢° exist and 2! are the com- ponents of a vector at a point P, then at a nearby point the components are A’ — Lj # dy". The conditions of the theorem are given by (Ly dt — Lh ¥ i af = 0. Proceeding with these equations in a manner analogous to that at the beginning of § 12, we obtain a 3 2 i = (14.2) Let n From these equations we have 3 i Th 2, — of Liss — or Ly). Contracting for ¢ and j, we have n—1 NJ) | = (14.3) Th and the preceding equations can be written as 1 on (144) 2 = Tr (05 Q, — 0% 0 9). Hence the connection must be semi-symmetric. * 1926, 3, p. 670. 38 I. ASYMMETRIC CONNECTIONS Conversely, if a connection is semi-symmetric and 2’ are a general system of coordinates and P is the point of co- ordinates 2%, when we effect the transformation (14.5) = = xi + bn (TL, ! 0) ad aff 0) # = Ay Gh + TRY we have at P (£5) ~ Sa 57, 02 2b ; ot faa = — (Tj), hoy 7 19 (21), + or (£2), and from the first of these it follows that (14.6) (14.7) (8), = (2). Making use of equations of the form (2.1). we have. = i 1 i . (Ly = (Zi)yt 7 16 (20), + 6 (2), ]. Since equations (14.4) hold for any coordinate system, we have in consequence of (14.7), Sms 9 ; Sd (5) = 1 dj Ly), from which (14.3) follows by contraction. Hence in the coordinate system defined by (14.5), the conditions (14.2) are satisfied. 15. Transversals of parallelism of a given vector- field and associate vector-fields. If for a given vector- field 4° the determinant |4%;| is not zero, a necessary and sufficient condition that the determinant [2°] for 2° = ¢ 4 be zero, that is, that the determinant | i 09 (15.1) p22 be zero, is that ¢ be a solution of the equation® * Cf. Kowalewski, 1909, 2, p. 84; Fine, 1905, 1, p. 505. 15. TRANSVERSALS OF PARALLELISM 39 | [ee 20 | oat oa | 1 1 ol / 2 “ee Ly (15.2 | Zl) lon : a 7 ig si. 2 Moreover, the rank of (15.1) is #—1 for each solution. Hence in considering any vector-field we assume that the components are changed by a factor ¢ if necessary, so that |4%;| is at most of rank n—1. We say that then the field is normal and that ¢ is the normalizing factor. This is a generalization of a unit, or a null, vector-field in a Riemannian space. For, in this case we have 14;4'; = 0, and con- sequently |2';| = 0. If the rank of [4% is m—r, there are » independent veetor-fields py (@¢ = 1, -.., r) which satisfy (15.3) Why = 0 and the general solution of (15.3) is (15.4) w= uly (e=1,.--, 7) where the ’s are arbitrary functions of the x's. When wp satisfy (15.3), the vectors A? are parallel with respect to each curve of the congruence defined by d x! dx" p Te as follows from (9.1). Moreover, it follows that the vectors Aig are parallel, whatever be ¢. Accordingly we say that each solution wp? of (15.3) defines a congruence of transversals of parallelism of the field 44% When |4';| is of rank %—r, we say that the field 4° is general or special, according as the rank of the matrix of the * Transversals of parallelism for a surface in ordinary space were con- sidered by Bianchi, 1923, 6, p. 806. 40 I. ASYMMETRIC CONNECTIONS last n rows of (15.2) is n— r-+1 or n—». When the field is special, and also when it is general and » >> 1, equation (15.2) is satisfied by every function ¢. When » = 1 and the field is general, equation (15.2) reduces to (15.5) il = Suppose that the field is general and that ¢ is a solution of (15.5) when » = 1, or any function whatever when » > 1. The equations (15.6) wo? +2725) — 0 29) are satisfied by all vectors uw’ defined by (15.4) for which the functions ¥* satisfy the equation j 0 Ploy 505 = 0 If there were a solution of (15.6) not expressible in the form (15.4), then from (15.6) we have equations of the form 2°—= a’2";, in which case the rank of the matrix of the last nm rows of (15.2) is n—r». Hence when the field is general, all the solutions of (15.6) are expressible in the form (15.4), that is, on replacing A’ by A%¢ no new congruences of trans- versals of parallelism are obtained. When the field is special, the determinant (15.1) is of rank n—r at most and there are at least » independent solutions of (15.6). Consequently if ¢ is such that not all of the equations ; oQ Wo od =r 0 (e = Lr vnsir) are satisfied, there is another solution, say Yop of equations (15.6). Evidently it is such that w/ oy ’ —1- 40. If wis any other solution of (15.6) not of the form (15.4), on £d 15. TRANSVERSALS OF PARALLELISM 41 eliminating 4° from (15. 6) and from the similar equations when wu’ is replaced by 2, Ln we have ” ; . 0log yg pO loge, faa (#12 WE = nr) 2 = 0, and consequently wf is expressible linearly in terms of His, (8=1,...,v+1). Hence for the given function ¢ all solutions of (15.6) are expressible linearly in terms of these r+ 1 vectors. For another function, say ¢,, there is at most one field wi, , other than wi, (¢=1,---,r). But in this case we have the equations obtained from (15.7) on replacing ¢ in the first term of the left-hand member by ¢, and w/ through- out by wh ,,- Consequently the change of the function ¢ does not yield new congruences of transversals of parallelism. Gathering these results together, we have: When a wvector-field 4° is mormal and the rank of | 2; jl 7s n—r, there are r independent congruences of transversals of parallelism, unless the rank of the matrix of the last n rows of (15.2) is n—r; in the latter case there are r +1 independent congruences of transversals; moreover, in either case any linear combination of the vectors defining congruences of transversals also defines such a congruence. When 2°; = A" a;, where o; is any vector, the vectors A? are parallel with respect to any curve in the space (cf. §§ 9, 10). We consider the converse problem: Given a vector-field u’ to determine the vector-fields 4° for which the former is a congruence of transversals of parallelism. We assume that the codrdinate system 2? is that for which p= 0 (¢ = 2,...,n).* In this coérdinate system the equations (15.3) for the ten mination of the A's are (15.8) b= 0, Any set of functions 4¢ satisfying these equations are the components in the x's of a vector-field with respect to which the congruence wu’ is the congruence of transversals. A set * 1926, 1, p. b. 42 I. ASYMMETRIC CONNECTIONS of solutions is determined by arbitrar y values of 27 for xt =0, that is, by » arbitrary functions of x2 -.., 2%. In particular, the n sets of solutions Bs, where «a, for om, chic determines the set and 7 the component, determined by the initial values (Ale) = mn 3. are independent, since the deter- minant |A(, | is not identically zero. Moreover, from the form of (15.8) it follows that A’ = ¢“ A{, is also a solution, where the ¢’s are any functions of x2 ..., z*. Hence we have: For amy congruence w' there exist n independent vector- fields Aw with respect to which the given congruence is the congruence of tramsversals of parallelism; moreover, the field 28 = go" Ton (a = 1 sie vi, Nn) possesses the same property. when the g's are any solutions of the equation oy for == 0, the coordinates x' being any whatever. When |4;| is of rank n—7, the equations (15.9) Vi 2; ==) are satisfied by +» independent covariant vector -fields v® (ae = 1,..., r) and the general solution is (15.10) Vi. == Ye, v po where the ¥’s are arbitrary functions of the x's. We say that each such field is associate to the given field A%* If the given field is general, there are »—1 fields of independent vectors given by (15.10) for which »; 2* = 0, and these fields are associate to the field A’¢ for every ¢ satisfying (15.2). If, however, the field is special each of the fields (15.10) is associate to 4’ ¢, whatever be ¢. * By normalizing the field we have that the rank is at most n —1 and consequently there is at least one associate covariant field. 16. ASSOCIATE DIRECTIONS 43 In like manner the components wu; of a covariant field can be chosen so that the rank of |u;;| is n—r (r > 1). Any solution 47 of A Wil; — 0 gives the components of a field of contravariant vectors associate to the given field.* 16. Associate directions. Consider a field of non-parallel contravariant vectors of components A’ and a curve C at points of which the coordinates 2 are expressed in terms of a parameter ¢. A family of contravariant vectors of com- ponents w«’ is defined at points of C by the equations da’ 1 (16.1) i : ne. If w' = f(#)2%, the vectors are parallel with respect to C. When this condition is not satisfied, we say that u? are the components of the associate direction of A' with respect to C. If 47 are replaced in (16.1) by 4? ¢, where ¢ is any function of the z's, and pu? are the components of the associate directions of the latter vectors, we have dy di: (16.2) pro RE J In this way we get a pencil of associate directions, determined by the given vector and any one of the associate directions. . Conversely it is possible to choose a function ¢ such that the associate direction of ¢ A? is a given one of the pencil other than the direction 4%, When the given vector-field has been normalized (§ 15). it necessary, and »; are the components of an associate covariant vector, we have »; ¢ =— 0. Hence we have: For a field of non-parallel contravariant vectors the associate directions with respect to a curve are pseudo-orthogonal to the associate covariant vectors of the field. * Hisenhart, 1926, 14. 44 I. ASYMMETRIC CONNECTIONS In particular, if C'is not a path of the space and 4’ are the 7 components of the tangent to C, that is, 4° = . equations (16.1) become d? xt i dxf dak gas Ely or If we change the parameter ¢, we get a pencil of associate directions as in (16.2). We note that associate directions of a curve are independent of the tensor £7. The associate directions of the tangent are evidently a generalization of the pencil determined by the tangent and first curvature normal of a curve in a Riemannian space (cf. § 24).* In a similar manner, if 4; are the components of any field of non-parallel covariant vectors, the equations yo Am) ic hg = define the associate covariant vector w; of 4; with respect to the curve, unless the vectors 4; are parallel with respect to it, that is, unless p; = 4; f(f). When A; is replaced by ¢ 4; where ¢ is an arbitrary function of the x's, we get a pencil of associate covariant vectors determined by the given vector and any one of them. Moreover, we have: For a field of mon-parallel covariant vectors the associate covariant vectors with respect to a curve are pseudo-orthogonal to the associate contravariant vectors of the field. 17. Determination of a tensor by an ennuple of vectors and invariants. Let in denote the componentst of » linearly independent vectors in a coordinate system a7’. Then the determinant (17.1) =a, *1926, 1, pp-60, 72. T As formerly the index with parentheses indicates the vector and the one without parentheses the component. This convention will be followed hereafter, and unless stated otherwise the indices take the value 1, -.., n; moreover, the summation convention is used for both sets of indices. 17. ENNUPLES OF VECTORS 45 is not identically zero. We denote by As” the »® functions defined by the equations (17.2) 12 kip = 03; as thus defined 4” is the cofactor Zon in 4 divided by 4. In any other coordinate system 2° the functions 2'{” defined 5 1(0) by #3 Af = 05 are such that 0 ne yO 0a ox" Consequently Z{” are the components of x independent covariant vectors. Furthermore, it follows from (17.2) that If we had started with the independent covariant vectors 2”, then equations (17.2) serve to define x independent contravariant vectors. Owing to the reciprocal character of the relations (17.2), we say that either set is conjugate to the other, and that the two sets constitute an ennuple. It is evident that an orthogonal ennuple of contravariant vectors in a Riemannian space™ and the associate covariant vectors form an ennuple in the above sense. If i, are the components of a tensor, then the quantities vf Gein) @,) 14, 5 (118) Blo =gln PE APE SL are invariants. If these expressions are substituted in the right-hand members of the equations dove, 2: Br 97 i, lo) (ey) (11.5) aj..; ol Ady - 43, fre . se 47 . these equations are identically satisfied because of (17.3). Hence we have: * 1926, 1, pp. 14, 40, 96. 46 I. ASYMMETRIC CONNECTIONS The components of any tensor are expressible in terms of mwariants and the components of an ennuple.™ . Ii In particular, we can express £5 in the form - : I 28) 2G) (17.6) 07 =m mm wy FL Ji rr % where wis, are skew-symmetric in the subscripts. We shall apply the preceding results to show that, if ay is a tensor such that agi Ae) Ale Mey = 0 for 0. ¢ and rv not equal to n, then fy: Py a( ( a ( ) (17.7) age = 2° a+ BY by + BD ey, where ay, by, cy are tensors. In fact, if we write a5: in the form (17.5), that is imu ijk =— Copy Ls we have hoy coir = 0. Hence (17.7) follows, where 7 dp == up” 4 and so on. Any other ennuple 7, 22 in given by (6) » 0 2 (G) G Ale) - Oo ~Z o 1 (1 7.8) hey = Ui hi) 5 +i = Aaty where the determinant «| is not identically zero, and the a's and A's are invariants in the relations (17.9) we AL = 0, as follows from (17.2). If ¢2'"" are the invariants for the Cis tensor ai with respect to this ennuple, we have Fired, = 2 . ; = 2 {17.10 Pe AT a r ) Tye Ts Py fs 43 Cr Ty Ts When for a given coirdinate system we take (17.11) Mey = O04 =O. 1926, 1, p. 97. 18. THE INVARIANTS Y,”, OF AN ENNUPLE 47 then (17.12) 1 em 3 as follows from (17.2) or (17.3). For this particular ennuple we have from (17.5) (17.13) wih that is, any component of the tensor in this coordinate system is equal to the invariant with the same indices as the component. We call the ennuple (17.11) and (17.12) the fundamental ennuple of the given coordinate system. 18. The invariants 7 + of an ennuple. For a given ennuple the invariants Yor + defined by (18.1) Yu = 1 FA Zh are a generalization of the coefficients of rotation of an orthogonal ennuple in a Riemannian space, as defined by Ricei and Levi-Civita.* From (18.1) we have because of (17.3) (18.2) Ko J Yuma Jon 7, If equations (18.2) be multiplied by 4’ and summed for . the resulting equations are reducible by means of (2.2) to 24 van (18.3) 75 ot “api i +d ” J io iw. Conversely, if we have any on and a set of invariants 7u'¢ and we define functions Lj; by (18.3) and 1h by corresponding equations for any other codrdinate system x'", we find that equations (2.1) are satisfied. Hence we have: An ennuple of vectors and any set of invariants u's de- termine a connection; and any asymmetric connection is so determined. ®1901, 1, p. 148; of 1926, 1, p. 97, T Of. Levy, 1927, 1, p. 307. 48 I. ASYMMETRIC CONNECTIONS When in particular, we take iw = 0, we have from (18.3) hy @ 0 Mey (18.4) Lii=—%4 and from (18.2) (18.5) Key y = 0.5 Consequently the n fields of vectors Ale are parallel fields and hence Lj; = 0 (§ 9). Conversely, if the latter condition is satisfied, we can chdose » linearly independent vector- fields satisfying (18.5) and consequently we have (18.4). Hence we have: A necessary and sufficient condition that the curvature tensor Lia of a manifold with asymmetric connection be zero is that the coefficients of the comnection be expressible in the form (18.4) in terms of an ennuple. From the form of equations (2.1) we have: Af Lix are the coefficients of a connection, so also are Lik + ajr, where aj, are the components of an arbitrary tensor. As a consequence of this theorem we have that for any ennuple the quantities 8 Ale) Ti (@) (18.6) Lip ==; 52 are the coefficients of an asymmetric connection. For from (18.4) and (3.3) we have (18.7) L,= 1h = 1+ 29, For the connection defined by (18.6) we have from (18.3) Vv ~(V j 87 j pis (18.8) Ju oo Ai ; (iii Mw 2, from which it follows that y,”¢ is skew-symmetric in @ and o. Equations (18.7) show that for the connections (18.4) and (18.6) to be the same, it is necessary that Lj; be symmetric in 2 and jy. In this case equations (18.5) become * Of. Weitzenbick, 1923, 2, p. 319. 18. THE INVARIANTS Vy + OF AN ENNUPLE 49 (18.9) Wer; = 0, where the covariant differentiation is with respect to the 7s. Then from (9.6) and (5.3) we have Bj; = 0. Conversely, when these conditions are satisfied, equations (18.9) admit n linearly independent fields of vectors parallel with respect to the I's. From (18.3) it follows that a necessary and sufficient con- dition that 7.” be skew-symmetric in the indices ¢ and o is that 8 Mey dah EE he Is the symmetric part of either of the coefficients (18.4) r (18.6) and consequently satisfies (5.6). In consequence of this result and the above theorem we have: A necessary and sufficient condition that the invariants y.” be skew-symmetric in the indices w and o is that 1 (1 (0) Pls 0 hier. 3 Ll (18.10) Ly = — |i pe a) + 8, where Qh is an arbitrary tensor skew-symmetric in I and j. If we denote by ’ Loy the covariant derivative of if for the connection La defined by (12.1), we have Ji ,0 on ,4 A) 2 4) Li 2 WV, ays from which it follows that (cz) i ; A; Chl 15 — 2 oir) == 0% Vy Consequently the mixed tensor a 1 3 . (1 8.1 1) ion by % Aon iP As | is independent of the choice of the vector ¥;. The same is true of the tensor H0) I. ASYMMETRIC CONNECTIONS (0) 1.) (3) Ailr— ns May if. If equations (18.11) be multiplied by 45” ns and 7 and /% be summed, we find that the invariants (18.12) en La 7d x are independent of the choice of the vector vy; in (12.1). Conversely, if we have two asymmetric connections Lj, and Lj; for which the invariants (18.12) are equal for a given ennuple, it follows from (18.3) that = i Lo © 1 ? 7 0 Ljg— Lj = 5 d; Go 3s p oD rs which evidently are of the form (12.1). Hence we have: A necessary and sufficient condition that parallelism be the same for two different asymmetric connections is that the corresponding invariants (18.12) for a given ennuple be equal Jor these connections. 19. Geometric properties expressed in terms of the invariants Vito In order that the vector-field Jr) of an ennuple at points of each curve of a congruence Aig be parallel with respect to the curve, it is necessary and sufficient that -J 1 i) AB) VA) J 0 Ae) . By means of (18.2) these equations are equivalent to (re 2 -_—0 0c) Jn =. Hence we have: A necessary and sufficient condition that the vector-field Pn of an ennuple be parallel with respect to the curves of a congruence Liz is that (19.1) veg == {(r==1,...,n574 0) As a corollary we have: 19. GEOMETRIC PROPERTIES 51 A necessary and sufficient condition that the curves of the congruence ie of an ennuple be paths is that (19.2) Pele — 0= (v sr Lhe ae 7; V 3% a), If we use the notation of i of ; et (19.3) nr © 5 then : [lot 2 51% 318 tf ot 3 2 ; yb San ; (19.4) 2, (hy Maypi— Mp Mei + 24) Ap 2) , a = (vg or ve 5 204" 5) x ; in consequence of (18.2) and (17.6). These equations are generalizations of equations due to Ricci and Levi-Civitat in Riemannian geometry. As an application of these equations we seek necessary and sufficient conditions that p of the congruences of an ennuple, say ig (6 =1, .-., p), generate a system of co”? varieties V,. In this case the equations of oy = (0=1,---,p) (19.5) i must form a complete system. From (19.4) we have: A necessary and sufficient condition that the congruences May for 6=1,..., p generate a system of ©"? parieties V,, is that vy ur prov Chi (19.6) Bu 7 B E 200, = 0) (or, 8 = 1, we ns 15 v=p-+1, Ta n). As a corollary we have: A mecessary and sufficient condition that there exist a coordinate system such that the curves of the congruences of * Cf. 1926, 1, p. 100: also, Levy, 1927, 1, p. 308. +1901, 1, p. 150: of. also 1926, 1, p. 99. o* h2 I. ASYMMETRIC CONNECTIONS an ennuple be coordinate lines is that equations (19.6) hold Jor all distinct values of «, 8 and »v.* We say that a congruence 4; is pseudonormal to a family of hypersurfaces F(a, .-., 27) = const., if : af hi = p=, ’ g Dat From the preceding results we have: A necessary and sufficient condition that a congruence 2 of an ennuple be pseudonormal to a family of hypersurfaces is that 97) Tilt Rely = 0 (nf... n—1)F “These two theorems for the case of a symmetric connection, in which case wl, — 0, are due to Levy, 1927, 1, p. 308. op +07. 11926, 1, p. 115. CHAPTER II SYMMETRIC CONNECTIONS 20. Geodesic codrdinates. When a codrdinate system can be chosen for which the coefficients of the connection vanish at a given point P(x), the vector at any nearby point P' (xi + da") parallel to a given contravariant vector at P has the same components as at P to within terms of the second and higher orders, as follows from (7.1). If in equations (2.1) we put Li = 0, we see that a necessary condition is that the coefficients in any other coordinate system be symmetric at P. In order to show that this condition is also sufficient, we imagine that the space is referred to a general coordinate system 2/ and we consider the transformation of coordinates defined by / i r dy) 1 qt 1k (20.1) x = xg +6 2’ — Tm 2" Low, 9 where 1? are any functions of the z”’s such that they and their rt and second derivatives are zero when the 7 's are zero.* From (20.1) we have at P 0 hea LEV b2 ar @02) | 2 = i, [= ra) = = i7LY. From these expressions and equations (5.6) we have at P (20.3) (Ife = 0. Consequently any coordinate system defined by (20.1) possesses the desired property. Hence we have: When, and only when, at a point the coefficients of a con- nection are symmetric in the subscripts, coirdinate Whee can * Of. 1926, 1, p. 56. 53 54 II. SYMMETRIC CONNECTIONS be chosen, with the point as origin, such that the coefficients are zero at the point. Weyl* calls a connection affine, when at every point a codrdinate system exists for which the components of a vector in this coordinate system remain unaltered by an infinitesimal displacement, to within terms of the second and higher orders, foe we use the term affine for asymmetric connections as fwo (cf. 5.7) A Any coordinate system for which (20.3) is satisfied has “been called geodesic by Weyl. From the foregoing results it follows that if the coordinates x’ are geodesic for a point P° as origin, other geodesic coordinate systems with the same origin are defined by (20.4) of = 0 where the ¥’s are of the character appearing in (20.1). It is evident that at the origin of a geodesic coordinate system first covariant derivatives reduce to ordinary derivatives. Consequently the use of such a system frequently makes for considerable simplification in any problem involving first co- variant derivatives. Moreover, when the results of such an investigation are stated in tensor form, their generality is not conditioned by the use of the particular coordinate system. Symmetric connections are characterized by another property. Consider a point P(x?) and two infinitesimal vectors d, 2? and dy 2* at P, and denote by -P, and P, the points of coordinates 2+ dy 2* and x + d, 2? respectively. When the vector da’ undergoes a general parallel displacement to P,, its components at P, are dio’ + dod 2’ + Lind, 2’ doo”, and the coordinates of the point FP, at the extremity of the vector are i i i $y yl fy oh 2+ dor +d’ do dy’ + Lj d, dex’. In like manner when the vector dy’ undergoes a parallel displacement to P;, the coordinates of the point P,, at the extremity of the vector are : 5 HN So dix Fda’ + did’ + Lidy 2’ dy 2” 21921, 4, p. 112. 21. THE CURVATURE TENSOR Dd Hence a necessary and sufficient condition that Pp, and Ps, coincide is that Lx be symmetric in j and %, that is, that the connection be symmetric.™ 21. The curvature tensor and other fundamental tensors. In § 5 it was seen that the quantities 2 ~ " = a + Ih he— Lh 21.1) Bu =— are the components of a tensor. This tensor arises when we express the conditions of integrability of equations (5.6). In fact, these conditions assume the form ’ Pp os 7 s 5 3x? 9x? ox 3 Ea (2 | 2) LT ps Nod Pars TT 7 Bj, Sal Oat Ox oz from which equations the tensor character of Bl is apparent. This tensor is a generalization of the Riemannian curvature tensor of a Riemannian space and we call it the curvature tensor of the space with symmetric connection. From (21.1) it follows that Bj is skew-symmetric in / and /, that is, : : (21.3) Bj + Bj = 0. Also the components satisfy the identities (21.4) Biu+ Biyj+ Bye = 0. This result is readily proved by choosing a geodesic coordinate system at a point P. In this case at P all of the I's are zero and (21.4) can be shown to hold at Pin this coordinate system. Since this is a tensor equation it holds at P in any coordinate system. Moreover, as P is any point, it holds throughout the space. In like manner at a point P in a geodesic coordinate system with FP origin. we have * Cf. Weyl, 1921, 1, p. 107; Levi-Civita, 1925, 5, p. 185. 56 II. SYMMETRIC CONNECTIONS ; 52 ry Bre Biju, m = aE Aa ad ge. ’ XE Bx Ox Da’ as follows from (21.1) and (6.1). Consequently (21.5) 2 Bj, m + Bim, I = Boni.1 = 0, Since these are tensor equations, they hold throughout space in any coordinate system. They are evidently a generalization of the identities of Bianchi for a Riemannian space, and are called the identities of Bianchi for a symmetric connection.” In a similar manner the following identities due to Veblen can be established: . : ol : 4 Bipt.m + Bim, + Boi; + Blamj1 = 0. mel 22. Equations of the paths. In § 12 it was shown that parallelism throughout a space with symmetric connection is uniquely defined, that is, that it is not possible to have two symmetric connections with respect to which parallel directions along every curve in the space are the same for both connections; thus each symmetric connection is a un- ique affine connection. However, as a corollary of the third theorem of § 12 we have: The paths are the same for two symmetric connections whose coefficients are in the relations (22.1) Lie = Ti+ 0) w+ 01. vy, where Wj is an arbitrary covariant vector.§ The paths are a generalization of the straight lines of euclidean space. Accordingly the properties of the space * Cf. Veblen, 1922, 5, p. 197; Schouten, 1923, 1. 71 c, p 197. i It is evident that the results of these two sections apply not only to the case of symmetric connections, but that they apply also to the symmetric parts of any asymmetric connection. § Weyl, 1921, 2, p. 100; cf. also Eisenhart, 1922, 2 and Veblen, 1922, 3. 29. EQUATIONS OF THE PATHS 57 which depend upon the paths and not upon a particular affine connection of the set (22.1) constitute a projective geometry of paths, whereas those depending upon a particular affine connection constitute an affine geometry of paths. In this chapter we consider the latter and postpone to the next chapter a study of the former. If we have a particular path, that is, an integral curve of equations (7.6), then Bat, dol dak dx Www am (22.2) where ¢ is a determinate function of {. If we define a parameter s by : ds Jou (22.5) 7 == an ; where ¢ is an arbitrary constant, equations (22.2) become Baki Ly det dak ; 22.4 — be [== 229 ds® TL ds” ‘ds 0 Thus the parameter s for a path, which we call an affine parameter, 1s the analogue of the arc s of a geodesic in a Riemannian space.* It is evident from (22.3) or (22.4) that, if sis any affine parameter, the most general one is given by as-b where ¢ and b are arbitrary constants. Further- more, by means of equations (5.6) we can establish the theorem (cf. § 38): When the coordinates xt undergo a general transformation, an affine parameter is not altered. From the form of equations (22.4) it follows that a path is uniquely determined by a point P, of coordinates x; and a direction at P,. In fact, if we put 292.5 Slim ; ( : 3 ds L where a subscript zero indicates the value at P),, we have from (22.4) £1926, 1, p. 50. hs II. SYMMETRIC CONNECTIONS ret — misma fst (Tl) BES. a ; Sy Ss n! \ ds" /o the coefficients of s* and higher powers of s being obtained from the equations which result from (22.4) by differentiation and reduction by means of (22.4). Thus we have (22.6) 2 dA? xt i dxdt dx* di Th JE eT = 0, ds’ + 2 "ds ds ds a i dx dxt dx dam at ol Jin hoa where irl oo 1 Pp 0 7 rs LL It rr hl = 3 is in skit jl 4 gy i) (22.7 . ) 1 0 = 3 P LE Th ark Ia), and in general : +i 1 AT 7 \ vi ie (22.3) Ia. Cmp = N 2 fot iT (N = iL) 1 5 mm Ir 3 P before an expression indicating the sum of terms obtained by permuting the subscripts cyclically and N denotes the number of Maa Hence we have in place of (22.6) (22.9) af = Zap g Sig 1, (Ii) §Jgkg2 lor ) SIEkELS 3 alates SEV RS Sess The domain of convergence of these series depends evidently upon the expressions for Z7 and the values of &. However for sufficiently small values of s they define a path, that is, an integral curve of equations (22.4).* 23. Normal codrdinates. In § 20 we saw that for a given symmetric connection there can be chosen coordinate * These results are an immediate generalization of a similar treatment for geodesics in Riemannian geometry, 1926, 1, p. 52; cf. Veblen and Z.Y Thomas, 1923, 1, p. 560. 23. NORMAL COORDINATES 59 systems for any point so that at the point the coefficients I are zero. In this section we wish to establish the existence of a class of coordinate systems possessing this property which are a generalization of Riemannian coordinates in a general Riemannian space.” They were eonsidered first by Veblen,& who has called them normal coirdinates. Let C be a path through a point F,, and s an affine parameter of the path which is zero at F,; then the con- stants & are uniquely defined by (22.5). To each point of (/ we assign coordinates y¢ by the equations (23.1) yl == Hg, Since equations (22.9) define the path in the x's, between the »’s and y's at points of " we have the relations (23.2) rH o-— fy 1 (I%) itis (Zr: ) wit =... FD.2) FE BTY go \Linhy’Y ghey’ YY 1 If we assign codrdinates in this manner to the points on all paths through 7 in a domain, such that no two paths meet again within it, we have a co¢rdinate system #¢ in the domain. Moreover, equations (23.2) are the same for all paths and consequently are the equations of the transformation of co- ordinates, FP, being the origin for the y's. Since the jacobian | 8% | Sloot : —— | of these equations is different from zero at 7, the series | oy’ | : can be inverted, and we have nL a (28.8) YF == Ab PU al, pay where [7 are series in the second and higher powers of {wf — af) for j==1,-.., un. Comparing (28.2) with (20.1) we see that the y's are a particular type of geodesic coordinates; we call them normal coordinates. : From the definition of the y's it follows that (23.1) are the equations in finite form in the y's of the paths through PF. =1926. 1. p. D3. T1922, 5, p. 193; also Veblen and T.Y. Thomas, 1923, 1, pp. 562-566. 60 II. SYMMETRIC CONNECTIONS Consequently the equations of the paths through the origin of a normal coordinate system have the same form in these coordinates as the equations of straight lines in euclidean space in cartesian coordinates. If we denote by Cj the coefficients of the connection in the y's, the equations of the paths in this coordinate system are (23.4) : a J ry Cl dy’ dy 0 ds ds Since these must be satisfied by (23.1), we must have (23.5) gf =9, and on multiplication by s* (23.6) 27 —0, which equations hold throughout the domain. Conversely, if these conditions are satisfied, equations (23.4) are satisfied by (23.1) and consequently the y's are normal coordinates. When we apply to (23.4) considerations similar to those applied to (22.4) which led to (22.9), we obtain y a &s TS 1 )o EE ol (Cita)o 5 § g $—... (23.7) y 1 =) S05 gh, esl. 1 ) ’ where a... are defined by equations of the form (22.8). Since these expressions must be equivalent to (23.1), we must have (23.8) (Che Jo === and ; (23.9) (F.0h = for all values of p. Equations (23.8) follow also from (23.5). since the §’s are arbitrary. From (23.8) also it follows that normal codrdinates are a particular class of geodesic coordinates (§ 20). 32. NORMAL COORDINATES 61 If instead of a general coordinate system z¢ we take any other coordinate system 2” and proceed as above and denote by #* the normal coordinates thus obtained, we have in place of (23.1) : oo (4) : Y. = ds /0 2 for the equations of the paths. Since dz’ dx dad AR OL To FTF == 2. &J egy (27 J fll J gE, where the a’s are constants, we have (23.11) y= dy. Hence we have: When the coirdinates x* of a space are subjected to an arbitrary analytic transformation, the normal coordinates determined by the x's and a point undergo a linear homo- geneous transformation with constant coefficients. From the definition (23.10) of the «’s it follows that when a transformation (23.11) of the normal coordinates is given, corresponding analytic transformations of the x's exist but are not uniquely defined. From the form of (23.11) it follows that normal codrdinates are fundamental in the affine geometry in the neighborhood of a point. If we differentiate equation (23.6) with respect to s along any path, make use of (23.1) and multiply the resulting equation by s, we obtain (23.12) Cla y’ a i = A), where : 1 8 Cj (23.13) Ch — + lan P indicating the sum of terms obtained by permuting j, k and / cyclically. Proceeding with (23.12) as was done with (23.6), we get a sequence of identities of the form 62 : II. SYMMETRIC CONNECTIONS (23.14) | cl. “ofp y" > ‘ey = 0, where r i 2 As thus defined the C’s are symmetric in the subscripts and they are the functions in the normal coordinates #¢ for which the corresponding functions in the 2’s are given by (22.8). From (23.13) and (23.15) we obtain i 2 5° Cr Ls 5 : Y J Citiim SE 3. 4 S dy oy™ eo S indicating the sum of the six terms obtained by the per- mutations of the subscripts, j, &, [, m, which do not yield equivalent terms. In general, we have O38) lL we red i = eres es J r(r—1) oy” ) in this case S indicates the sum of »(r—1)/2 different terms. If y* are a system of normal coordinates with a point P as origin and y° are the normal coordinates corresponding to the y's with the point P’, of coordinates dy’, as origin, we have from (23.2) k y= dy ty’ —— Ciy'y (23.17) =i 1J kyl Chit y yey =i. fi DO bt wo where Cj. and so forth are the values of the corresponding (’s for the y's at P’. Because of (23.8) we have 8 Cie Ce | — yt J td EE Cie (2%) a oor ay dy 24. Curvature of a curve. let C be any curve in a Vu, not a path, the coordinates 2 being expressed in terms of a general parameter ¢. The equations 24. CURVATURE OF A CURVE 63 d? xf : dat dot (24.1) TE jk Td = u define a contravariant vector u’. If we change the parameter we get in general a new vector, which is of the pencil determined by wu’ and ~. We single out one of these vectors by choosing for the parameter an affine parameter s of the path tangent to C at a given point and we choose s so that s = 0 at P. Accordingly we write dA? at 4.7 dx’ dak ek T7.7 Lajk TTR 4, dst ds ds ; which in fact are equivalent to Bal x dxl dak (24.2 =i: =, 212) ds? +7 ds. ds ik that is, the vector wp’ is not affected by the choice of the tensor 2. Since s is determined to within a constant factor (§ 22), the same is true of u'. If we take for the z's a set of normal coordinates with origin at P, these equations reduce to dA? at dst = =n (24.3) . These equations are an evident generalization of those in euclidean three-space which define the first curvature vector of the curve at P. Accordingly we call ¢* defined by (24.2) in general coordinates the first curvature vector of the curve at P. From (24.3) we have for C ; sooo daly (24.4) phy m= [5) + 9 (1) s* + ! Sy and the equations of the path tangent to C at P are = qf Ss. 0 64 II. SYMMETRIC CONNECTIONS Consequently the values of wp?’ at P determine the departure of the curve from the path at a point near P. It is readily shown that The surface formed by the paths through a point of a curve in the pencil of directions determined by the tangent and first curvature vector to the curve at the point osculates the curve.™ 25. Extension of the theorem of Fermi to symmetric connections. The following theorem was proved for Rie- mannian connections by Fermit and in this section is estab- lished for symmetric connections: For a space with a symmetric connection it is possible to choose a cobrdinate system with respect to which the coefficients Ti are zero at all points of a curve, or of a portion of it. Suppose that the curve C is defined by 2 = ¢(f) and that at a point PF, of it we take n—1 independent vectors io for « = 1, ..., n—1, which also are independent of the tangent to the curve. that is, at P, the determinant 1 AN dy --- dw (25.1) m= An—1)" ay A(n—1) 1/ ’ @¢ + aie pt is different from zero, primes indicating differentation with respect to ¢. From these vectors we obtain »—1 families of vectors Aw) by parallel displacement along C. It follows from continuity considerations that there is a portion R of C about P, for which 4 + 0. At P, the components of any vector depending upon the given » — 1 vectors are of the form (25.2) P= LE, tas. n=l) If this vector undergoes parallel displacement along C, we get a family of vectors whose components at each point are * Cf. 1926, 1, p. 62. T1922, 7; the method followed is an adaptation of a proof of the theorem for Riemannian connections given by Levi-Civita, 1926, 4. 25. EXTENSION OF THE THEOREM OF FERMI 65H given by (25.2) in which the A4’s remain constant. Since any point can be taken as F, the components of any vector at any point in the (r—1)-fold of vectors at the point can be expressed in the form (25.2). If we put onl) Boy = FOO + Mey S20) and express the condition that the functions 2% are a so- lution of (7.1), we have (25.4) tpt lr =, where u’ are defined by (24.1). In the region B functions a(?) and a“(f) can be determined such that or i : (25.5) Pa EL since A 4 0, and the functions f and f“ are determined by the quadratures wo rm ect p— frearee where the ¢’s are constants. If in particular the given curve isapath a> =0fov a =1... n-=—=1. Consider at any point P of C the (n— 1)-fold of vectors defined by (25.2) and the paths of the space through P in these directions. The locus of these paths is a V, 3. The equations of any one of these paths are 25.7) of = 9H) + Athy 5— (Php 4% tly 48 1lsy oo, where ¢ is the value at P of the parameter along C, (Th)p are evaluated at P and s, the affine parameter of the path, iz chosen (§ 22) so that at P we have + wy Ae ion A new set of coordinates y’ is assigned to each point of C by means of equations y = yr (0, 66 II. SYMMETRIC CONNECTIONS where the Y's are any continuous functions of ¢ In like manner along each path through P we associate cotrdinates y’ with each point by means of the equations (25.8) yt = de (Nd A%, y= yr, where the A’s are the constants in (25.2) which determine the direction of the path at P;, thus s=0 at P. We assume that v™ 4 0, so that the last of (25.8) can be written (25.9) t= a(y). We eliminate the A's from (25.7) by means of (25.8) and replace 7 by its expression (25. 2 This gives equations of the form at = Fy) + bo (of — 9) 25.10) fa ur Un) Hy ka (ye — (YP UB) + sles where the A's, ¥'s and (I%) are functions of y*. If this process is followed out at each point and for each direction, we have coordinates jy’ associated with every point of the family of V,—1’s as defined, and equations (25.10) are of the same form for all the points. At points of C, that is where y“ = the jacobian of (25.10) is reducible to 64 or A/yY", Ys A is given by (25.1). Hence for a domain of the space in the neighborhood of the portion 22 of the curve for which A +0, equations (25.10) define a transformation of coordinates. ; It we denote by Aw the components in the i's of the vectors Alm, we have at points of C, by means of (25.10), - NES ; da ) Ap) + Aer La he | er ee iy d yy" (B d y" 3 gid 7 0 at og Me) i= Ala) 5 yy Ee When these equations are written in the form kp (Ady — 0a) + Aiey (97 — Ai) oh r= a, 1 25. EXTENSION OF THE THEOREM OF FERMI 67 we have that (25.11) lon ==100. In order that a family of vectors Aq may have the components (25.12) hy = of in the y's, it is necessary that ; ; Bat 3 xt ; i 1 al - J Ee Sr 7 ~7 An) = Ain 3 yl a dy = (p* — Aer) Pe’) : Wy . Comparing this result with (25.3), (25.4) and (25.6), we see that, if in the above definition of the y's we take where the f’s are given by (25.6), then the components in the y's of the vectors (25.3) will have the values (25.12), and will be parallel along the curve because of (25.6). From (25.10) we have at points of C 0 xt nl 0 — Ae) s at EN o) an ix Bol Gof = — (Ly) Hey py = — 7) dy oy by byt’ Hence from equations of the form (5.6), we have in the y's (513) Th =0 (n=l ru=1ii=1 ....0) Since by hypothesis the vectors of components (25.11) and (25.12) are parallel along C, we have —: dy rs; 2 — 9, Gm ? Hj=1,..., n—1, we have in consequence of (25.13) and yw + " dot I'én = 0, and then for j =— n that I, = 0. Consequently we have established the theorem. B® 68 II. SYMMETRIC CONNECTIONS We observe that only the first three terms of (25.10) have entered in the above discussion. Consequently, any expressions differing from (25.10) in terms of the third and higher orders of (y* — ¢®) define transformations of the desired type. 26. Normal tensors. Because of the conditions (23.8), when the functions Cj; are developed in powers of the y's, we have 7 > / 1 i / Ch = Aud + 5 Am, . rt (26.1) 1 7 { / +A yt Yr where 3 at pre (26.2) 7 [ot 0 Yy adn] yr 0 From the equations of transformation (23.2) we have 2) = ad; oe == = (TY, 0 y/ J . 2 0 ( J)o ) (26.3) HE Le Jy oye Lee In consequence of the first set of these equations it follows that, if at the origin the numbers (26.2) are taken as the components of a tensor in the #’s, the components of the same tensor in the x's have the same values. If we take any other coordinate system 2” and the corre- sponding normal coordinates y/* with the same origin as above, we have a new set of constants defined by | . or Ch (26.4) ol, _— |] . Py: 30h We will show that the 4's and A4”s are the components of the same tensor in the y's and y's respectively, and also in the x's and 2's respectively. From (23.11) it follows that 72 9g ri 0% \ BY. Ley (26.5) oy iT a; , oy oy TE 26. NORMAL TENSORS 69 Consequently the functions Che and CJ. satisfy the relations [ef. (5.6)] yy” ay r AE he iopd : (26.6) cr 2 _ 0 Since af in (26.5) are constants, we have from (26.6) by differentiation et ay l TL ThE 2h... 04" OY a7) 7 Y : | GY TE TRE dy" ou dy" oy’ oy 9 dy" . At the origin of the two sets of normal coordinates (26.8) LEME a Bile" \pyB Bye Balle oy’ ’ in consequence of (26.3) and similar equations. Consequently at the origin we have from (26.2), (26.4), (26.7) and (26.8) i 3 G G 1“ ox gf 2” 37 4% ox LTT omen = TTT Sa : Flys Provo pad dak gh Dat Hence if at each point in space we obtain the numbers Aja... and A'3y6...c, by the processes (26.2) and (26.4), these are the components in the 2's and zs of a tensor. Being defined at each point of space, the A’s and 4”s may be regarded as functions of the z's and z's. In fact, we shall show presently what the functional forms of certain of them are. Following Veblen and T. Y. Thomas,* who have developed this theory, we call them normal tensors. If we differentiate the equations ) ve. do 32 of i ul ak. 26.9 (E22 wm Lp OW ( ) ) Jh 9 y” a yy 9 yr ai By Fl y/ 9 iy * 1928, 1, p. DAT. 70 II. SYMMETRIC CONNECTIONS with respect to 7’ and make use of (26.3), we have at the origin 5 Z a1}, +2 0 pg (26.10) Af = Se. a ve — I's I. Since any point may be taken as the origin, equations (26.10) define Aj irotihoat space. If we differentiate (26.9) with respect to ¢* and y”, and proceed as above, we obtain 32 I}, i 8 Th 8 Ii J Atm = Pi da T— Lom TR hr Lim Ta da = Tr 8 Ij jh 5 In 30% i | J (26.11) "oan rh— 5 2m rh nr Il — Th lh, 7 pe bg Fi rh, + Ajit Lim + Ajiom Li 7 h 7 h 37 + Le (Tf Dion + Lin 172) - By continuing this process the components of a normal tensor of any order can be obtained. a From equations (26.2) it follows that the components A,..1, are symmetric in j and % and in the last » indices. In con- sequence of (23.6) we have that, if Cj as given by (26.1) is multiplied by y/#* and j and k are summed, each term on the right must be zero, that is (26.12) Pidly 0) = 0, where P indicates the sum of all the terms obtained by permuting the indices. However, because of the above ob- servation concerning symmetry, this equation can be replaced by (26.13) SA...) = 0, where S indicates the sum of the (r-+1)(r-+2)/2 terms obtained by the permutations of the subscripts which do not yield equivalent terms. Thus for » = 1 and r = 2 we have respectively ; ; ; (26.14) Aji + Ay + Ajj = 0, (26.15) Ajpam + Ajpmn + Ajmia + Aktjm + Alomjs + Aim = 0. 26. NORMAL TENSORS. 71 Because of (23.9) the functions hid; as defined by (23.15), are expressible as power series in y's of the form i j / E 20616) C.., — Ay...n + g Alun, yryt where because of (23.16) and (26.2) [ 20 M hue], B= fn Tey oy! Sa oy* 0 0 va mt lob BAL r(r—1) ( 2 ; 4, >of (26.17) 1) the functions Al. J 0,1, being components of a normal tensor and S indicating the sum of » (—1)/2 different terms obtained by the permutation of the indices ji, ---, j». From these results it follows that A(; ...;)z,...;, are components of a tensor in the 2's, symmetric in the subscripts ji, ---, j- and in the subscripts /, ---, ls. When we apply to (26.16) reasoning similar to that which led to (26.12) from (26.1), we have (26.18) Pral sg)i== giving identities connecting the components of these tensors. Since I, as defined by (22.7), are symmetric in % and /, it follows from (5.4) and (26.10) that (26.19) Bi = Al— An. The identities (21.3) follow directly from this result; and (21.4) because the A’s are symmetric in the first two indices. From (26.14) and (26.19) we have ; 1 on ; 1 : : (26.20) Aj; = n (2 Bj. + Bij) = 5 (Biju: + Bij) - From (26.19) by covariant differentiation we obtain Bixi,m = Ajik,m— Ajit, m - 2 II. SYMMETRIC CONNECTIONS In normal codrdinates 3’ we have because of (26.10) and (26.17) dim = Ajam — Arm - Consequently we have (26.21) kl = Ajuem = Ajtitm in any coordinate system. In like manner we obtain Bl, mm, — Sm m, GE tm, ol Atom, Lom, — Aun, A, + Ahm, Abiim, To Ao, Aen, -L Lon, (Al — Abn = a m, (Af nT Abr) ab Am, (Al on Am) 1-4 a. (4 ba fii hr) . By this method we are able to express any covariant derivative of the curvature tensor in terms of normal tensors.™ 27. Extensions of a tensor. In this section we define a process of obtaining tensors of higher order by differentiation, suggested by the method of obtaining normal tensors. Consider a tensor of components Te and Tas in ceneral coordinates 2¢ and /, and let t: on oe it denote the components of this tensor in jhe ner) covrdinates yt and y £3 corresponding to z¢ and 2‘ respectively, for the same origin P. Accordingly we have 2h: . i ail £ yey 0 yh 1h 0 yr 0 y" SE 0 y" dds By Bs oy“ oy" oy” Dy’ i dy" dy“ Since the quantities = 7.ond : z are constants (§ 23), we y have from the preceding equations on differentiating with k k respect to y'*, ..., y*, * For further developments concerning normal tensors see Veblen and Z. XY. Thomas, 1928, 1, pp. 576-580. 27. EXTENSIONS OF A TENSOR 3 Th gi) Wi af a >< oy" eli oy oy" lois dy 5 ys 5 yh 94 dy" : If we put 27.2) 100, : cs ( 2; i i i and similarly for the #’s then at P in consequence of (27.1) and (26.8), we have Co . 0, pr 8 oy a. pyr Jo pty ay gly ox : LE Madr 2 Jill A = EN Yn £1 Eels ST Vp 5 x1 oa ‘p i 3% : Hence at P the numbers so defined are the Sompanenis of a tensor in the coordinate systems 2" and 2°. Since P is any point, we have thus at every point the components of a tensor in the two coordinate systems, and thus the 7"s and 77's of (271.2) are functions of the z's and z's, as in the case of the normal tensors. Following Veblen and T. Y. Thomas* we call them exfensions of the pth order, when there are p additional subscripts as indicated in (27.2), From (27.2) it is seen that an extension is symmetric in the subscripts indicating differentiation, whereas this is not the case for covariant derivatives. When p = 1, the right-hand member of (27.2) is equal to the first covariant derivative of ig at P and con- sequently the left-hand member is the first covariant derivative of Cay in the 2's, evaluated at P. However, when p > 1, the pth extension is not equal to the pth covariant derivative, since the second and higher covariant derivatives involve derivatives of the coefficients Cis which are not zero at PF. #1923, 1, p. 572. We use a different notation in that a comma followed by p subscripts indicates the pth covariant derivative and a semi-colon followed by p subscripts the pth extension.. 74. II. SYMMETRIC CONNECTIONS In order to obtain an expression for the second extension . 00 OL. . of Cad we observe that at P the second covariant 3 i”s .0 derivative of az is given by [ef. (6.1)] 1 s EEE y Lr i 2 Cyr Cy gi Pi oe Bs 3 o, .e oy 0 Cig oe -a,, oy m= rE a Me, EE io de oy 0 i 01 y. cy Y T a1, cer 8 c : as x» or 200 Pir I— fr A D 2 or — UBB; 1Thiy Ps 3 J ely 29° From the preceding observation and those of § 26 we have in the »'s at P Lose, ; ey. Cy a oy Cp LL Cpe Qi 16011 Op [0 7 OD J! 2 Ts! Ass Bi Buyd BrP fs (27.3) ? Tr Oy: Uy T = 2 1B Tig Bs Aira: where the A’s are normal tensors. Since P is any point, we have thus the general relation connecting covariant derivatives and extensions of the second order. Evidently this process may be extended to any order. In general, the difference between a pth covariant derivative and an extension of the pth order is expressible linearly in covariant derivatives of orders p—2, p— 3, ---, 1 and the tensor itself, the coeffi- cients being normal tensors covariant of orders 3, 4, - --, p+1 respectively.* The form of these expressions is not so important as the fact that there exist tensors whose components reduce to the derivatives alone at the origin of normal coordinates, as in (27.2). Moreover, we remark that both the covariant derivatives and the extensions are generalizations of ordinary derivatives in euclidean space referred to cartesian coordinates, since both expressions reduce to ordinary derivatives in this case. 28. The equivalence of symmetric connections. The question of whether a set of functions Ij; of codrdi- * Cf. Veblen and T. Y. Thomas, 1. c., for a number of examples. 98. EQUIVALENCE OF SYMMETRIC CONNECTIONS 5H nates »' and a set I'j. of coordinates »’* define the same symmetric connection reduces to the problem of determining whether the equations 92 at & Vi 0u LC 2d 0: ak ge oat mp Ct BTR - == Ig re 328 ax" px" ba 7 an” (28.1) . . | ) 1 . . admit a solution 2’ = ¢' (z’', ---, 2’™), such that the jacobian of the ¢’s is not zero. The conditions of integrability of these equations are (ef. § 21) rc / i oe (28.2) Baro Ue = Biju uz u, ug, where we have put : 0x ; (28.3) TE oa’ : When equations (28.2) are differentiated with respect to =’, the resulting equations are reducible by means of (28.1) to ¢ : /C 3 SES ane J m (28.4) 350, a, le: == Bia, mm, Ug silstieigy EL J Continuing this process, we obtain the infinite sequence of equations : 0 o’ yt ap m, Bs, G10, be == Bj, mu, Ug oie "a, 3 (28.5) 5/0 7 Laie »e x J ni Bs, Gy: +0; We = Bij, m,n, ita ee Wo! ; By means of (28.3) equations (28.1) can be written as ny 7 == JI e015 1} up off These equations and 2% 3) consiit je a system of the form (8.1), such that the »? quaniities we and the 7 quantities 2 are the functions 6% the z's being the independent variables; consequently M — n®-+n. The equations (28.2), 76 II. SYMMETRIC CONNECTIONS (28.4) and (28.5) are in this case the sets Fy, Fy, --- of § 8. Accordingly we have: A necessary and sufficient condition that two symmetric connections of coefficients Lj, and Th, be equivalent is that there exist a positive integer N such that all sets of solutions of the equations of the first N sets of equations (28.2), (28.4) and (28.5) satisfy the (N+ 1)th set of these equations; if the number of independent equations of the first N sets is n®-n—1p (p = 0), the solution involves p arbitrary constants. From the considerations of § 8 it follows that, if the equations (28.2), (28.4) and (28.5) are consistent, then N=1 and the solutions are gi for e =1,....7, then (29.10) 9; = 4,99, where the A’s are arbitrary constants, is a solution. In general the A’s can be chosen so that the determinant |g;;| is not zero. When this solution g;; is taken as the fundamental tensor of the Riemannian space, the tensor Bj; becomes the Riemannian curvature tensor Rj;. For this space the other r—1 sets of solutions are tensors whose first covariant ~ derivatives are zero. Hence we have the theorem: A mecessary and sufficient condition that there exist for a Riemannian space p (—1) tensors a, other than the funda- mental temsor qi, such that their first covariant derivatives be zero, 1s that there exist a positive integer N, such that the first N sets of equations (29.4), (29.5) and (29.6), in which Bj is the” Riemannian tensor Rj, admit a complete set of solutions, Gijy OFs = +, a, which also satisfy the (N-1)th set of the equations. t A space with a symmetric connection is said to be flat, or plane, if the curvature tensor Bj is zero. In this case equations (29.1) are completely integrable. Hence we have: * Cf. Eisenhart and Veblen, 1922, 4, pp. 22, 23; also Veblen and T. Y. Thomas, 1923, 1, pp. 590, b9l. T Cf. Eisenhart, 1923, 3; also, Levy, 1926, 5. 30. SYMMETRIC CONNECTIONS OF WEYL 81 A flat space is necessarily a Riemannian space. From the last theorem of § 9 it follows that a preferred coordinate system exists for a flat space such that the co- efficients I for this coordinate system are everywhere zero. In this coordinate system the solutions ¢;; of equations (29.1) are constants. Consequently the preferred coordinates are generalized cartesian coordinates.” 30. Symmetric connections of Weyl. We consider the symmetric connections for which there exists a vector ¢; and a symmetric tensor ¢; such that (30.1) Jij, k +2 ij $1: — 0, and the determinant ¢ is not zero. We remark that it follows from (29.9) that if ¢; is a gradient the space is Riemannian. We assume that ¢; is not a gradient. If we substitute in (30.1) the expressions (5.9) for the I'’s, we obtain 9 Iin a, +g, af == 20 In i @ From these equations we obtain (30.2) a = 0 ont 0 gi —gin eg’, where (30.3) 9 = gq, Consequently the coefficients of the connection are (30.4) Ph ww Ll ob pA HU. nw | jf 2 ax + 0 9; "9% 9 Symmetric connections of this kind have been proposed by Weyl+ as the basis of a combined theory of gravitation and electro-dynamics. From the remarks at the beginning of this section we observe that in a sense it is an immediate generalization of a Riemannian geometry. If we put 06 (30.5) =m B=, * Cf 1926, 1, p. 34. T1921, 1. pp. 125; 296. ]9 II. SYMMETRIC CONNECTIONS where 6 is an arbitrary function of the z's, we have that gi and ¢; also furnish a solution of (30.1), when g; and 9; do. We may speak of two Weyl geometries whose fundamental quantities are in the relations (30.5) as conformal to one another. Weyl refers to the effect of changing 6 as a change of gauge. When we express the conditions of integrability of equations (30.1), we have in consequence of (6.4) (30.6) guj Bia + gin Ba + 295 (ori — 91.0) = 0. Multiplying by ¢¥ and summing for 7 and j, we have : 0 ow 0.7) Bu=nlpr—m)==n Edy = J In § 5 it was seen that for any manifold Bj is the curl of a covariant vector which is determined to within an additive arbitrary gradient. In consequence of (30.7) we may consider gr in (30.1) as. a definite function of the 2's namely a, where a; is a vector whose curl is equal to Gon fel. (510M). Hence equations (30.1) are of the form (8.1). : By means of (30.7) equations (30.6) may be written in the form (30.8) hj Bat gm Bi = 0, where I n 1 Lj Bir — Bim reel Bij. Equations (30.8) constitute the set F} for the theorem of § 8, and the sets Fy, F;, --- are obtained from (29.5) and (29.6) on replacing the B's by B’s; we call them (30.8)" and (30.8)" respectively. Hence we have by means of § 8: A necessary and sufficient condition that equations (30.1) admit a solution is that gi be a vector such that Bi is the curl of the vector my, and that there exist a positive integer N such that the first N sets of equations (30.8), (30.8)" and 31. HOMOGENEOUS FIRST INTEGRALS 83 (30.8) admit a complete system of r(Z 1) sets of solutions which satisfy also the (N+ 1)th set; then the complete system can be chosen so that the functions gi; of each system satisfy equations (30.1). When » = 1, we must add the further condition |g; 4 0, in order that a given connection be that of Weyl. When r > 1, ordinarily by a suitable choice of the constants 4 in equations of the form (29.10) we can obtain a solution g; for which |¢;| + 0 and thus have a Weyl geometry. When more than one such solution exist, not conformal to one another, we have several geometries of Weyl, which have the same symmetric connection, and consequently the same paths. As in § 29 we have that N and r are invariantive numbers of Weyl connections. 31. Homogeneous first integrals of the equations of the paths. If each integral of the equations of the paths (22.4) satisfies the condition dz’t dx’ dz'™ 31.7 een Ty CRT ST - 0 i, (31.1) feet ds ds ds SONAL the equations (22.4) are said to admit a homogeneous first integral of the mth degree. From the form of (31.1) it is seen that there is no loss of generality in assuming that the tensor a, ...,, is symmetric in all subscripts. If we differ- entiate (31.1) covariantly with respect to 2%, multiply by ak 2 1; , sum for ~ and make use of the equations of the paths ds written in the form : d xk ed (31.2 Ta 51.2) ds to Jk ? we obtain dx’ dx’ da* 31.3 pv Jy ——a a ) rye Tin! ds ds ds = 0. Since this equation must be satisfied identically (otherwise we should have all the solutions of (22.4) satisfying a differ- ential equation of the first order), we must have 6% 4 II. SYMMETRIC CONNECTIONS (31.4) Plar,..r,u) = 0, where P indicates the sum of m+ 1 terms obtained by per- muting the subscripts cyclically. Conversely, if equations (31.4) are satisfied, equations (31.3) are and the left-hand member of (31.1) is constant along any path. (Cf. § 43). In particular, if the integral is of the first degree, that is, d at di— — const, ds the conditions (31.4) are jr Oj 4 = 0. The question of linear first integrals is considered in § 44. We consider the case when the equations of the paths admit a quadratic integral, namely drt dxl (31.5) rae = const. In this case the conditions (31.4) are (31.6) gi.k + gi + gry = 0. From § 29 it is seen that Riemannian spaces are a sub-class of spaces with symmetric connection for which a homogeneous quadratic integral exists. From (31.6) we have (31.7) gigi + jx, a+ grip = 0. Interchanging % and /, we have (21.5) gi, + gp, in + gui je = 0. If we subtract from the sum of these two equations the sum of the two equations obtained by interchanging 7 and % and J and k in (31.7), the resulting equation is reducible by means of (6.4), (21.3) and (21.4) to (31.9) gy. m—git,ij = Gej Bli + gia Biig— ger Bjir— gre Bia. 31. HOMOGENEOUS FIRST INTEGRALS 8H If & and / are interchanged and this equation is subtracted from (31.9), the resulting equation is satisfied because of (6.4). Thus we are unable to solve equations of the form (31.7) for each of the quantities gj. However, if (31.7) be differentiated covariantly, we obtain equations which can be solved for ¢; xm and then the further conditions of in- tegrability can be obtained with the aid of (6.4); and the determination of whether a given space admits one or more quadratic integrals is reducible to an algebraic problem somewhat after the manner of § 29, as has been shown by Veblen and T. Y. Thomas.* Instead of developing this question further we consider the problem of determining symmetric connections for which there is a given quadratic integral, such that the determinant ¢ is not zero. With the aid of the tensor g; we write the I's in the form (5.9); then ij. lo = ary —%ki; where (BL10) my = gp oly, of = Pay, Poy= 0; we remark that an is symmetric in jy and %, as follows from (5.9). Hence the conditions (31.6) become (31.11) ijk + Aji + ary; = 0. If cj is any tensor symmetric in ¢ and j and we put (31.12) ijk = 2 Cijk— Cikj — Cjki, the condition (31.11) is satisfied. Hence if we have any tensor cj symmetric in 7 and j, and define aj by (31.10) and (31.12), then the symmetric connection given by (5.9) is such that the equations of the paths admit the first integral (31.5). Conversely, if (31.5) is satisfied for a given connection and consequently a; are given, the tensor ¢; is not uniquely defined by (31.12). In faet, from (31.11) it follows that *1923, 1, pp. 599-608. 86 II. SYMMETRIC CONNECTIONS ay; — 0 and consequently ¢;; are arbitrary. When two of the indices are the same, we have from (31.11) and (31.12) CH) Gi = 9 Ai) == = ij. Consequently either one of these ¢’s can be chosen arbitrarily and the other is then determined; hence there are n(n—1) arbitrary choices. When all of the indices are different and have given values, there are two independent equations (31.12) for the determinination of the ¢’s with these same indices. Consequently any one may be taken arbitrarily and the others are uniquely determined. Hence we have: A tensor gi for which g + 0 and a tensor cj, symmetric in i and j determine a symmetric comnection for which the equations of the paths admit the first integral (31.5); con- versely, if a geometry of paths is given whose equations admit a first integral, n(n-+1) (n+ 2)/6 of the components ci: are arbitrary and the others are uniquely determined.” * Cf. Eisenhart, 1924, 2, p. 884. CHAPTER 111 PROJECTIVE GEOMETRY OF PATHS 32. Projective change of affine connection. The Weyl tensor. In § 22 it was shown that the paths are the same for two symmetric connections whose coefficients are in the relations vi i i i 52.1) Ip = Ig 6 vo 4, where ¥; is an arbitrary covariant vector. We say that the affine connection of coefficients ry, is obtained from that with the coefficients 77; by a projective change of the connection. If we write the equations of the paths in the form Bt | = ded dF . 2.2 Tig I; i i 522 ds® =r ds. ds 0, analogous to (22.4), we have from these equations and (32.1) that s is given as a function of s along any path by (32.3) 2 == cf 2 Lk ds. For the expressions 77; the components of the curvature tensor Bj, analogous to (21.1), are reducible to 32.9 T= Bi = of (Wi — Wiz) + 62 Yi — Ol; Wir, where ; (32.5) Yu = W—Y Wi. Contracting for ¢ and / and for ¢ and j, we have (cf. § DB) (32.6) . Bi. = B+ np— guy, Bu = But 23 (Wr — Yur) wil £ BY J 9 A 5 at 5 1k 87 (32.7) : = Bu 38 III. PROJECTIVE GEOMETRY OF PATHS In § 5 it was shown that 8; is the skew-symmetric part of B; and that it is the curl of a vector a;/2. Consequently if we choose (32.8) ad aa “J. where o is an arbitrary function of the 2's, we have 8; = 0. Hence we have: By a suitable projective change of the affine conmection the tensor By; for the mew comnection is symmetric and the tensor Bij is a zero tensor.* From (32.7) we have also: When the tensor By; is symmetric, a necessary and sufficient condition that the tensor Bj; for a projective change of conmection be symmetric is that VY; be a gradient. From equations (32.6) and (32.7) we have 1 - 2 5 Wi = — = (Bj Bp (Bi.— Bi); (32.9) Yeni = ToT (Bix — Bir). When these expressions are substituted in (32.4), the latter are reducible to Ww} = = Wk, where Wa Sn Bu “ 2 0; ‘ Bur ert EER TF Bp— 0; Bj: ) (32.10) i gt ri} Bir — 0% Bj). Hence the tensor Wi is independent of the vector v;, that is, it is unaltered by a projective change of affine connection. It was discovered by Weyli, and was called by him the projective curvature tensor. We shall call it the Weyl tensor. * Hisenhart, 1922, 2, p. 236. 71921, 2, p. 101. 32. PROJECTIVE CHANGE OF AFFINE CONNECTION 9 Since (32.11) Bi— Bji = 284, equations (32.10) can be written in the form Whi = Bha+— Tard J; (Bii— Bu) + = [0%(n Bj + By) — 0 (n Bj + By). In consequence of the identities (21.3) and (21.4) we have the identities ; (32.13) Wika + Win = 0, (32.14) Wi + Wij + Wii = 0. Also from (32.10) we have by contraction (32.15) Wea = Whi = 0. If we ditferentiate (32.10) covariantly with respect to the I's and make use of the identities (21.5), we obtain Wit, m= Wont: + Wha he [0% (Bjt,m — Bim, 1) + 04 (Bjm,i— Bite, m) + dn (Br. i— Bp.0)] + DN . [0% (Bim,1 — Bit, m) + 07 (Bie, mu — Bim, 1) + oh Binr—8r7. Contracting for 7 and m, we have in consequence of (32.15), (S210) Wh; == pod —% |. By, rr 2 1 (Ba n— gies) An invariant, such as the Weyl tensor, which is unaltered by any projective change of the affine connection is called a projective invariant. By processes analogous to those used in § 18 we establish the theorem: 90) III. PROJECTIVE GEOMETRY OF PATHS j « If Mey and 27 are the components of any ennuple, the tensors 4 1 3 SO) Lr ial an i i ; v7 ah i La),j— nt 1 (Ae) yo Aigy.j + 0; Me Ly; 23), n) and the invariants 1 : : (dy Pins O eT ad (0.7.5 T 927." are projective invariants.™ A tensor which is not a projective invariant may, however, be invariant under certain projective changes of the connection. Thus from (32.4) and (32.5) we derive the theorem: A necessary and sufficient condition that the curvature tensor be unaltered by the projective change defined by a vector Vy; is that the latter satisfy the condition (82.17) Wij — Wi = This is also a necessary and sufficient condition that the tensor By; is invariant under the change. Equations (32.17) can be written in the form 0 Yi i ow =, where ~ os 1 : 7 (32.18) Fh = 1h+ 2 y+ 8 up). Hence the space with the affine connection defined by r; admits a field of parallel covariant vectors ¥; (§ 11). Consequently the problem of the theorem and that of spaces admifting fields of parallel covariant vectors are equivalent. Similarly from (32.5) and (32.6) we have: A necessary and sufficient condition that the symmetric part of the tensor By be unaltered by the projective change defined by a vector Y; is that (32.19) Vit Wi—2¢ 4; = 0% *CL 7. Y. Thomas, 1925, 10, p. 319; also Levy, 1927, 1, p. 310. T Cf. Schouten, 1925, 6, p. 4563; also, J. M. Thomas, 1926, 8, p. 62. 1 Of. J. M. Thomas, 1926, 8, p. 62. 33. AFFINE NORMAL COORDINATES 91 For the affine connection defined by (32.18) equations (32.19) become Wii +: = 0 that is, the equations of the paths admit the linear first integral vy; " = const., where s is defined by an equation ls of the form (32.3) (cf. § 43). Also we have from (32.7): A necessary and sufficient condition that the skew-symmetric part of the tensor Bj, and consequently the tensor Bhi, be unaltered by the projective change defined by a vector WY; is that yy be the gradient of an arbitrary function. The second theorem in this section is a corollary of the above theorem. 33. Affine normal codrdinates under a projective change of connection. If we denote by 4 and 3’ the affine normal coordinates corresponding to the same coordinate system 2 for two connections in the relation (32.1), the equations of the paths through the origin 7 in these coordinate systems are given by (23.1) and y' = (22) s Y= de la Since a projective change of connection leaves each path individually invariant, it follows from the above equations of the paths that along each path 3° is proportional to #7, Moreover, throughout the domain under consideration y" is a function of the y's. Consequently these functions must be of the form sR ie yw where f(y) is a function of the y's regular in the neighbor- hood of the origin and not vanishing at the origin. Similarly we have (33.2) i 1% ff == ol, Fw) 92 111. PROJECTIVE GEOMETRY OF PATHS where fis of similar character and f(y) - f(y’) = 1, because of (33.1) and (33.2). It 0% and CO ii are the respective coefficients of connection in the y's, we have (33.3) C= — 210 of 0; Yr = or Y;, where ¢; are the components in the y's of the vector defining the projective change. If Cf; are the coefficients in the 2's of the second connection, we have from equations of the form (5.6) Gay Zp=o 00 yun 2y/? ay” ay'* ay’ Bofs Since the y'’s are normal coordinates, we have Cj; yPyT = 0 (§ 23). In consequence of (33.4) these become (33.5) (2 3 a) ou DU yay — 0. DO. Jk dy a dF oy? oy" yy From (33.2) we have oy’ 1 on ! iE (oh Ly" ! oy J oy dll Dwi, 7 El pm 3 9 2 0 Yak Duy) of ir Ee J] wr Fa op Taga? Yr oy / “ ; of a Since the equation /’— oy? y'” = 0 does not admit a solution regular at the origin and not vanishing there, and since Chey’ yf = 0, the equations (33.5) are reducible to (33.6) [ow ed i) Sa A+ pts pied 33. AFFINE NORMAL COORDINATES 93 Since Uy are assumed to be regular at P, we put yt iiayd YP — bio + bri yy + ar biij vy Uy = > f=1 Fay yay + where without loss of generality the a4’s are symmetric in the indices and the 0's in all but the first. Substituting in (33.6), we find that a; = —by and that the other 4's are uniquely determined. Thus when vy; are given, the function f is determined. There is also the converse problem of giving f and finding the ys from (33.6). We consider in particular the case when (33.7) f= 1+ av where the a's are constants, so that the transformation (33.1) is linear fractional. Now equation (33.6) reduces to gy tay Although this equation gives the condition on the ¥’s in the y's, we are interested in finding their components in a general coordinate system so that we may have a means to knowing when the case (33.7) is possible. To this end we differentiate this equation with respect to z¢, multiply by # and sum for /; then from the resulting equation and (33.8) we eliminate a; This gives (33.8) ve + : oy (53.9 Py (Gr — we) =o, which because of the relation Ch 1 yy = 0 and (23.1) can be written thus dy “dy = YY i UY) = ds ds In the general coordinate system x? corresponding to the y's this equation is 94 111. PROJECTIVE GEOMETRY OF PATHS | ok ] ! Ji oe Ls (Whe, 1— Wk: yy) = 0, where 1}; are the components in the z's of the vector i in the y's. If this condition is to be satisfied. at each point in space, it is necessary that (33.10) Wii + Wie— 20k; = 0. Since this is a tensor equation, if it holds in one coordinate system, it holds in all.- Conversely, if there exists a vector satisfying (33.10) for a given space, and we choose a normal coordinate system #¢ with a given point P for origin, equation (33.9) holds at P. If we put —1 . 7 differentiate with respect to #/, multiply by 2 and sum for /, we have in consequence of (33.9) (33.11) Sumit eq ; wad 0 Yury w+-1)+ = 21 Y=0 Y By means of (33.11) this is reducible to aa 219 ee 1 If the function 7 is regular at P then f= 1 at P and the integral of (33.12) satisfying this condition is f= 1a; 7’. From these results and the fifth theorem of § 32 we have: In order that the affine normal coordinates at every point undergo a linear fractional transformation when the affine connection undergoes a projective change, it is necessary and sufficient that the symmetric part of the tensor Bj; be unaltered by the projective change.”* 34. Projectively flat spaces. We may interpret the results of § 32 as giving spaces with corresponding paths. * The question of this type of projective change was raised by Veblen, 1925, 7, p. 131, and the theorem was established, in a different manner, by J. M. Thomas, 1926, 8, p. 62. 34. PROJECTIVELY FLAT SPACES 95 Weyl* has called a space Vy, projectively flat when its paths have the same equations as the paths of a flat space V,, (§ 29). This is equivalent to saying that for V, there exists a pre- ferred coordinate system in terms of which the finite equations of the paths are linear. Since for V, we have (34.1) Bl =, Bye=0, Bi=0 it is evident that a necessary condition that a space be pro- jectively flat is that the Weyl tensor be zero. We: shall show that this condition is also sufficient, when »n > 2. From the first of equations (32.9) we have : 1 2 (34.2) Puy == Wily Byte 5: The conditions of integrability of these equations, namely I Wi i— Wikg = Yn By, are reducible, by means of (32.11) and the expression for Bj, obtained by equating to zero the right-hand member of (32.10), to : 2 (34.3) Bi. j— Bur + rT {Byr—8x)) = 0. From (32.16) it follows that these equations are a consequence of the vanishing of the Weyl tensor, when n> 2, as was to be proved. When n= 2, we have, because of the identities (21.3), 2 ; 1 ; 2 1 Bu = Bie, Bw = Bin, Ba = Bun», Bx = Bm. Hence from (32.12) we find: The Weyl tensor vanishes identically when n = 2. Accordingly we have the following theorem of Weyl:+ *1921, 2, p. 104. 11921, 2, p. 105. 96 III. PROJECTIVE GEOMETRY OF PATHS A mecessary and sufficient condition that a space V, with an affine connection be projectively flat is that the Weyl tensor vanish when n=>2 and that equations (34.3) be satisfied when n=2, From (32.1) it follows that, if a space V, is projectively flat and 2¢ are cartesian coordinates in the projectively related flat space, the coefficients of the affine connection in V,, are given by ; : (34.4) ry ee = (a Wi. + 0 Yj). Conversely, the most general projectively flat space is ob- tained by taking the 77s in the form (34.4), where ; is an arbitrary vector. When the expressions (34.4) are substituted in the equations of the paths (22.4), the latter can be integrated in the form Toten? x — a” 2 fv, az* (34.5) NR eal ny == fe Juv, az as, the integral Ju dz" being taken along a path, which result is in keeping with the remark at the beginning of the section. From (34.2) it follows that a necessary and sufficient con- dition that V,, be a projectively flat space for which the tensor Bj; is symmetric is that ¥; in (34.4) be a gradient. ; a oY If we replace ¥; in (34.4) by a curvature tensor are expressible in the form the components of the hn wis po Ba ad. ijk — z Je io er (3 6) Bij e Doo k Pr bm Contracting for 2 and %, we have 2 e? (34.7) B= (1 — ne? Sr From these equations we have 1 1 v v (34.8) Bl + nal (0 Bi. — Oi By) = 0. 34. PROJECTIVELY FLAT SPACES 97 The left-hand member of this equation is the expression for Wh when Bj; is symmetric, as follows from (32.10). We consider now the question of determining the Riemannian projectively flat spaces, that is, spaces whose geodesics can be put into correspondence with the straight lines of a flat space. In this case Bj; are symmetric and are in fact Rj, the components of the Ricci tensor. For n>2 we have equations (34.8), which are equivalent to 1 (34.9) Bur. = Er (gn; Ba: — gr: By) - When in (34.9) we put 2 = i, we find that (34.10) Bs = I,(1—n) zp, where K, is the factor of proportionality thus obtained. By reason of (34.10) equations (34.9) are reducible to (34.1 1) Rhijrc = K, (gn; gi — Ynk gi) . Consequently V,, for n > 2 is a space of constant Riemannian curvature K,.T When n = 2, it follows from the definition of Rj; that Ly i Rs a pin) ue Faqs J11 J12 Wi 22 gq Since R,i3/g is the Gaussian curvature K, of the surface, it follows that (34.10) holds also when » =— 2. When we apply the conditions (34.3) to (34.10), we find that K, is a constant. Hence we have: A necessary and sufficient condition that a Riemannian space be projectively flat is that its Riemannian curvature be constant.} From (34.10) we have for all values of nn, Bj; = O. Conversely, if we have * 1998, 1, p. 21. +1926, 1, p. 83. + Weyl, 1921, 2, p. 110. -X 98 111. PROJECTIVE GEOMETRY OF PATHS (34.12) Dy == Bj, Bij x EEE 0, gy defined by (34.10), where. K, is an arbitrary constant is such that gx =— 0. Hence we have: A mecesssary and sufficient condition that a projectively flat space be Riemannian is that (34.12) be satisfied. In the codrdinate system for which R;; takes the form (34.7) we have from (34.10) 1 02 e? (34.13) gC aan From this expression and (34.4) in which vy; = gad it follows that the conditions ¢;,x = 0 are reducible to 53 2 TTT ee ( - 0" Ox oak } Consequently (34.14) AY = wat al +20; 2 +c, where the a’s, b's and ¢ are constants. : 35. Coefficients of a projective connection. From equations (32.1) we have (35.1) Ti = Tita +1) ws, from which and (32.1) we find that the quantities 1 n—+1 are independent of a projective change of affine connection.” We call 7); the coefficients of a projective connection. In order to find the relations between the functions ji; and the analogous functions /7f; in a coordinate system 2/°, we remark that from equations of the form (5.6) we have (35.2) ul, = rj,— (0% Is + 05. Ti) J 110 dx g H ) 0 log A xd ’ *CL 7. Y. Thomas, 1925, 2, p. 200. 35. COEFFICIENTS OF A PROJECTIVE CONNECTION ' 99 where 4 is the jacobian of the transformation, that is, (35.4) 4 = | Then from (35.2) and analogous expression for my and from (5.6) and (35.3) we obtain Py” a n dx pe i Bald 0 2 } ox" 06 BE bm Uh Br xf oa Bul Bf (35.5) | pal Be he Pal Amd’ where for the sake of i we have put (35.6) p = She log A. When we express the conditions of integrability of equations (35.5), we obtain : dat bat oa 0 i 7 GBI) Uo J, oe EE Dnt dp 7) ijk pro 5 7° 5 xt 3 2) 5 2k J Cike is ke Cijy where aN (35.8) ml = oo + he I — I; TO oa) and oe a 06 0 04. 24 a 026 (35.9) Cy 1; ah Vat Sxl bat dad’ Contracting (35.7) for 7 and k, we have Ra 1 px’? oa” (35.10) Cy — 5 3 [Bm 1s, 5 ra 27 , where by definition (35.11) HH; =e ia, When the expressions (35.10) are substituted in (35.7 d we have J a dah 8x 8x7 ba? Wie = Was rien ves ) ul". Dut Bad. BaF 100 III. PROJECTIVE GEOMETRY OF PATHS where 85.12) Wh = ml. + * iy, — OF my). From (35.8) and (35.2) we have a hn 1 I hn h (35.13) Hy — By wor? 07 Bj — 0x. Ay + 05 Aw), where by definition or; 0x and from (35.13) 1 yy. == iy 4 h rr I+ - Lr, 1 (35.14) I; = B,;— ET [28+ 0—1) 4). When these expressions are substituted in (35.12), it is found that Wik so defined are the components of the Weyl tensor (32.10).* Substituting the expression (35.10) for ¢;; in (35.9), we obtain _n — 0 8 rr 2? 0 e trier (35.15) Sl A = IT a n—1 Geils Bx dal Expressing the condition of integrability of these equations, we obtain dx" ox? 90 (35.16) Ly, = sy mre ox ia i + (1 n) Wii Ba where oll; oll; (35.17) IT; Rn A m= 5 ” + 17}; 11; ll IT. 36. The equivalence of projective connections. If we put ox” 06 (36.1) - a == 7, RT * Cf. J. M. Thomas, 1925, 3, p. 208. 36. EQUIVALENCE OF PROJECTIVE CONNECTIONS 101 equations (35.5) and (35.15) become , ou? he = 70 uf — Hf wl Wt gb ul 9, (362) 1 7) = - 9; 9; + 5 i =r (11, — 113, u? ur). These equations together with the functional relation (35.6) must admit a solution for given expressions of ZZ; and /7 jf; in the 2's and 2"’s respectively, if the two projective connections so defined are to be equivalent. From the preceding results it follows that the conditions of integrability of these equations are (36.3) Whe ty = Wes uf uf uf (36.4) a, = ns wy uf? wet Al~ WW fy If we denote by wu! the cofactor of «% in the jacobian | od | divided by the jacobian, we have from (35.6) 56 1 0 us 0a’ n+1 "98x which is satisfied identically because of (36.1) and (36.2). If we differentiate equations (36.3) with respect to 2%, the resulting equations are reducible by means of (36.2) and (36.3) to nd Wiki in 2Wi $1 i gi — Wino j — Wh Pr) (36.5) 7re B 0 +uw Who, = W Brio wu uf, where Wi denotes the projective derivative of Wi, that is, the covariant derivative with respect to the 77 be we remark that the projective derivative of a tensor is not in general a tensor. In this notation equations (35.17) may be written in the form (36.6) Hj =— Hy — Hy. We observe from (35.12) and (32. 10) that Win is of the same form in i as it is in Ij, when the tensor Bj is 102 III. PROJECTIVE GEOMETRY OF PATHS symmetric. Since (32.16) follows formally from (32.10), we have from (35.12) and (36.6) n—32 — 1. (36.7) Wii == oy If we multiply (36.5) by «! and sum for « and /, we obtain Whit (0 —2) Why, = Wt i prole When n > 2, these equations are reducible by means of (36.7) to (36.4). Consequently in applying the results of §8 we denote by Fi equations (36.3) and proceed with these equations to get the sequence Fy, Fj, --. of derived sets. Hence we have: A necessary and sufficient condition that two projective con- nections for n=>2 be equivalent is that there exist a positive integer N, such that equations (35.6) and the sets of equations Fy, ---, Fy are compatible in 0, ui, x'° and ¢; as functions of the z's, and that the (N --1)th set is satisfied in consequence of the preceding ones.™ When» = 2, Wi vanishes identically (§ 34). The above theorem applies to this case with the understanding that the sets Ii, ..., Fly consist of (36.4) and the derived equations. As in § 28 it can be shown that N is an invariantive number for all manifolds with the same projective connection, and likewise p, where (n-+1)*>— p is the number of inde- pendent equations in the sets Fy, ..., Fy. When nn >>2, the Weyl tensor and its first, -.., Nth (NV < (n-+1)% projective derivatives form a complete system of projective invariants (§ 28) for the manifold. When n — 2, the functions 77; and their first, -.., Nth (N < 9) projective derivatives form a complete system. Another interpretation can be given to the preceding results. Thus let 7Zj; be the coefficients of projective connection of a manifold V, in coordinates z’ and similarly I; in any other coordinate system 2’, the equations of the trans- formation being : (36.8) glam obit, vue, 2). * Cf. Veblen and J. M. Thomas, 1926, 6, pp. 288, 290. Bait Uw; us Uy, . 36. EQUIVALENCE OF PROJECTIVE CONNECTIONS 103 We consider an associated manifold *V,;, of coordinates a at, ..., z* and in the x's define a set of functions *Ig, 1 * Voit by 1+ n : 15, ) 5, = 1—n : x i = i wml 2 Go0) 27h = Wh, HL = hh where greek indices take the values 0,1, ...,n and latin 1, ---, n. For the transformation in *V,y; defined by (36.8) and (36.10) 2° = #-Llog a, where A is given by (35.4), we find that the coefficients (36.9) and similar expressions in the 2's satisfy the relations A 4A daft | Bia¢ on 2 Bu (36.11) py = ley Te 528 fal In fact, when «, 8, y take the values 1, ..., n, equations (36.11) reduce to (35.5); when 8 or y = 0, the equations are satisfied identically; and when « — 0, and #8 and y take values 1 to n, the equations reduce to (35.15). Thus the problem of equivalence of projective connections is reducible to a restricted problem for affine connections, as shown by T,X. Thomas.t In order to consider the problem more fully from this point of view, we denote by *Bios the curvature tensor formed with respect to the *I'’'s. In consequence of (35.12) and (35.17) we have * Bl == Wha, * Bo = tl amie ”7n—1 and that all the other *B’s vanish identically. If we take the functions so defined and apply the reasoning of § 28, equations of the form (28.2) (interchanging z's and 2's) become ntl 3612 What Gra ly = Wi, uf ul, 1926, 10. 104 III. PROJECTIVE GEOMETRY OF PATHS (36.13) 11, Ji pa 1 Sw m= I, 1 uy, Co 1 (36.14) Wi Wa went = 0, jy up went = 0, where one or more of the indices o, 0, 7 is O. For the relations (36.8) and (36.10) between the coordinates, equations (36.12) and (36.13) reduce to (36.3) and (36.4) respectively, and (36.14) are satisfied identically. 37. Normal affine connection. If for a given affine connection and a given coordinate system we choose for the components ¥/; of a projective change the values TE 1), it follows from (35.1) that fhe coefficients of the new connection satisfy the conditions I, = 0. We call this uniquely determined connection the normal affine connection for the given coordinate system.” Hence we have: Among all the affine connections with the same projective connection there is a unique normal affine connection for any coordinate system. From (5.8) it follows that Bj is symmetric for a normal connection. Conversely, if B;; is symmetric for an affine connection, we have from (5.8) in any coordinate system (37.1) ory ofl : oar oat If we put 89 ph Br = £7 these equations are completely integrable in consequence of (37.1). When we define a coordinate system 2’ by the equations (37.2) 7 — [er dot, oll==gf Home] v.00), we have for the jacobian of the transformation | 7 i by ri e? . | dat | * This definition is equivalent to that adopted by Cartan, 1924, 3. p- 223, as pointed out by J. M. Thomas, 1926, 3, p. 664. 37. NORMAL AFFINE CONNECTION 105 Consequently log 4 0g rh gad. 6xl. and from (35.3) we obtain 77 — 0. Hence we have: A necessary and sufficient condition that there exist for a given affine connection a coirdinate system a with respect to which the connection is normal, that is (37.3) rj; =o, is that the temsor Bj be symmetric. Furthermore we have from (35.3): The normal affine connection for a given coordinate system is the normal comnection for all coirdinate systems obtained Jrom the given one by transformations of constant jacobian and only for these. When equations (37.3) hold, we have from (35.2) for the normal connection : : (37.4) Ij; = Tj. Then from (35.8) we have nl, m= Bl and from (35.14) II; = B;. Hence the equations (35.12) become " 7 1 7 (815) Wi = Bint Bu—0i By). Since this is the form which (32.10) assumes for a space when Bj; is symmetric, we have thus another proof that the tensor defined by (35.12) is the Weyl tensor. The first theorem of this section is a corollary of the theorem: Fach of the affine connections with a given projective con- nection is uniquely determined by the values of rj. In fact, from (35.2) it is seen that the coefficients of a projective connection must satisfy the conditions (31.6) 7} = 0. * Cf. Cartan, 1924, 3, p. 225 and J. M. Thomas, 1926, 3, p. 665. 106 III. PROJECTIVE GEOMETRY OF PATHS From (35.2) it is seen also that when 9 are given, IT are uniquely determined, and furthermore that the corresponding functions 77; are equal to the given values of these functions. 38. Projective parameters of a path. When the ex- pressions for Ij; are obtained from equations of the form (35.2) and substituted in the equations of the paths (7.6), the latter become del ide oe dr” oo np it TT ar a Si IE (0 i AY, Var Yam all Consequently along any path we have Bo : dt dnt dx? 8. ye = (382) aml arr where ¥ is a detetminate function of #. If we define a parameter p, to within an additive constant, by the equation : dp Wdt (38.3) rod : dt where a is an arbitrary constant, equations (38.2) become Bal |, ,.: Awl dak . 2 ~ Hp — = 384 Ap? i & dp dp 0 From the form of these equations it follows that the para- meter p is not altered by a change of projective connection. It has been called a projective parameter of the path by T. Y. Thomas who established its existence.* From equations (38.2) and (22.2) we have for a path in consequence of (35.2) 2 n dx’ Y Sg n-+1 hj dt Consequently by means of (38.3) and (22.3) we find the following relation between projective and affine parameters of a path: * 1925, 2, p. 200; cf. also Veblen and J. M. Thomas, 1925, 4, p. 205. 39. COEFFICIENTS AS TENSORS 107 Bg (38.5) pe=b fe wrt | dk ds, where 0 is an arbitrary constant; in the integral on the right it is understood that the 2's are expressed in terms of s. If we make use of equations (35.3) and denote by p’ the parameter defined by (38.5) for a cooérdinate system »’“ we obtain the relation (38.6) A ea where b is a constant. Consequently when we speak of a projective parameter it is associated with a particular coordinate system:* in this respect it is different from the affine parameter s (§ 22). From the form of equations (38.5) and the results of § 37 we have: ; A projective parameter p for a covrdinate system a is an affine parameter of the paths for the normal connection for the «'s. When 4; in (32.1) are the components of the gradient of a function ¥, equation (32.3) becomes f= I AY ds. e In consequence of the results at the close of § 37 the above results are consistent with equation (38.6). 39. Coefficients of a projective connection as tensors. It is seen from equations (5.6) that the coefficients of an affine connection are components of a tensor for affine transformations of coérdinates, that is, qe i r i / 2" = af 2+ ¥, where the a's and 0's are constant, and only for such trans- formations. *CLZ Y. Thomms, 1925, 2, p. 201. 108 111. PROJECTIVE GEOMETRY OF PATHS We seek the types of transformation for which the co- efficients of a projective connection are components of a tensor. From (35.5) it follows that in this case we must have ai 3s 80 24 Bo (39.1) ox 0x’ Jz). Dot Daf Dx) Expressing the conditions of integrability of these equations, we have ox" 0. no = oak \oal dal oxt ox Coa Bh 0 aan dal \ Bat oak at r= 2 which, since they must hold for e« =1, -.., n, are equivalent to fq 00 D4 oxl 0a dxt dxt The integral of these equations is (39.2) a0 = yal}, where /# and the a's are arbitrary constants. When this expression is substituted in (39.1), the resulting equations can be written in the form . Ba Balt (ap ol-L 1) + a; Te ra 0, 0x 0x 0 0x of which the first integral is . 0 x” re « (ay ak hh) AL + ai r = Ci, where the c¢’s are arbitrary constants. Integrating again, we have ’ & x + a” (39.2) 7 == aL h the d's being arbitrary constants. The jacobian of this transformation is 39. COEFFICIENTS AS TENSORS 109 lb gril galt | 52" | 1 @ ad ee. Ligh 204) |r] = etal, ar rel | | a, © ore n n n | This result is in keeping with (39.2) and (35.6), when it is observed that in (35.6) A may be replaced by cA, where ¢ is any constant without altering (35.5). Hence we have: The coefficients of a projective connection are components of a tensor under linear fractional transformations of the coordinates and under these alone.t If there exists a coordinate system for a given space for which the coefficients of the projective connection 77; are zero, the space is projectively flat, as follows from (35.12). Conversely, when the coordinate system of a projectively flat space is such that the 77s are given by (34.4), then /Zj; = 0. Hence we have: A necessary and sufficient condition that a space admit a coordinate system for which ils coefficients of projective con- nection are zero is that it be projectively flat. When the coordinate system is such that 77; = 0, the most general transformation of coordinates such that 77 fr == () 1s any which satisfies equations (39.1). Hence we have: When the coordinate system of a projectively flat space is such that the coefficients of the projective conmection are zero, the most general transformation of cosrdinates preserving this property is linear fractional. From (38.4) it follows that in these codrdinate systems the equations of the paths are 2 = aly, =a where the a's and b's are constants. When 77; = 0, it follows from (35.2) that (39.5) ry =o Gd 14 8) * Of. Kowalewski, 1909, 2, p. 84; Fine, 1905, 1, p. 505. 7 Cf. Veblen and J. M. Thomas, 1926, 6, p. 284. 110 I1I. PROJECTIVE GEOMETRY OF PATHS and that vi 1 Sl 7 1% == oy (1-+4), where ¢ is not summed. Hence we have in general (39.6) ri=g0-19, ; J 1 = where ¢ and ; are not summed and ¢; = ol rk. Con- versely, the expressions (39.5) and (39.6), in which ¢; are arbitrary functions, define the most general coefficients of affine connection for which 77; — 0. 40. Projective coodrdinates. Let P be the point of coordinates x and consider the transformation of codrdinates defined by : el Eb 5. (40.1) x = xb +0e oC (ip ZF + of, where ¢’ are any functions of the z’’s such that they and their first and second derivatives are zero when the x's are zero, and (Mes)p indicates the value of Is at P. From (40.1) we have oat : 92 xt : (40.2) bo = de, — rr) = — apr. | 3a If A denotes the jacobian ia , it follows that 10 (40.3) [Tm ae fi pe | Se —(ts)p = 0. dx P wt Palin in consequence of (37.6). Therefore if we substitute these values in equations obtained from (35.5) by interchanging the x's and z's, we obtain (40.4) Tipp = 0. Hence we have: When a transformation of coordinates of the form (40.1) is effected, the coefficients of projective commection in the 2s are zero at the point P of codrdinates x. 0 41. PROJECTIVE NORMAL COORDINATES 111 We call the coordinates «'“ projective coirdinates. A particular system of projective coordinates is obtained, when we proceed with equations (38.4) in a manner analogous to that which yielded affine normal coordinates in § 23. In this case the equations of the paths through the point 2 of coordinates x? are a cl d xt (40.5) yl = [2 ) where p is the projective parameter for the x's, and the equations of transformation of coordinates are at Ere x: zh y' — > U1) p iy’ y (40.6) r . Te 21 A 00p yt yu" y's he vif, where nl. are the same expressions in the //’s as (22.8) are in the I's i If we denote by 1; the II's in the y's, we have from (40.5) and the equations of the form (38.1) in the #’s that the equations (40.7) lay — Hh) oy = must hold throughout the domain for which equations (40.6) define a transformation of coordinates. From the theorem of § 38 it follows that the y's as defined by (40.5) are the affine normal coordinates corresponding to the a's for the space with normal connection for the x's. Moreover, equations (40.7) follow from (23.6) and equations of the form (35.2) for Hj. 41. Projective normal codrdinates. We have remarked that p in (40.5) is the projective parameter for the z's and not for the y's. We seek a system of coordinates 27 such that the equations of the paths through P (27) shall be dx? 41, gl — [ZN (41.1) (on » 112 III. PROJECTIVE GEOMETRY OF PATHS where p is the projective parameter for the z's. Moreover, we require that the z's be Sh coordinates, that is, WD P= a LU where the terms of order higher than the second are as yet. undetermined. From (40.6) and (41.2) we have 02. 22) = [22 = an (2: = Ly r raziip When the expressions (41.1) are substituted in equations of the form (38.4), we have (41.4) Phrld = 0, the P’s being coefficients of projective connection in the z's. In order that there may exist a transformation of the ys, defined by (40.6), into z's such that (41.4) hold, we must have JE od or) Rak : = 8 oft Bylagf ! By By’ By By! oy 2 P= 0, 028 024 (m; pd. op 5. 80, 8¢ 2) as follows from (35.5), where a | 8 3 Ings, d=7% 1 = 6 = — — (41.5) TE The above equations may be written in the form 36 dy 0p 022 a6) (22 al \ ar =o. gel 0z¢ T= 9 2P z Since equations (40.5) and (41.1) define the same paths, it follows that the transformation is of the form (cf. § 33) (41.7) Y= where ¢ is a function whose expansion must be of the form (41.8) = 1+a;2 2+. 41. PROJECTIVE NORMAL COORDINATES 113 in order that (40.6) and (41.2) be consistent. From (41.7) we have oy Lie S28) (41.9) 2 ON 0? (v Of == 52] which are consistent with (41.3), in consequence of (41.8). Accordingly we have 0z/ ¢ 2 ak i 2) sh ee = iy zJ oh Bei pak = 7 ¢3 dz) 02k og ) 0 q | Vlp. 28 oH * d20 ©) ook ? and Twa pti 2 EE EY TT a (41.10) y 0g og 1 ry 2 1 d 1 1 1 a P m= ont Zz 1 dg; | Pras (v oo 2) In § oo & 1 n Consequently 1 By +, 0 = logp———10 — 2), pez +1 210 aa) ir Py 2) Zk 39 pn AI, PTT TATE pT TT eon” by Pome Dz When these expressions are substituted in (41.6), we obtain er : 0 z/ 1+n" 082/ 82F We remark that from (41.7) it follows that aah \2 Soy \ n2 ; ri Jk 172... Hz? Heyy “Cf. Kowalewski, 1909, 2, p. 84; Fine, 1905, 1, p. 505. 1i4 III. PROJECTIVE GEOMETRY OF PATHS and from (40.7) that the value of the right-hand member is the same for each 7, it being understood that 7 is not summed. From the form of equation (41.11) it is evident that it admits a solution of the form (41.8), and thus there exists a pro- jective coordinate system z’ associated with a given coordinate system 27, for which the equations of the paths through P (xo) are of the form (41.1), p being the projective parameter for the z's. ‘Following Veblen and J. M. Thomas,* who established their existence in a different manner, we call them projective normal coordinates. It instead of starting with a coordinate system x’ we had used another general coordinate system »’‘, we should have obtained another projective normal coordinate system z‘. Then in place of (41.1) and by means of (41.1) we have tt (] or [iz dz ) 5 vires) = dp ol “Mog ap AP of — Sas or (42) my r ! dp’) (41.12) where p and p’ are projective parameters for the respective coordinates z/ and Z’', and in consequence of (38.6) we have 2) els dp’ lo : Jaf (FE Between the z's and z's we have a relation of the form : 2,7) (41.13) Lae WT ¢ (2) as follows from (41.12), where Je pT 2 r a ? n+1 Differentiating with respect to p and making use of (41.1), we have * 1925, 4, p. 205. 41. PROJECTIVE NORMAL COORDINATES 115 pl ap 29 Yr Ty = me ele ogi dp 02 Jajah By the method used in (41.10) we find for the jacobian of the transformation (41.13) | 52" | 1 / 3 = — 29 Vd Z| | ql | ies git! (v pr’ J la; |. Substituting in the preceding equation, we obtain, in conse- quence of (38.6), 1 og \ 15% We must find the solution of this equation such that [82 ; 5 7 = @j; consequently we must have (p)p = 1, 2. /r 0 ; ; (2%) = a;, where a; are constants. The unique solution g/p of this equation satisfying these conditions is ¢ (2) = 1 + a,2". Hence we have: When the coordinates x of a space undergo a general trans- Jormation, the projective mormal coirdinates at a point P associated with the a's undergo a linear fractional transformation a'zl 41.14) == ( 1+ ay 2" From (41.3) and analogous expressions in the primes and from (41.14) we have 0 rd 5 2 J 2 82° oa’ 57 02" and consequently : ; 9: I ) (41.15) : 5 = * This result has been established in a different manner by Veblen and J. M. Thomas, 1925, 4, p. 206. 8* 116 III. PROJECTIVE GEOMETRY OF PATHS Also from the law of multiplication of jacobians, from equations of the form (40.3) for transformations of the form (40.1) and from (41.13) we have 0 52" \ : J) = 92% log Lill] Ji ~~ {n-11) op. Consequently equations (41.15) and (41.16) give the significance of the constants in (41.14) and the original transformation in the z's and xs. 42. Significance of a projective change of affine connection. In consequence of equations (40.2), (40.3) and (35.3) we have for any system of projective coordinates associated with a coordinate system 2’ and with 7 (x7) for origin (42.1) THe = (Che, Bo dad (41.16) ol logis where Cj; are the coefficients of affine connection in these projective coordinates. Suppose that the latter are the pro- jective normal coordinates z’ associated with the 2's. If we introduce homogeneous coordinates, putting (42.2) ==, equations (41.14) become (42.3) = where 24) Y=af, B=0 B=1 W=a.r If we put (42.5) (CHp= —(n-+1)u;, (CHp= — (Duy, wo= p= 1, from equations of the form (35.3) in the z's and 2's we have, in consequence of (41.16) and (42.3), TT = Wp Hence the «u's are transformed contragrediently to the z's. * Here it is understood that greek indices take the values 0,1...., n and latin 1, ..., n. 43. INTEGRALS UNDER A PROJECTIVE CHANGE 117 Accordingly if at 2 we look upon the z's as homogeneous coordinates of a projective space, each choice of an affine connection singles out a hyperplane, which justifies the use of the term affine. In § 37 we saw that among all the spaces with the same projective connection there is one for which 77; = 0 in the given coordinate system x’. From (42.1) and (42.5) it is seen that at every point in the coordinate system z/ at the point associated with the #’s we have w; = 0 (j=1,..., n). Consequently in this coérdinate system the plane at infinity is Z°=0. Accordingly at each point the 7’s are homo- geneous cartesian coordinates, defined by (42.2) in which the z's are cartesian.* 43. Homogeneous first integrals under a projective change. If for a given affine connection 7° z the equations of the paths (22.4) admit the homogeneous first integral : ; dx’? da’ a ... = const., time le ds ? it follows from (32.3) that for a projective change of con- nection defined by (32.1), the equations (32.2) of the paths admit the first integral 7 2m fp, 1a : ax ax e i sn bimrme === CONSE, , S where the integral | Wr dx” is taken along the path in question. Conversely, if the equations (22.4) of the paths admit a first integral ay... a ~- = const., 43.1 J ) 1m Js das then for the affine connection defined by (32.1), in which 43.2 Wem Pl Yn y 2m’ * Cf. Veblen and J. M. Thomas, 1926, 6, p. 295. 118 III. PROJECTIVE GEOMETRY OF PATHS the corresponding equations (32.2) admit the first integral da’ dz'™ i Tr te TE Ct TE = const. When we express the condition that (43.1) be a first integral of equations (22.4), we get (43.4) ; Par, “Tom li = Ur, Tm $i) _— 0 s where 7” denotes the sum of the terms obtained from those in parenthesis by cyclic permutation of the indices (cf. § 31). Hence we have: A necessary and sufficient condition that a projective change of affine connection can be effected so that the corresponding equations of the paths shall admit a homogeneous first integral of the mth degree is the existence of a symmetric tensor ar, .. r,, and a vector ¢; such that equations (43.4) are satisfied; then the projective change is defined by (32.1) im which VW; = — ¢;/2m and the first integral is given by (43.3).* When, and only when, ¢; in (43.1) is a gradient, equation (43.1) is of the form (43.3). Hence the results of § 31 may be stated as follows: A necessary and sufficient condition that the equations of the paths for a given affine connection admit a homogeneous integral of the mth degree is that there exist a tensor ay...» and a gradient gj. such that the corresponding equations (43.4) hold; then the first integral is dx’ da i sve == eonst.T 7 1 im ds ds The condition (43.4) is satisfied by the tensor g; and the vector ¢; of a Weyl geometry, as follows from (30.1). Conse- quently the equations of the paths for the given affine con- nection admit the first integral Near det da’ 2 i == congl. Ads ds = Ot Fisenhart, 1924, 2, p. 381; also J. M. Thomas, 1926, 7, p. 119. 7 Cf. Veblen and 7. Y. Thomas, 1923, 1, p. 583. 44. SPACES WITH LINEAR FIRST INTEGRALS 119 From (43.2) it follows that for the projective change defined by (32.1) in which ¥;, — —¢;/2 the new affine connection is such that its equations of the paths admit the first integral {xt dat (43.5) gij oe HE w= const, It gi, denotes the covariant derivative with respect to the new connection, from (30.1) we have (43.6) Gis + ge i+ gn Oi —2g95 Yr. = 0. From these equations we have (43.7) aps = 2 —1) Ys, and consequently (43.6) can be written as 43.8) 2(n— VD gui + 92g Gpa, 7+ Git Ppa. — 2 Gii Gpa. 7) = 0. Conversely, if the equations of the paths of an affine connection I admit a first integral (43.5) and equations (43.8) are satisfied, for the vector vj defined by (43.7) equations (43.8) reduce to (43.6) and by means of (32.1) with ¢; = —2vy;, we get (30.1). Hence we have the following theorem due to J. M. Thomas:* A necessary and sufficient condition that an affine geometry whose paths admit a quadratic integral (43.5) have the same paths as a Weyl geometry is that equations (43.8) be satisfied, covariant differentiation being with respect to the given connection. 44. Spaces for which the equations of the paths admit » (n+ 1)/2 independent homogeneous linear first integrals. In order that the equations (22.4) of the paths admit a linear first integral dx* (44.1) ty —— ds = const., it is necessary that (§ 31) (44.2) Wj = 0. * 1926, 7, p. 122. 120 III. PROJECTIVE GEOMETRY OF PATHS Differentiating covariantly with respect to #*, we have (44.3) + a3 = 0. If from the sum of this equation and the first of the following ar, ij + Oily — 0, aj, ri + an. ji — 0 we subtract the second, the resulting equation is reducible by means of (6.4), (21.3) and (21.4) to (44.4) Ui fn == — wm Bl. The conditions of integrability of these equations are reducible to Hl 7 a5, {Bry,1 — Bhi.) 3 Dp pl 2 lk » ; oh 4 an, p (07 bij — oF; Buy; | dj By ot Bia) = If we put (44.5) Ig Rh —= == ap Ig I by, these equations and (44.4), written as Z bij, i = = Brij, constitute a system of equations of the form (8.1) in the n quantities «; and the »® quantities b;. In this case we have a system Fj; of n(n 1)/2 equations b;; + b;; = 0 which follow from (44.2). Equations (44.5) are the set F) for this case. Hence we can apply the results of §8 to get the conditions to be satisfied, in order that there be one or more first integrals.” When (44.5) are satisfied identically in virtue of (44.2), the solution admits n(n -+1)/2 arbitrary constants; it is this case we consider in what follows. From (44.5) we have the following equations of condition: (44.6) Bi —B: = 0, * Of. Veblen and T. Y. Thomas, 1923, 1, pp. 591-599. 44. SPACES WITH LINEAR FIRST INTEGRALS 121 Vi L J 7 ph Vu Pp Pp ph Jo OF Bly— oF BE, — of Bij + 0 Bf; + 0 Bia— 07 Bh (44.7) is — df Bly 0 Bl = 0. Contracting (44.7) for p and /, we obtain (ef. § D) / h 1 h 1 Vu o A ; (44.8) Bry — Sd (0; By— 0; Bj; +1 20; Bi)- Contracting this equation for % and j, we have : : 2 Bri— By =— ——— Bu. n-—1 Since gj: is the skew-symmetric part of By (§ 5), it follows from these equations that 8; = 0, and consequently Bj; is symmetric. Hence (44.8) reduce to 5 1 Aas (44.9) pk SST (0% Bi be or Bj). When these expressions are substituted in (44.7), we find that these conditions are satisfied identically. Again when they are substituted in (44.6), we obtain (44.10) Bwi—DBar = 0. Comparing these results with those of § 34, we have: A necessary and sufficient condition that the equations of the paths of a space admit homogeneous linear first integrals imwvolving mn (n—+1)/2 arbitrary constants is that the space be projectively flat and the tensor Bj be symmetric. In §34 it was seen that any such space is determined by taking (44.11) Ii = — © ¢+ 0 wy, where ; is an arbitrary gradient, and that the coérdinate system 2¢ for which the 7s have this form is cartesian in the corresponding flat space. In this coordinate system and for the 77s given by (44.11) equations (44.2) become (44.12) LEI Ty ox’ Ox Eee Tr TT 1 i 1h, ami, Sn SM St rer 122 111. PROJECTIVE GEOMETRY OF PATHS where (44.13) bi = a; Equations (44.12) are the form which (44.2) assume in a flat space referred to cartesian coordinates. In this case equations (44.4) become , 92h; 44. ree vise 4), {iy RY ? From (44.12) for i = j it follows that b; is independent of x’. Then from (44.12) and (44.14) it follows that the general solution is (44.15) bi = cj x) + di, where ¢;; and «; are constants, subject only to the condition that ¢; is skew-symmetric in the indices. Hence there are n(n -+1)/2 arbitrary constants, as desired, and for the given space a; are given by (44.13) and (44.15).* 45. Transformations of the equations of the paths. Equations (35.5) may be obtained in another manner. In § 22 it was shown that the affine parameter of a path 'is not changed by a general transformation of coordinates. Con- versely, if we take the equations of the paths in the form (22.4) in two coordinate systems 2’ and x'* and assume tbat s is unaltered by the transformation, we obtain (5.6). We wish now to consider the more general case when s is not invariant under the change of coordinates. To this end we take the equations of the paths in the form (7.6), and seek the con- ditions which the 7s and 7'’s must satisfy in order that (7.6) are transformable into J [ J) « 9 5 ; da’ ul ordre? PRET R She v x ce A hi Zi dr pear ai 45. . : 5 _ A? Bet ede? ad, dt pa A at by a change of coordinates. * Ct. Fisenhari, 1926, 9, p. 3386. 45. TRANSFORMATIONS 123 If we effect the transformation 2'° = (2%, ..., 2*) on (45.1) and express the condition that (7.6) be satisfied, we obtain (or sg. 2a? ‘) da’ da) de 9 at 0 dak TW dr di dt ? where ; a 10 3 3 « WD A= pp he set te 2) Af = nal / = 4 Dok Bd OF pat, py 4 BE Since the above conditions must be satisfied for all the paths, we must have re / a EY du” qo 07 48 BE) dak TU Tn dt TI (45.3) J8 ore ol ll 2 Ae 0 at Jl. 1 0 ol ki oat ki re oak ; on Multiplying by oo and summing for & and «, we obtain in consequence of (45.2) (41) AL == 1 2 i + 1% 2 ie rt) sa 5x'P 0 log A a 3 ie St ve tl, where A is the jacobian (45.5) tl When the expressions (45.4) are substituted in (45.3), the latter are satisfied identically. Hence the conditions are given by combining (45.2) and (45.4); this gives equations (35.5). From (22.2) and (22.3) we have ds dz a be dat dat di: da’ 5 6 . / ¢ op ie — (45.6) at Bar ds di dt 124 111. PROJECTIVE GEOMETRY OF PATHS for the determination of the affine parameter s of a path. By means of (35.5) we obtain from (45.6) and analogous equations in the 7”’s and the affine parameter s’ gr. 22 dat it dt? ln (2 i 4) da wan) TT 0% anv amr Tie dt dt 2 d log A a n+l di If the affine parameters s and s” are to be equal, we must have nh 7 Lo r x (45.8) wm = tay = in which case equations (35.5) reduce to equations analogous to (5.6); we have seen that equations (45.8) are a consequence of these. If we consider the most general solution of equations (35.5), when the coordinates are not changed but only the affine parameter, we have /j; = 11, which shows the invariant character of the /’s under a projective change. 46. Collineations in an affinely connected space. The results of §45 may be used to define transformations of points of an affinely connected manifold into points of the manifold such that paths are transformed into paths. We call such transformations collineations. The conditions to be satisfied by a space in order that it may admit one or more collineations arise from equations (35.5) on the assumption that each pair of coefficients 77; and Ij with the same indices are the same functions of 2¢ and 2" respectively. This is a particular case of the problem considered in § 36 and may be handled in that manner. However, if the finite equations of the transformation involve » (=> 1) parameters and possess the group property, they define a finite continuous group of collineations. In this case the transformations may be 46. COLLINEATIONS 125 considered as generated by 7 infinitesimal transformations.™ Accordingly we consider infinitesimal ecollineations as the basis of another method of obtaining affinely connected spaces which admit collineations. An infinitesimal transformation is defined by (46.1) 2 = du, where & are functions of the 2's and du is an infinitesimal. Since by hypothesis the 77s and /”’s with the same indices are the same functions of the »’s and z"’s respectively, the same is true of the 7's and II ""s, as defined by (35.2); hence by Taylor's expansion we have 5 IL), (46.2) nj; — Ij + Tor du, neglecting infinitesimals of the second and higher orders; this will be done in what follows. From (46.1) it follows that the determinant A of the transformation is given by 1+ : a and consequently olog A 32 5k 46.3 °= = — > Ju. 43.9) ox’ dak aa? When these values are substituted in (35.5), we obtain, on neglecting the multipler du, pg o o&" o 9 &k 01; k ps Pr Das I +o vo 1; DT +4 ¥ Br Ty dak (46.4) bl 1 5 pw 0 s¢ 02 2 5h a nd-1\" sate" path bmi] Because of (35.2) these equations are equivalent to pig ou O28 v DEE oly; OE —t Lp LTR Len SlU pk (46.5) oxox’ + I 0a + Li 0 at +3 dah J dak = 0 9; + 07 ¢;, “The reader is supposed to be conversant with the Lie theory of groups as contained in the treatise of Lie, 1893, 1 or Bianchi, 1918, 1; - a résumé of this theory is given by the author, 1926, 1, pp. 221-227. 126 III. PROJECTIVE GEOMETRY OF PATHS where by contraction we have ; 1 52 &h 7 o&k pL oS = Sh, (46.6) ¢; ay na mpd TE Kquations (46.5) may be written in the form (46.7) £i=8Bytdetdy,. Contracting for 2 and #, we have (46.8) £o=FS5%+0+ny. In order that the affine parameter s be unaltered by the infinitesimal transformation (46.1), it follows from (45.8) and (46.3) that ¢; as defined by (46.6) are zero. In this case equations (46.7) become (46.9) = FH. Hence when a set of functions & are a solution of (46.9) equations (46.1) define an infinitesimal collineation which preserves the affine properties of the space, and when they are a solution of (46.7), where ¢; 4+ 0, the collineation preserves the projective properties. Accordingly we call them infinite- simal affine and projective collineations respectively.” Consider the case of a projectively flat space and assume that the coordinates z’ are such that Hj; = 0 (cf. § 39). Under these conditions equations (46.4) may be written 02 Le (46.10) = Got Gy, dai oa The conditions of integrability of these equations are oF 0; c 09) oi hi 0¢; oe 09 E ! 2 UE i 1 ie onl hy RA Contracting for « and 7, we find that ¢; is the gradient of ; . 0 lay ; a function ¢, that is, 9; — oo Substituting in the preceding 02g dat dal * Of. Eisenhart and Knebelman, 1927, 2. equations, we have = 0 and consequently 46. COLLINEATIONS 127 o = apa +d, where the a's and J are arbitrary constants. Then from (46.10) we have Y= matteo, where the 0's and ¢’s are arbitrary constants. We recognize these expressions as defining the most general infinitesimal projective collineation in a projectively flat space.” If, on the other hand, we consider equations (46.9) for a flat space o , . mn < referred to cartesian coordinates, we have gee 0 and consequently $0 dt al #==agatl, which define the most general infinitesimal affine collineation.t Thus as defined affine and projective infinitesimal collineations are generalizations of these respective collineations of a flat space. Suppose that we have a solution & of equations (46.7) and that the coordinates 2° are chosen so that in this coordinate system (46.11) He), FP =0 {a=2... 900 In this case equations (46.4) reduce to ; oll A (46.12) = 0. By means of these equations we shall prove the theorem: When an affinely connected space admits an infinitesimal projective or affine collineation, the transformations of the Jinite group Gy generated by it are collineations. In fact, for the chosen codrdinate system the equations of the finite group are (46.13) 2! = ta =" {or == 2, o.oo Ct Lie, 1893, 1, p. 24. 701 Lie, Lc. p. 35. 31986, 1p. 293 128 III. PROJECTIVE GEOMETRY OF PATHS where a is a parameter. For this transformation equations (35.5) reduce to fj; — Hj. In consequence of (46.12) this condition is satisfied for a projective collineation. For an affine collineation (46.11) we have from (46.5) that I Zz is independent of x', so that the theorem follows in this case also. Moreover, we have shown incidentally that The most general affinely comnected manifold which admits a finite group Gy of affine collineations is given by taking Jor Ij. functions of n—1 of the coirdinates. In consequence of (39.5) and (39.6), we have that equations (46.12) are equivalent to i Hu —0 G4 iD and oly 9 ole __ : 8x ae Th $e Wel, somo=1, n;0%i) where 7 and « are not summed. By means of these equations we are in a position to choose the coefficients Ij, of an affine connection so that the manifold shall admit a group G, of projective collineations. This result is seen also from (46.5). If & is a solution of equations (46.7) for a given connected manifold, it follows from (46.4) that it defines a collineation for every manifold in projective correspondence with the given manifold. If the coefficients of any such manifold are given by (35.1), we have (46.14) Fh — Itt Ds, and consequently from (46.6) we have that the functions ¢; in this case are given by oY; oxh’ Sk he &h 215 ll at . 08 (46.15) = 9+ Yi; Da Nh. . . . « If we denote by &'7 the second covariant derivative of § with respect to the /’s, we have 47. INFINITESIMAL COLLINEATIONS : 129 of / i ja WLR tax Fr =84t+00T,+- GF ut Eu) + & (i; — Pi ¥). In consequence of (32.4) these expressions and (46.15) satisfy equations of the form (46.7). From (46.15) it is seen that the collineation determined by & is projective for the connection of coefficients 77, unless Yr; satisfies the conditions Ir. > 2, equations (47.7) and (47.9) are equivalent in consequence of (32.16), which may be written +4 WW Wi: = aT 2 (Cn i— Cnn). Hence as observed in § 8 we may apply the theorem to this case taking (47.6) as the set Fi, (47.8) as the set F, and so on. Since all of the equations are linear and homogeneous in the dependent functions, we have: A necessary and sufficient condition that an affinely connected space for m=>2 admit r (— 1) infinitesimal projective colline- ations is that there exist two positive integers N and r such that the matrices of the equations Fy, ---, Fy and Fy, -- -, Fy are of rank n®-+2n—r; when r = 1, the solution involves a quadrature; when r>>1, the general solution is a linear Junction with constant coefficients of r fundamental sets of solutions. When n = 2, the Weyl tensor vanishes identically, and consequently equations (47.6). The above theorem applies to this Jase with the understanding that equations (47.7) with Win = = 0 constitute the set Fj, and the other sets are derived from this one. 9 132 III. PROJECTIVE GEOMETRY OF PATHS From the form of equations (47.6) and (47.7) and the results of § 34 we have: The maximum number of independent infinitesimal projective collineations which a space can admit is n® + 2mn; this is the case when, and only when, the space is projectively flat. The determination of spaces admitting infinitesimal affine collineations reduces to the solution of equations (47.10) and (47.11) in which ¢; = 0. In this case we have a theorem analogous to the first of the above theorems for which the sets F, and F, are obtained from (47.1) by putting ¢, = 0 and from (47.8) by replacing Wis, by BL. Since there are n? +n functions & and 4} in this case, we have: The maximum number of independent infinitesimal affine collineations which a space can admit is n*—+n; this is the case when, and only when, the space is flat. The forms of the solutions & for projectively flat and flat spaces in cartesian coordinates have been obtained in § 46. A special type of collineations is that for which the path curves of the collineations, namely the congruence determined by &, are paths of the manifold. In this case the functions & must satisfy the conditions (cf. (7.5)) 47.12) FETE) — 0 3 In applying the existence theorem we take these conditions as the equations F|, referred to in § 8. Differentiating (47.12) and reducing by means of (46.7), we obtain a new set of conditions which together with (47.1) and (47.4) constitute the set F, of equations; and so on. Since the equations (47.12) are homogeneous and of third degree, the existence theorem assumes the more general form of § 8, and not that applying to the cases when all the equations are linear and homogeneous. If the coordinates are such that the com- ponents & are of the form (46.11), equations (47.12) reduce t0. 2% = 0 (0 = 2, .+-, un). . Combining this result’ with those of § 46, we have a means of defining the most general affine connection admitting a group ; of the type under discussion. 48. CONTINUOUS GROUPS OF COLLINEATIONS 133 48. Continuous groups of collineations. If 0 for « = 1, ..., r determine infinitesimal collineations, we call E Xe f=§ n= = r the generators of the collineations. Further- more, we Ch by (Xe, Xj) f the Poisson operator, that is, - rN ol 0 «J of il 0 J of (48.1) (Xo, Xp) J — Se) dat (¢6s rs TD) At Se) Bi . We establish the following theorem: If Xeof for « =1,.-.,r are the generators of infinitesimal collineations, so also ave (Xe, Xg) f fore, =1,... r(aF 8). Consider the case when « = 1, 8 = 2. From (48.1) it follows that rw Np am A (Xy, Xo) f = ¥ pat’ where i 208m on 0S oh oh i (48.2) & = &,, Tr fat = sm fon —<» ayn. From these expressions and (46.7) we have in consequence of the identities (21.3) and (21.4), ) Lh ld 7 & = = Jj Sen — A i Sw. + Eo Eo Bu J hy 3 tf + 3 EY von — En van) + En go) — Pj. If we differentiate these equations covariantly with respect to 2* and in the reduction make use of (46.7) and (21.5), we obtain on = g Bi, + 6 or. + 0 Pj, iy ow where ha lh is pseudo-orthogonal (§ 11) to every contravariant vector tangential to V,, at P. From (50.2) and (50.4) it is evident that the vector », is independent of the choice of the functions ¢’ in (50.2). We call it the covariant pseudo- normal to V,. *In this and the following sections greek indices take the values 1,-.-.a+1andlatinl,..., nn. 137 138 1V. THE GEOMETRY OF SUB-SPACES We define also a contravariant vector »“ by the equations oy“ EY Ty: (50.5) yp m= As thus defined »* are the components of the vector tangential to the curves of parameter #1, that is, the curves along which all the x's except z*t! are constant; we call them tramsversals of the hypersurface. Evidently »“ depends upon ~ the choice of the functions ¢’ in (50.2). From (50.4) and (50.5) we have (50.6) Ya2% = | and : Eno Ba” (50.7) TT wm If we change the curves of parameter 2”, we get a new vector of the type »“. Calling it »%, we must have o if we require that »“v», = 1. Suppose that we have a set of functions uw“ (yi... y*™1) such that ye 2 1, that is, Wy, =1. If we puto” = 9", the condition (50.6) is satisfied. Moreover, if for the func- tions ¢’ of (50.2) we take nm independent solutions of the equations : dy % da n+41 (50.9) 2 —... = Lo then in the coordinates x“ the integral curves of (50.9) are the curves of parameter 2”. Thus for a given congruence »“ not tangential to V, we can define a coordinate system x“ satisfying the requirements of this section. When equations (50.2) are solved for the y's, we have (50.10) YZ == Jo ot, -~, l) 50. COVARIANT PSEUDONORMAL. VECTOR-FIELD 2% 139 or, as expansions in powers of aH, B01) yg" = filet, , +f sv, ari. Consequently V7 is defined by the parametric equations (50.12) yt = 0, 2), and at points of V), the expressions for »“ in the x's are (50.13) Or) —— i (7, ia a7). If in (50.11) we put r= x, ali od Fi, virial) 2, then the curves of parameter z’**! are the same as those of parameter x1, but in place of (50.13) we have 2% = I tl, ry wz, Thus we see in what manner the coordinate xt! may be changed, if the vector »“ at points of V, is not to. change in direction. In order to consider the effect of a change of the vector- field »“, we consider a transformation of coordinates of the form plats x A (z), ..., 2") pH (50.14) + Wl, 277 = dr ge (wh, ppt Ll. that 1s, a transformation such that in the two coordinate systems the space V, is defined by 2" = 2" — 0, and x'' = 2" at each point of V,. From (50.5) it follows that in the 2's the components of the field »“ are (50.15) pl = 0, phil La 1 From (50.14) and (50.15) we have that the components of this field in the z’’s are 140 : IV. THE GEOMETRY OF SUB-SPACES = = 2s Ht. (50.16) sig 1-420 +1 ty It follows at once that the curves of parameter zx”! and of parameter x’ have the same directions at each point of Vn, when, and only when, (50.17) i(xly 2") = G=1%. 0) If we denote by y“ a general coordinate system, we have from (50.14) 9 y” : 9 y” a 5 Wi Sb, RR 0 yy 0 wy HIND | | ; =h gm an (or : ) eee), (50.18) % hey 0 by {ot he 2 Wg Zn +1 + .. 9am 57 rd Suntan x The result(50.17) follows also from the last of these expressions. Also we have 1 ; 7 \ ox — ox’ (4+ 0 Wy a 10 fii oy“ By’ J ord : n-+H1 + 2 hae +. 1 (50.19) m1 . n-+1 dx = 0x { ye ( gd 0 ii 0 YY 0 x SET 0 xt LL 2g antl ni From equations (50.18) and Li ) we have that at points of 1, 9 ye ne 3 yy” 3 y“ a y” ; 5 y° 3 xt rs 5 7 ’ Fart rn 3 2 Wy 5a n+l? (50.20) 5 tr 0 7° 9 LH 5 tt Kk ea 11 ee v 4 == . UY _— — oy” By” By” 12 0 y“ 34” oa 51. TRANSVERSALS WHICH ARE PATHS 141 Also from (50.18) we have that at points of V, If we write the last of equations (50.18) in the form P% = is (2p ant +...) 0x (50.22) +P (AF 2 ei ant Ly then at points of V, «2° by” io AytsB a , 1D 7 50.28) ——r == yf ol he = (50.23) 0a’ Poel 't' pa Bud | 3 51. Transversals of a hypersurface which are paths of the enveloping space. We consider the trans- formation of coordinates in the space V,.1 defined by ot = fl fla x Lah £7 ye. (51.1) — EE at BEY where f° and £i* are functions of «”, ..., #™ Ig, are the coefficients of the affine connection in the 4's in Vy, iq, 15 Brim are defined by (22.8), and the zero subscript indicates that in these functions y” have been replaced by fi. From the considerations of § 22 it follows that these expressions satisfy formally the equations ia 7 FL 0 aa 0 yy ep (i Ba J+ iy 5 a” Sane 5 Ni no . Hence when constant values are given to 2. ... wv”, g ; equations (51.1) define a path of Vy, the parameter being 2". The hypersurface 2/7" = 0 is defined by 142 IV. THE GEOMETRY OF SUB-SPACES (51.2) y“ Lr hr ('", ta) Zz"), and the tangent vector »“ at a point of this hypersurface to the path of the above set through the point has the components (51.3) (v = g1 bolle, 2%), When we effect upon the z's a transformation of the form (50.14) in which ¢? = 0, we get (50.11), in which the functions f° .and f°” are of the same form as in (51.1). Moreover, equations (51.2) and (51.3) are the same as (50.12) and (50.13) respectively. In what follows we shall use the equations in the form of § 50, when we are considering relations between the 7's and z's, and the same notation with primes when the coordinates 2’ are under consideration. ‘Thus in place of (50.2) we have BLY af = 0 0h ty, a = Bl, In particular, we remark that ¢’ = 0 is the equation of the hypersurface V,. From the last of (50.14) it is seen that go =oFG . ..,M), where F is of such. a form that F=1 when 9 = 10, Differentiating the last of the equations (51.4) with respect to 2’, we have og’ 0G (51.5) oy 2% == 1 where, in consequence of (51.1) (51.6) 7 w= fT E+ Differentiating (51.5) with respect to 2’, we have at points of Vy (61.7) ('" y'P TN = () Again differentiating (51.5) with respect to z/’, we have : 22g’ Byf go! op" 51.8 [ve | ] =, ( dy“ oyP pa? ot oy“ axle 52. TENSORS IN A HYPERSURFACE 143 Proceeding in like manner with the last of equations (50.2), we have : 2 8 : o (51.9) (ve oy a) = N lo oy” oax* /o ai : 7 dy" ye Since at points of V,, (»'°)% = (»*) and (2) m= (22) 0) [ef. (50.20)], we have from the equations (51.5) and we 20 by BY 0 SLB oy" EE ’ dy” Art BL) ’ By” Bo’? that (=) = [2 . Consequently from (51.8) and (51.9) oy“ 0 0 yy? 0 v ) we have at points of V, = 0 ps ; / \ (51.10) ve (pap— ap) = 0. From this result and (50.3) it follows that (51.1 1) *) (9,08 9’, a3) = (¥8)o £1 where, in consequence of (51.7) and (50.6) (51.12) SEA apd = f. An application of these results is made in § 56. 52. Tensors in a hypersurface derived from tensors in the enveloping space. Let &, and 2, be the components in the y's and o's of a covariant vector-field in V4; then . Dy” (52.1) bi = fuls and 0 a ays = (62.2) Jn 11 = Qa er — &a At points of Vy (2! = 0) the functions &. are expressible as functions of »',..., 2% and for each value of « the 144 IV. THE GEOMETRY OF SUB-SPACES i ay : ; quantities are the components of a covariant vector in V». Hence 4; given by (52.1) are the components of a co- variant vector in V,,, which we say is derived from the given vector in Vy41. In particular, the vector ¢»., where ¢ is an arbitrary function of the 7's, is the most general covariant vector whose derived vector in V, is a zero vector. Also we have at once that all vectors of the pencil & + ov. have the same derived vector in V, whatever be go. In like manner, if aq ..., are the components in the y's of a tensor in V,4;, then o a, (52.3) = man - Se 1 r 1 7 ox! ow” evaluated at points of V,, are the components of a tensor in V,, derived from the given tensor in V,::. From (52.1) and (52.3) it is evident that the tensor in V, derived from a covariant tensor in the enveloping V,:: is independent of the choice of the vector »“ This is readily seen also by observing that the quantities b; ...; possess tensor character under transformations of the form (50.14) whatever be the functions vi, as follows from (50.20). The same is true for the general transformations = ) 7 + + =1= +i (52.4) ale ¢ (", ny Pp % 1 i ns 1 at 77 oC el nl 3 ; nn : +1 where F and its first derivatives are finite for 2’ = 0. Let &* and 4% be the components in the y's and x's of a contravariant vector-field in V1; then we have is Bat (52.5) 4 = SC = \ oy” and (52.6) Ant Ee sv Vero At points of V,, the functions & are expressible as functions Ss ox’ of z!, .-., 2” and for each value of e the quaniiijes —— ) ) oy” 52. TENSORS IN A HYPERSURFACE 145 for 2 = 1, ..., n are components of a contravariant vector in V, under transformations of the form (52.7) qt = Yl (2, us 2"), atl — n't: We say, that 2’ defined by (52.5) is the vector in V,, derived from the given vector ¥* in V,.i,. A contravariant vector in V, is necessarily one in V,.;. As a vector in Vy, the vector defined in V, by (52.5) has the components 2 ; oat 1 52.8 lem PL JNO (52.8) Joys 0 Accordingly it follows from (52.6) that when a contravariant vector in V,1, is tangential to V,, it is identical with the derived vector in V,. If it is not tangential to V, and we denote by &“ the components in the y's of the vector (52.8), we have ) = AB oy" = = oy” ois zp 02° DY” oe & Bg, 52.9) &¢ = J & £52. 4 af 0 af iy where (52.10) Bs = 05—»" vg. Following Schouten® we say that the derived vector of a contravariant vector is the tangential component in V, of the given vector. From the form of (52.9) and (52.10) it follows that unless the given vector is tangential to V, its tangential component depends upon the choice of the vector »“. ; > Ox , This may be seen also from (52.5). For, 2 being the 97% ol cofactor of . in the determinant A = | . divided by 4, ps : it evidently depends upon »¢ When in (52.5) we repace &* by »%, we find that the derived vector is a zero vector, and ¢»“ is the only vector possessing this property. Consequently, when a vector »* has been chosen, in the system so defined its tangential component *1994, 1, p. 134. 10 146 IV. THE GEOMETRY OF SUB-SPACES being zero, this vector is analogous to the normal vector to a hypersurface in a Riemannian space in the sense that the tangential component is zero. This fact has led certain writers to refer to »“ as a pseudonormal.* We do not use this term for a general choice of »“ because in a Riemannian geometry it would be confusing (ef. § 56). In like manner, if «“ "“ are the components in the y's of a tensor in V,.1, the quantities (52.11) Pr = gh £1 al Oy oy evaluated at points of 1}, are components of a tensor nv; under transformations of the type (52.7), that is, general transformations in 13, but which in 7,41 do not change the vector »“. We call this tensor the derived tensor in V,. The tensor in V,i; with the components ¥""* and °° " = 0, when one or more of the ¢’s is n+ 1, is called by Schouten the tangential component with respect to V,. If a™ “ are its components in the y's, we have (52.12) Be pn LL iT, where the B’s are defined as in (52.10). Since a covariant vector in V,, is equivalent to the bundle of contravariant vectors in 1, pseudodrthogonal to it (§ 11), it is evident that a covariant vector in V, is not one in an enveloping V,4+1. If A; are the components in the z's of a covariant vector in 1.1, the vector of components (52.13) em ly = is equivalent to the bundle of contravariant vectors deter- mined by »* and the contravariant vectors in V, pseudo- orthogonal to the derived vector of 4, in V,. In this sense the derived vector is the tangential component of the vector * Cf. Weyl, 1922, 6, p. 154 and Schouten, 1924, 1, p. 134. 52. TENSORS IN A HYPERSURFACE 147 (32. 13). If & are the components of the given vector in the y's, the components of (52.13) in the y's are Fe = 20 BE In general, the derived tensor in V, of a tensor Ber, .cor, IN Virb is the tangential component of the tensor (52.14) Biri, = ags,.. Br aon, . LBD, i an asa : 5 . Again if a3’ ave the components in the y's of a mixed 1 is tensor in Vj, .;, the quantities : a pa pa’ py oy (52.15) be Pi ont En : 2 i i Jy Js 1 Ps oy! By” oxi os evaluated at points of 1, are the components of a tensor under general transformations (52.7) and we call it the tensor derived from the given tensor in V,;. This derived tensor is the tangential component of the tensor whose components in the y's are ORI ELT. a Bl oy fs We call each of the tensors defined by (52.12), (52.14) and (52.16) the associate of the given tensor with respect to 17. From (52.10), (50.3), (50.6) and (50.7) we have 3 oat d 3 dant ne 2 BLT 0, ; oy’ oy" oyP (52.17) n.,8 3 2 By* no oF bf Bx’ PF gantt Because of these identities we have from (52.15) (52.18) = ag vk dy 0, Bxh iy 0 Ps el = By Bn 10* 148 IV. THE GEOMETRY OF SUB-SPACES Similar results follow from (52.3) and (52.11). Hence we have: The derived tensors in Vy, of a given tensor im V,i1 and of ils associate with respect to V, are equivalent. 53. Symmetric connectioninducedin a hypersurface. If I'fy and I's, are the coefficients of a symmetric connection in a V,41 in coordinates z“ and y” respectively, we have ~ from equations of the form (5.6) Rye « OYB 2) Bal ns Th = ( dat +7 xl oy“ Pr owl oak At points of V, (°F == 0) the values of I . depend upon the choice of the vector »* as is evident from (53.1). However if I's, are the coefficients for V1 in the zs defined by (52.7), from equations analogous to (53.1) we obtain aw : yt @ 8x”? La i Sy : 2 Dud dal 0k! px BY I = ; Consequently rs and I, 1 are the coefficients of the same connection in Vy; we call it the connection induced in V, by ‘that in V,1 for a gwen choice of the vector v“. This qualifi- cation is seen to be necessary from (53.2); for, in case of transformations of the type (52.4) there are in the last member wil bu” ox’? oat of (53.2) the added terms Ip, = et, ood Bak 2” For an asymmetric connection in V,.;, equations (53.1) hold and the skew-symmetric part of the induced connection is given by : ; —n Dyl By. Bat ma hm ~ |! that is, the tensor 2 of the induced connection is the derived tensor of the tensor 25 of the connection in V1. Consider, in particular, the case when the affine connection in V,4, is that of a Riemannian space, determined by the *In § 56 we obtain the relations between the coefficients of two induced connections for different choices of »%. 53. INDUCED CONNECTION 149 fundamental tensor of components wes and gop in the y's and z's respectively. Then we have 0 y” 0 YP (53.4) ’ Ue Ba J Pa = Uk From these equations we have* at points of V, (z*T! = 0) ’ 2 Yeu] {2 oy 2 (63.5) dap 2 ol Sx iil, vl, 0a dad where = are formed with respect to the fundamental form ep In Vypyr and / 2 i with respect to the fundamental form g;; in V,. From ns of the form (5.6) we have le a rial pn A SE) — ERT re — 1 i = oat ox’ lun, dat Bal Y dal J gant Substituting in the preceding equations, we obtain , h DAL (13— y | J ga + grata : if = 0, tg Al In order that I, " == J If , 1t is necessary and sufficient that g Jin+1 = 0 or I; 7 — 0. In the former case the curves of parameter x"! it y, orthogonally. In the latter case we have ye 2) By” | Lal By” By Data Tl, dat luvl, dxf xs from which it follows that V,, is a totally-geodesic hyper- surface of V,1.i Hence we have: A necessary and sufficient condition that the coefficients of the induced connection in a hypersurface of a space with a Riemannian connection be the Christoffel symbols of the second kind formed with respect to the derived fundamental BOL, 1926.1, p. 127. T Cf, 1926, 1, equations (43.4) p. 147 and (54.1) p. 188. 150 IV. THE GEOMETRY OF SUB-SPACES tensor us that the vector v“ be orthogonal to the hypersurface, or that the latter be totally-geodesic. We indicate by one or more subscripts preceded by a semi- colon covariant differentiation with respect to the induced connection. From the definition (§ 52) of the associate of a given tensor in V1 with respect to V, it follows that the components in the z's of the associate tensor for which one or more of the indices are n--1 are zero. Consequently from the general law of components of a tensor in two coordinate systems we have from (52.18) BGT =a 2 i pr Ce 2 : oy Py” ox dx’ 0x Hence we have: The first covariant derivative in a hypersurface of the derived tensor of a given tensor im the enveloping space Vy. 1 is the derived tensor of the covariant derivative in Vyui1 of the associate of the given temsor with respect to the hypersurface. It should be remarked that although the derived tensor of a covariant tensor is independent of the choice of »% its covariant derivative in the hypersurface as a derived tensor does depend upon »“.* Equations (53.6) do not hold for derivatives of higher order. However, the second covariant derivative of br 5 is the derived tensor of (ag 7 Jo By sontinang his process we obtain derivatives of any ordliy (cl. § 54), 54. Fundamental derived tensors in a hypersurface. We denote by ;; the tensor in Vj, derived from the tensor Ve, In Via, that is, : by” bys (64.1) Wj = Va, 8 rT From the form of these expressions it is evident that this tensor is independent of the choice of »“ In the x's the components of the vector ». are * Cf. Schouten, 1924, 1, p. 137. 54. FUNDAMENTAL DERIVED TENSORS 151 (54.2) Yi — 0, Yotil — 1, as follows from (50.4). Consequently in the 2's equations (54.1) are Wn-1 (54.3) ny) == J . Evidently ow; is symmetric. For a given choice of »“ we have the tensor » 3 and the vector »“ gv... We denote by /; and /; the components of their derived tensors in V,, that is, ve 02 8y8 (54.4) Lom Pay ir and ; , 3 3 . 9 (54.5) L= SH Se In the z's the components of the vector »® are given by (50.15) and consequently in the x's we have (54.6) = Ip, and (54.7) == rotl, Because of (50.6) equations (54.5) can be written as RA Q XY oy? (54.8) li = —Va,g le With the aid of the tensor w; we are able to express the covariant derivative of a derived tensor in terms of the derived tensor of the covariant derivative of the given tensor and other derived tensors. Thus if b;; are the components of the derived tensor of as, we have from (53.6) a opi. 0 By By bine lop Bp D3he 5p Sir BT With the aid of (52.17) we obtain iy yt By pal Bal Bak 01 p RT 0p (re 3 y wi + VP oi) : bij kT dup, a 152 1V. THE GEOMETRY OF SUB-SPACES 55. The generalized equations of Gauss and Codazzi. If I's, and I's, are the coefficients of an affine connection in a Vu: in coordinates xz” and y” respectively, we have from (5.6) v y oyr a Oy” re 5 a1) ou aa +75 rind de pr d At points of the V,, (z* = 0) these equations can be written, in consequence of (54.3), in the form : oy oyr (65.2) y" Li Is; 52° 3x) — =p; 2 the first term being the second covariant derivative of y“ with respect to the induced connection in Vi. When in equations (55.1) we replace ¢ by n 1-1, we have 2 Se pa Ferd I == Paty gab At points of V; these equations become, in consequence of (54.6) and (54.7), pli ie 3 oyr 01 (55.3) a + Igo a =U v ty, We desire to find the conditions which the tensors oj, I and /, must satisfy in order that equations (55.2) and (55.3) To comsstons, The conditions of integrability of (55.2) are of the form [ef. (6.4)] = % t : oy“ (55.4) y —y a = 5a 4 Bs where Bj is the curvature tensor of V, formed with respect to Th: When the expressions (55.2) are substituted in (55.4), the resulting equations are reducible by means of (55.2) and (55.3) to oy“ am Te oyf oyr 31° oy » in 2 (65.5) 9a™ Ta = Bir 3 pai Bax Bak | nlf =o Th) 29% (0ij;— Wi; j +} Wj l,— og; 4, 55. EQUATIONS OF GAUSS AND CODAZZI 153 where Bs is the curvature tensor of V1 in the y's evaluated : Sn dal at points of V,. If these equations be multiplied by oy and by ». and e be summed, we have respectively, in con- sequence of (50.3), (50.6) and (50.7), 3 ; J ll oy ; ze BYP Byi By Ba / : (58.6) Bil = Bp EI lj — wij li, el =u. By Buf Bf 55.7) @f—— Oil j — Biya a Ve + @i — uy ly. The conditions of integrability of (55.3) are [ef. (6.3)] »u—v' n= 0. Substituting from (55.3) in these equations and proceeding as above, we obtain sur 01° Hx" 8 ay Y x cr D J i lu li He ae (55.8) lj = li + 4 lj sf 4 + Bgyo V Bak Bl Py” == 0, J G5 Li —L 0 opp I op; + Bs Ver’ eo Equations (55.2) and (55.3) are generalizations of well-known ‘equations in the theory of surfaces in euclidean space,* and of equations in a general Riemannian space, ; being the components of the second fundamental tensor and »“ the unit vector normal to V,. In these cases the processes followed above lead to the Gauss and Codazzi equations. Equations (55.7) and (55.8) are equivalent in these cases and (55.9) are satisfied identically. Accordingly we call (55.6) the generalized equations of Gauss and (55.7) and (55.8) the generalized equations of Codazzi.: *1909, 1, p. 154. +1926, 1, pp. 147, 148. 1 Cf. Schouten, 1924, 1, p. 140. 154 IV. THE GEOMETRY OF SUB-SPACES From the equations of Gauss it is seen that the curvature tensor of the induced connection is not ordinarily the derived tensor of the curvature tensor of the enveloping space. When equations (55.6) are contracted for / and 7, we obtain O50) Thm Tp mr, IL (55.10) Dijk = Dayo ar Ya i] Bop) Po + wg | hy lk. By means of (55.9) and the theorem of § 5 these equations are equivalent to eae 2 0h Bh bad oak TU pak dat’ (55.11) Bj = : Byf: 2 When the connection is asymmetric, the results of the preceding sections are essentially unaltered and the corre- sponding equations are obtained on replacing Ii: by Tins then the tensor w; is not symmetric in 7 and jy. The generalized equations of Gauss for this case are obtained from (55.6) on replacing Bj. and Bg. by Ll and L§,5. In the right-hand member of (55.7) there is the added term —?2 2, wy; and in the Betts hand members of (55.8) and (55.9) the added terms 2 Of; & and 217, o respectively. 56. Contravariant pseudonormal. We consider the effect upon the coefficients of the induced linear conection and upon the derived tensors // and /; of a change in the vectors »“ at points of Vj, and to this end make use of the transformation (50.14). For the coordinate system z’* the coefficients of the induced connection in V7, are given by a yr ww Oyf ls ba 8 Weoley (56.1) Jh 52" 85" Pr or rd n 0 uy’ In consequence of (50.20) and (50.21), it follows from the above equations, (53.1) and (54.3) that at points of V7 (56.2) I= 15— on. IH Bij denote the components of the curvature tensor in 1, we have 56. CONTRAVARIANT PSEUDONORMAL 155 J J ] 1 Bij = Bij + Yi [wij — oi; j + Yh (04 0; — 05; og] (56.3) ) ) + ®;j Yh kT Mg (12 J From (50.22) and (50.23) we have at points of 1, ELT TY Bo pd Fad dal Dad rag [By rs oy 2 ; he Te gr oat ouxt 1 « oy? ’ == py 8 7 —_— Y's .j a i Uh P* : In consequence of these equations, (50.20) and the definition (54.4) of the tensor /; we have (56.5) Uj = G—it + i+ ong 0h). Since ve == », at points of V,, as follows from (50.20), we have in like manner from (54.5) (56.6) Lo=="l;+ wy of. It is readily found that equations (55.6), (55.7), (55.8) and (55.9) and similar ones for the induced commection 77 are consistent in view of the above relations. As an immediate consequence of (56.6) we have When the determinant | w;;| is mot zero, a wvector-field v* at points of Vy is uniquely determined with respect to which the vector I; is zero™ Also we have: When the determinant | wy; is of rank n—r (r = 0), a vector- Jield v* at points of Vi, with respect to which the vector 1; is zero cannot be obtained unless the matrix l 11 Sais "1p [1 li | | ; I Wy sje ®D np In | * This theorem is due to Schouten, 1924, 1, p. 143. 156 IV. THE GEOMETRY OF SUB-SPACES is of rank n—r; in this case the determimation of v¥ involves arbitrary functions. It is seen from (54.8) that the vector /; vanishes at all points of V,, when, and only when, (56.7) eve, = frp at points of Vj, where f is a function of the 3's which may be zero. These equations are, in consequence of (50.4) (56.8) YE up = TO, from which it follows that Yor on = f. If the determinant |¢,.g| is of rank »--1, by a suitable choice of f a unique vector p¢ is given by (56.8) so that 0g “ = 1. apply the method of § 51 to obtain a coordinate system z'“ so that the curves of parameter 2” are paths, whose tangents at points of V, have the direction of »% it follows from (51.11) and (51.12) that »* ¢/¢3 = 0. This is a general- ization of the situation in a Riemannian space when a family of hypersurfaces ¥ = const. are geodesically parallel, the function ¥ being chosen so that® id Yq Ys = 1 : From this we have gf ¢ « ¢ 5, = 0. Hence the vector »“ in this case is the normal to the hypersurface. Accordingly when there is a field »“ satisfying the conditions of the first theorem, we say that the vectors »“ are the contravariant pseudonormals to the hypersurface. When for a given choice of the field »“ the vector /; is a gradient, and we put gloge blog 6h oy~ l; — = TT ——— eee 0x byt Ont’ If we retain this field at points of V,, and * 1926, 1, p. 57. 56. CONTRAVARIANT PSEUDONORMAL 157 we have from (54.5) at points of Vj, nisl (Ov ).8 Ve Dat — 0. Hence, if in accordance with the results of § 50 we take the field #»* at points of 17, the corresponding vector /; is zero. Accordingly we have: When for a given choice of the field v¥, the vector I; is a gradient, there exists a function 6 such that for the field 6 »* the vector l; us zero. Thus the case where /; is zero is equivalent to that where it is a gradient so far as the direction of the field »* goes. YH #{ in (56.6) are chosen so that 4 == 0, from (53.11) we have ol; Dak SIR a of dad re gal Bak’ 1 Bij = Bip— Consequently if Lo — 0, we have Bj — 0. Conversely, if By = Bug = 0, it follows from (55.11) that % is a gradient. Hence we have: When for an qffinely connected space Bo = 0, a necessary and sufficient condition that the directions of a vector-field v* at points of a hypersurface be such that I; be a zero vector is that the corresponding tensor Bj be zero.* When equations (55.3) are written in the form i ga Ae F hl the quantities on the left are the components of the associate direction of the vector »“ for a displacement in the direction of a curve of parameter x/ in V,,. Hence we have: When a hypersurface admits a contravariant pseudonormal, the associate direction of this normal in the enveloping space Jor any displacement in the hypersurface is tangential to the hypersurface. * This theorem has been established by Schouten, 1924, 1, p. 142. 158 IV. THE GEOMETRY OF SUB-SPACES This property of the contravariant psendonormal is possessed also by the normal to a hypersurface in a Riemannian space, which justifies further the term pseudonormal. 57. Fundamental equations when the determinant w is not zero. In this section we consider the case when the determinant o is not zero at all points of V,, and we under- stand that the unique vector »“ has been chosen for which li = 0. In this case a tensor ¢¥ is defined by the equations (57.1) Poy = L. We assume that the determinant (57.2) y == [4] is not zero.v Then it follows from (57.1) that the determinant ¢¥ is not zero, and a tensor ¢;; is uniquely defined by the equations sh Ji gl = JF, 97 ar == or. Krom these equations and (57.1) we have sr. py k (57.4) oi; = gi lj. If we put dat dx’ nw ee ee TE (57. ) ep bij 51" dyP | Ve V3, it is evident from the form of these expressions that as are the components in the #’s of a tensor in V, 1. From (57.5) we have, in consequence of (50.4), (50.6) and (50.7), ia dy“ dy (57.6) Che he Yi irr cr 11 ; 01 g (57.7) Hog — or VP = ang 1" ak = * 1926, 1, pn. 148. T This is an additional assumption. For it follows from (56.5) that for a transformation (50.14) preserving » (i. e., 1 = 0) I; are unaltered. 57. FUNDAMENTAL EQUATIONS 159 and 57.8) Hep pt Vd zm 1 . From these equations it is seen that for the determinants (51.9) a = | tte ; fg = | vi] we have I By” 2 57.10 a | =| == gg, ( ) | 9af | g Consequently « 4 Oin the case under consideration. Accordingly a tensor a“? is defined by 51.11) Me a = or, a? ape = or. As a consequence of (57.7), (57.8), (50.3) and (50.6) we have {p7.12) deg V¢ = ge Vv" = vg. Moreover from (57.11) and (57.12) we have (57.13) ; Pt = pg aft = yz, These equations are a generalization of the equations of a Riemannian space connecting the contravariant and covariant components of the normal to V/,. If equations (55.6) are multiplied by gs, and summed for 7, we obtain, in consequence of (57.12), by oy (57.14) Bri m= Bega 3" Lita ok + oy 0g — oy; 0, where we have put wig oo : Hic (51.15) By: = "wy, BY, Begs = fos Bj.s. In the present case equations (55.7) become 5 = oyh 0 y° ‘ Sari i 2 Or “dL SZ ge (57.16) ij; 0p) — Boss y 828 ae ak 160 IV. THE GEOMETRY OF SUB-SPACES If we substitute in these equations the expressions (57.4) for w;;, we have that equations (55.8) are equivalent to oyr 01 byh p27) Yir;k Ij — Yir;j i= Ea Bis Le 0 fr. 2 Ve = ly JirV By” (Bb7.17) ’ : = y 83 oy” ot x = Bes 5; ” fo y rh 1 the last expression being a consequence of (57.6), (57.7) and (57.13). To these equations must be added (55.9) which reduce to nd 5 hi ol) at Ye) tg) o - -— (B1.18) 7 (op; wg — py; 00g) Bupa? vB Gg BE 0. In a Riemannian space V1 for which aes is the fundamental tensor, it and ¢; are symmetric. Also ¢;.x = 0 and Boss is skew-symmetric in « and B. Consequently in this case equations (b7.17) and (57.18) are satisfied identically and (b7.14) and (57.16) assume the form of the Gauss and Codazzi equations.™ 58. Parallelism and associate directions in a hyper- surface. Let A’ be the components in the 2's of a field of vectors in V,, and &“ the components in the 7's of this vector- field in V,;. Then we have oa Al oy” (58.1) m2 A Teg If we differentiate these equations with respect to z/, we have, in consequence of (55.2), o oyP i Oy ae al Bd J opat — thy ligy2, (58.2) gE“ At points of a curve Cin V,, whose coordinates are expressed in terms of a parameter {, we have dyf i dd By” oda gr = Ly wy nr *1926, 1, p. 149. (08.8) §& 1 58. PARALLELISM AND ASSOCIATE DIRECTIONS 161 These equations may be written in the form yy . oy” d (58.4) NC = wt GR = Al i” where 2¢ and pu’ are respectively associate directions (§ 16) of the vector in V,1 and V, with respect to C. If the vectors are parallel in V,,; with respect to C, 9¢ = f(t) & (cf. § 7) and from (58.4) and (58.1) it follows that da’ (58.5) W=f0r aii=0 Hence we have: When a family of contravariant vectors in a hypersurface are parallel in the enveloping space with respect to a curve, they are parallel in the hypersurface with respect to the curve, Jor every choice of the vector v“. From (58.4) and (58.5) it follows that a necessary and sufficient condition that a family of vectors in V, at points of a curve C be parallel in V, with respect to C, when they are not parallel in V,.q, is that there exist a function f(#) such that of (58.6) 7% aA) SC — B= = ny Li pe Conversely, if a vector field A’ is such that mw; A’ Li ==) and (58.6) hold along a curve, it follows from (58.1), (58.3), (8.4) and (58.6) that [7.5 a S(@) yor oo = 0. ae] Since the rank of the matrix [= 7] the vectors of the field are parallel in V, with respect to the curve. Two hypersurfaces V, and V, are said to be tangent along a curve (, if they have the same covariant pseudonormal (§ 50) at points of (. Since the components of the pseudo- normal are determined only to within a factor, there is no is n, we have that 31 162 IV. THE GEOMETRY OF SUB-SPACES loss of generality in assuming that eo; and »“ are the same for V, and V, at points of C. Hence from the above results we have: When two hypersurfaces are tangent along a curve, contra- variant vectors parallel in one with respect to the curve are parallel in the other.* 59. Curvature of a curve in a hypersurface. When in equations (58.3) we take for & the tangent vector to C, that is, 2° = y , these equations become rr Fe 4 YP dyr shi) ar Tran (p01, — (= +7 dod dof | a do? dad, : dt dt | dat Wodt. df From these equations and (7.6) we have: When a path of a space lies in a hypersurface, it is a path of the latter, for an arbitrary choice of v*; and it is a curve Jor which (59.2) wy dot dud = 0, This is a corollary of the first theorem of § 58. If C is not a path, we choose for the parameter an affine parameter s of the path tangent to C' at a point P. Then at P we have (59.3) go = Te where 2“ are the components of the first curvature vector of at P for Va+1 (§ 21), : i dxt dad so > 5 ds ds and oy“ Px? i dxd dak A Te . helio ° or = uf, (59.5) 4 vy ds? + Li ds ds gr OF. 1926, 1, p.- 75. 60. ASYMPTOTIC LINES 163 The vector x’ is in the pencil determined by the tangent to C and its first curvature vector in V,. If it is tangent to C, then C is a path in V, and »® is a vector of the pencil determined by this tangent and the first curvature vector in Vat1. When »“ is the contravariant pseudonormal to V, (§ 56), we call 1/R the normal curvature of the curve, and 4“ the relative curvature vector of the curve in Vn. If aes 99° and aeg 7% qf are not zero (§ 57), we put (596) laren? = 5, lawyer] = 0 9 and call 1/¢ the first curvature of C in Vy, and 1/9, the relative curvature of Cin V,, as in the case of a Riemannian space.* 60. Asymptotic lines, conjugate directions and lines of curvature of a hypersurface. The associate covariant vector in a space V,.1 (§ 16) of the pseudonormal », of a V, with respect to a curve C of V, is given by Va, A / =, Wer From these equations and (54.1) we have: A necessary and sufficient condition that the associate co- variant vector of the covariant pseudonormal to a hypersurface with respect to a curve of the latter be pseudoirthogonal to the curve at a point is that the direction of the curve satisfy the condition (60.1) wij dat dx’ =, As this is a property of asymptotic directions of a hyper- surface of a Riemannian space,t we call the directions defined by (60.1) the asymptotic directions. A curve whose direction at every point is asymptotic we call an asymptotic line. We * (Of 1926. 1, p. 131, 71926, 1, p. 187. 11% 164 1V. THE GEOMETRY OF SUB-SPACES note that asymptotic lines and asymptotic directions are independent of the choice of the vector »“.* From (59.1) and (59.3) we have the theorems: When an asymptotic line is a path of the hypersurface, it is a path of the enveloping space and conversely. The first curvature vector, in a space, of a curve in a hyper- surface is tangential to the hypersurface at a point, when, and only when, the direction of the curve at the point is asymptotic. If C is a path of V, and s is an affine parameter in V, for the path, equations (59.1) become Cy" | yo dyf dy’ ds? 3: ds Hence we have: A path of a hypersurface in an asymptotic direction at a point has contact of the second or higher order at the point with the path of the enveloping space through the point in this direction. As in Riemannian geometry, we say that two directions at a point of a hypersurface are conjugate, if dat da _ — Wj; = p% de ds 5 og dt 8x = 0. Thus asymptotic directions are self-conjugate. From § 58 we have: In order that a family of vectors at points of a curve of a hypersurface be parallel both with respect to the hypersurface and the enveloping space, it is necessary that the direction of the vectors be conjugate to the curve. If, whenever possible, the vector »“ is chosen in such a manner that the vector /; is zero, we have along any curve C, in consequence of (55.3) . . dyP = : 0° da’ (na ew = Saw The left-hand member of this equation is the associate direction in V,41 of the vector »* with respect to the curve. In order * Cf. Schouten, 1924, 1, p. 148. 60. CONJUGATE DIRECTIONS 165 that this direction may coincide with the tangent to the curve, we must have : Lola, : Since the rank of the matrix or in n, these equations are equivalent to ; Lda’ (60.3) (elj — 9j) or Conversely for each root of the determinant equation = 0. (60.4) loli — 8 = 0 a direction is determined for which the associate direction of »* in V4 coincides with this direction. When in particular the conditions of § 56 are satisfied, »“ is the contravariant pseudonormal and the curves of V, defined by (60.3) are ~ an evident generalization of the lines of curvature in a Riemannian space. Accordingly we call the curves defined by (60.3) the lines of curvature of Vy. : If the roots of (60.4) are real and distinct, there are » uniquely determined families of real lines of curvature. If o is a real root of order r, it is possible to find » linearly independent families of lines of curvature corresponding to this root; moreover, any family of curves linearly dependent upon these families also satisfies (60.3). Each choice of a vector »* determines a tensor l and con- sequently leads to equations of the form (60.3). However, if /; 4 0, the associate direction of the vector »“ satisfies the i ; dr above condition only in case /; 5 (55.3). We reserve the term lines of curvature for the case when J; = 0.7 If equations (60.3) are multiplied by gn; (ef. § 57) and summed for /, we have from (57.4) #1926. 1, p. 157. T Cf. Schouten, 1924, 1, p. 148. = 0, as follows from 166 1V. THE GEOMETRY OF SUB-SPACES dx’ (60.5) (or; e— gui) = 0. If as in § b9 we use for parameter an affine parameter s of the path tangent to a line of curvature at a point and dda : : put gj. a we ==. then o = Ro, A discussion of equations (60.5) can be made similar to that of the corre- sponding equations for a hypersurface of a Riemannian space.* In particular, if A] and 43 are the directions defined by (60.5) for two distinet roots of (60.4), we have gid = 0 wy 2a = 0. From the second of these equations it follows that the two directions are conjugate. 61. Projectively flat spaces for which B; is sym- metric. Consider for a space V, with a symmetric con- nection the system of equations (61.1) 8, ij = a;; 8. The conditions of integrability (6.4) of these equations are reducible by means (61.1) to n j 0.1 (Bai + OF au — Or aij) — 0 (aij. — ax) = 0. In order that equations (61.1) be completely integrable, and consequently that the general solution admit 2-1 arbitrary constants, it is necessary that Ju hy h Bir+-0f op— 0k ayy = 0, aij, — i,j = 0. Contracting the first for 2 and k, we have iL (61.2) Ai — etl Bij, so that the above conditions are Nb 1 , op 3 : (61.3) Bi + ai (0) Bi, — 0; Bp =40, By,—Bu;=0, * Of. 1926, 1, p. 153. 61. PROJECTIVELY FLAT SPACES 167 Hence V,, is a projectively flat space (§ 34) and Bj; is symmetric as follows from (61.1), (61.2) and (6.3). Since each solution of (61.1) is determined by initial values 00 ; : of 6 and To there exist #1 solutions #%(z!, .--, 2®) for which the determinant | 89 26" y Ppt 0H dan (61.4) A =] ny | 00" h agri! gril 3x! Ba? is not identically zero and the matrix of the first » columns is of rank n. Hence the jacobian of the equations (61.5) yt = get, ---, a we=11,....n-1-1) Y is not zero, and these equations define a transformation of coordinates in a space V,+1. We define a connection for this Va+1 in the coordinates 2%, by taking for 7%(i, j,k =1,--., n) the expressions for these functions for the given V,, and in addition of 1 ; (61.6) I" t= Ty Lrenp=05, (¢,8=1,-,n+1). If I';, denote the coefficients of the gonncetion in the 7%, it follows from the equations By AE 2 pp PY dab oar “dxB dxr Br 5pm (2, B, 7, 0, v = 1,:+»:;n+1) and from (61.1), (61.2) and (61.5) that =r oy4 oy” M oxB dar = 0. Consequently rr = 0 and V,41 is a euclidean, or flat space. From the definition of the affine connection in V1; it follows 168 IV. THE GEOMETRY OF SUB-SPACES that the induced connection in the hypersurface 21 = 0 is that of the given space V,. Hence we have A projectively flat space of wn dimensions for which Bj; is symmetric can be immersed in a flat space of n-+1 dimensions.” In order to investigate the situation more fully, we observe that by suitable linear combinations, with real coefficients, of the #’s the fundamental form of V,.; is reducible to Dee(dy®)®, where the ¢’s are plus or minus one. If we o denote by aes the coefficients of this form in the »’s, we have n-+1 0 Gg“ 0 go 911 YY 3 0 oY thi) = or nr a gaat in+t1 =— € i” So pe - ane (61 7) «o 0x gr « ox 2! v dntintl == 7 2 vel A%)> 1f we put (61.8) Dea (892 = 24, « the successive covariant derivatives of this equation are reducible by means of (61.1) and (61.2) to @ a0 5 = al 0 = Lis « = B+ 2 ee 6:0"; = 24. n—1 « 2 From the results of § 53 it follows that the coefficients of the induced connection in the hypersurface +1 —= 0 are the Christoffel symbols of the second kind formed with respect to gr = Set 0%, o only in ease wy = 0, that is, when 4 is a constant. In this case we have 1 —n By == rr 1s and consequently the hypersurface is of constant Riemannian curvature.v * Eisenhart, 1926, 9, p. 338. T1926, 1, pp. 135, 208. 61. PROJECTIVELY FLAT SPACES 169 Consider now any hypersurface of a flat space V1 for which the coordinates y“ are cartesian and the fundamental form is > ee (dy®)®, the equation of the hypersurface being (61.10) FQ, ..., y+ = 0. This equation may be replaced by (61.11) ye = 0% (2, ..., an) where the functions 6¢ are arbitrary, except that (61.10) is satisfied and the jacobian of any n of them is not zero. When, and only when, the function /' is not homogeneous in the ¢’s, the determinant A defined by (61.4) is different from zero; that is, when the hypersurface is not a hypercone with vertex at the origin, or a hyperplane through the origin. Consequently, with these exceptions, the functions 6“ can be chosen so that equations (61.5) define a transformation of coordinates in V,1 such that »*t! = 0 is the equation of the given hypersurface. The coefficients of the induced con- nection in the given hypersurface are given by [2] lj wl. where the latter are defined with respect to the a's given by (61.7). Since the Christoffel symbols formed with respect to the y's are zero, we have (61.12) Th == fa) sty or BL \871.= af bar 5,0 From these equations and (61.12) we have ay 7 oy* 1 In-+1| 2/0 iy loll = ) le un [ a which reduce to the form (61.1) because of (61.5). Con- sequently By; Le (n — ON I ¥ i a 170 IV. THE GEOMETRY OF SUB-SPACES and for the induced connection the hypersurface is project- ively flat. Also from (61.5) it follows that oy” ophvtl =u by means of which and (61.14) we get the equations (61.6). Returning to the consideration of the cases excepted above, we see that by a suitable change of origin of the y's, the hypersurfaces excepted for one coordinate system are not excepted for another. Hence we have: The vector-field v“ can be chosen at points of any hyper- surface of a flat space so that for the induced connection the hypersurface is projectively flat. 62. Covariant pseudonormals to a sub-space. When a space Vy, is referred to coordinates y“ a sub-space, or sub-variety, V, is defined by ; (62.1) 9° (y', cee y™) = 0 {p= n+1, wes, m). If we put : (62.2) 7 a @’ La hy 7), 7% = dd where the functions ¢’ are arbitrary except that the jacobian of the m ¢’s is different from zero, equations (62.2) define a coordinate system for which the given V), is defined by the equations 2° =0(e=n-+1, ..-, m). Any displacement in V), satisfies the conditions G (62.3) By dy? = 0, and consequently the covariant vectors »¢’ in V = defined by i 0 ¢° d= (62.4) 10 Sy =a oy oy * In this and the following sections latin indices take the values 1, - - -. n, letters at the beginning of the greek alphabet values 1, ..., m and those at the end n41, ..., m. 63. DERIVED TENSORS. INDUCED CONNECTION 171 are pseudoorthogonal at any point of V), to any displacement in the latter. We call them covariant pseudonormals to Vy; evidently any linear combination of them is the most general covariant pseudonormal to V,. If we put 210 (62.5) pC — 0 y (0) 2%’ v(., for a given value of ¢, at any point are the components of the contravariant vector tangential to the curve of para- meter 2° at the point, that is, the curve along which all the a's but 2° are constant. As thus defined the functions »§, depend upon the choice of the functions ¢? in (62.2), whereas »@ do not. From (62.4) and (62.5) we have (62.6) pe PD = 4 and , 0 20 62.7 ye EIT ee 0 ( ) (0) oy” When equations (62.2) are solved for the #'s, we have (62.8) y = FY (2, Sa, 27) or as power series in 2”, ... 2™ (29) 7° = 00s +s VPS oon YE Consequently V,, is defined by the parametric equations (62.10) y= Ll a) and at points of V, (62.11), Ye = 12 {= ee x"). 63. Derived tensors in a sub-space. Induced affine connection. For a tensor in V,, the quantities given by (52.3), (52.11) and (52.15), where greek indices take the values 1, ..., m, evaluated at points of V, define a derived 172 IV. THE GEOMETRY OF SUB-SPACES tensor in V,. In particular, the derived vector of any co- variant pseudonormal is zero, and the derived vector of any 7%, is zero. (©) If we put (63.1) Bi = 05— vig vy), we have ou Ox 9 xt y 32% : ns. = mr in = 8 0 y* oy ’ Bj dy“ 0, n8y By nln? by 3:7 8a’ Bj La 0. Hence with this interpretation of Bj equations (52.12), (52.14) and (52.16) define tensors in V,,, associate to the given tensors, such that in the x's any component involving one or more indices n+ 1, ..., m is zero and the other components are those of the derived tensor. Furthermore, the last theorem of 852 holds.-for a V, in a V,,. If I's; and I's, are the coefficients of an affine connection in V,, in the x's and y's respectively, we have equations of the form (53.1). If I's, are the coefficients in coordinates x'“, where all == Wl, ul), 2° = af then equations (53.2) hold, and thus the quantities I and I’; evaluated at points of V), are the coefficients of the same connection, which we say is induced in V,. From the form of equations (53.1) it is seen that this induced connection varies with the choice of the vectors »(. In what follows we indicate by a semi-colon followed by indices covariant differentation with respect to the induced connection. We remark that for a V, in a V,, there is a theorem similar to the last of 3 53, 64. Fundamental derived tensors in a sub-space. We denote by wy the tensor in V, derived from the tensor vas in V,,; that is, 64. FUNDAMENTAL DERIVED TENSORS 173 o oy® 01 8 (64.1) wy} —r Vig Sy . 3 [0 In the +'s the components of the vectors »y and »{ are » 3 7) a XY (64.2) ve = or, Vi = 4 as follow from (62.4) and (62.5). Consequently in the z's equations (64.1) are (64.3) wf = TT, We denote by I; 5; the tensors in 1, derived from « « T) Vv, h . Yo), Bg and Yo), Ve 1 “my t at 1S, » 3 i oe La oy and : x) Oyf (64.5) 12), — 2 “i B pT) Pe gz . In the z's the components of these tensors are (64.6) log = Taj, Uo = I%. In consequence of (62.6) we have from (64.5) yp (64.7) lai = — vis ols) CL =. The geometrical significance of these tensors is pointed out in the next section. In order to study the effect of a change of the vectors o . . JS . Vi, We consider a transformation of coordinates of the form (64.8) a’ = of + lan, 2% = gn where the ¥’s are functions of z', --., x» - At points of V, we have id IV. THE GEOMETRY OF SUB-SPACES Df Ty oat oy oat a7? ax a dy“ Sy, oa’ 32 0x0 = 3a'C fo at es oy“ Ee r+ Pr 000 9 2 i wl 0 2 © G i ’ 59 oa" Cl ay: yy. nr Py i oy” GEV eBy0 © a Ea or Ww) 1 6 1 z dx'ox da’ da "ox px’ Ox oy” : bx! bx) 0 x" 0 x > From the fourth of these equations we have that »,.% — » 3 that is, the covariant pseudonormals are unaltered. The second and last sets of these equations are reducible by means of the others to . dys (64.10) Voy = = 5 7 oo : and ey _ wily By, oy 9 3 3 xt dxtoxl T° pxd sat” From these two sets of equations, (64.9) and (53.1) we have Je dd a La oy i on SL en rr YOM i dat da’ pat oad) Ya dy“ oy’ (64.11) 3 oR J By Buf = eS ot TT +e oR vey If we define Lis and 1 by equations analogous to (64.4) and (64.5), we have (6412) Uj = ly — hej + 0} 18); + WE wh 0d (64.13) ly = lo + wh ofl. There is also another element of indeterminateness due to the fact that the covariant pseudonormals are not uniquely 65. EQUATIONS OF GAUSS AND CODAZZI 175 determined. In fact, the sub-space defined by (62.1) is like- wise defined by the equations ¢° = 0, where ¢® = Az ¢T, the A’s being arbitrary functions of the y’s subject to the condition that the determinant |A47| is not zero as a con- sequence of (62.1). From the above we have (0) 4° (T) (T) a (T) Vo B= _— ¢° 300 = — Ar ve, Fig A7 75 Ye + 47.4 rg + Az apo" From these equations and (64.1) we have at points of V, (0) (7) (64.14) ni = AZ Wij, where now A; may be interpreted as arbitrary functions of xl, ..., 2" such that the determinant |A7| is not zero. An application of the foregoing results will be made in the next section. 65. Generalized equations of Gauss and Codazzi. At points of a sub-space V, of a V,, as defined in § 62, equations (55.1) may be written by means of (64.3) oy By oo od Ae (65.1) Y u = Igy ~ oa When in equations (55.1) we replace j by ¢, the resulting equations may be written 3 ia) x i 2 8° ox? +1 re 0x = dah’ and at points of V, these equations become, in consequence of (64.6), 9 Ve) 8 2 (Ty 0 A =e 0. Yr (65.2) Toa == Is, Yio) pl = lioyi os + lo)i Vay With the aid of the identities (55.4) for the present case, we obtain as the conditions of integrability of equations (65.1) 176 IV. THE GEOMETRY OF SUB-SPACES oy“ (0) (6) h Oo oy oyr ay’ a Bl + yj Loi Wj; Yn) — Bj,» oF Do 55 om o (6) (0) T) 4(6) (T) 15 + vey (03) 5 — wi;; + of lok — oi In) = Ju If these equations are multiplied by Se and by »® and i « is summed, we have the respective equine yf Ba = 30 (6) (65.3) Bi ar Bs, PE wie oy @; Low + wy Yi y 2 (©) (CD) 1] oy (a) 79) (T) Io (65.4) wij; — wip;; = Bgs le of lo + of Ig); In like manner we find that the conditions of integrability of (65.2) are hy Ti (T) z (T) qh l6)i.j — li): i + ligyi l Ef lic) lei (65.5) Pep oy oy’ sa + Byrd Vo oat ors oy“ oT ? 1s = le)si + Losi J 2) me FA ol (65.6) J 20.0 By oy + Baa ve VN op Bw 0 These equations are evident generalizations of equations obtained by Voss and Ricci for a sub-space of a general Riemannian space*. From (65.3) we have on contracting for / and ¢ ny oy” oy’ w©® Six EE (Spo — Bin VE y xl nd Oyj (0)k =f wi) 1 Rl? which is reducible by means of (65.6) to ay a 01 oy’ (©) (@) (65.7) iy = Ss To Br Fly: k Ey los) From (64.13) it is seen that if the determinant of any one of the sets of functions of is different from zero, or can be * 1926, 1, p. 163. 66. PARALLELISM IN A SUB-SPACE 177 made such by (64.14), the functions y., and consequently the vectors »(), can be chosen so that the sums 4S) are zero. Consequently we have from (65.7): In general the vectors vi, can be chosen so that when the tensor Beg in the enveloping space is symmetric, the tensor by; of the induced connection in the sub-space also is symmetric.” It is an algebraic problem to determine by means of equations (64.14) whether m—mn independent covariant pseudo- normals can be chosen so that all of the determinants {} are different from zero. When this condition is satisfied, m—mn independent vectors »¢ can be determined by (64.13), ( . so that Is; = 0, where ¢ is not summed. In this case, as follows from (65.2), the associate direction in V,, of each vector ve for a displacement in V,, does not have a component in the direction Yes (cf. $ 56). As this is a property of the normals to a sub-space of a Riemannian space,i we say that the corresponding vectors »( ave contravariant pseudonormals to the sub-space. The process of determining these pseudo- normals is not unique since each choice of covariant pseudo- normals satisfying the desired conditions yields a set of contravariant pseudonormals. When the connection is asymmetric (ef. § 55), the equations analogous to (65.3), (65.4), (66.5) and (65.6) are obtained from the latter on replacing Bj and Bis subtracting the term 2 oh nD from the right-hand member of . Jc (65.4). and adding the respective terms 2 oF Lon and 2 £2; 518 to the left-hand members of (65.5) and (65.6). 66. Parallelis;n in a sub-space. Curvature of a curve in a sub-space. The results of §§ 58, 59 can be generalized to the case of a general sub-space. In place of (58.3) we have (6) bo by Lin and Lg, ad j n Y rae a AYE si del. By wy ar dal (66.1) & 2 15%, 7 So — my; A Er Vo) * Cf. Schouten, 1924, 1, p. 162. T1926, 1, p. 161, equations (47.9). 178 IV. THE GEOMETRY OF SUB-SPACES Consequently we have When ao family of contravariant vectors of a sub-space are parallel in the enveloping space with respect to a curve, they are parallel in the sub-space, for every choice of the vectors v® In particular we have When a path of a space lies in a sub-space, it is a path of the sub-space, for every choice of the vectors vi; it is a curve for which wo) dit dz’ =— 0 {0 = n-t+1,.-.,m). 0)" From this theorem we have also: A necessary and sufficient condition that every path of a sub-space be a path of the enveloping space is that 0) oy) = 0. These sub-spaces are the analogues of totally-geodesic sub- spaces of a Riemannian space.™ When in (66.1) we replace §% by a we have the dt.’ equation obtained from (59.1) on replacing the last term R dx dn! ; de’ dn’ : by » Yor i Consequently wf Sr Bb the component of the curvature of the curve in the direction »{,. If we multiply (65.2) by 4 for a curve, we have y dt Pp i |p © Jt EP Va) ro : 2 are the tangential components of the associate direction of the vector »{, the components in the directions »{, i T 67. Projective change of induced connection. In order to determine the effect upon the induced connection of *1926, 1, p. 184. 1 Of. Weyl, 1922, 6, 1. 156. Hence Ui i for the curve and {®@ dx’ i 77 are 67. PROJECTIVE CHANGE 179 a projective change of connection in the enveloping space, replace Is, in (53.1) by expressions of the form (32.1) and understand that «, 8, y take the values 1, -.., m. From the resulting equations and (53.1) we have (67.1) Ti = I+ Sout diy, where v7.0 oy’ (67.2) gp; = Ws Tr Ys being the vector in Vy, determining the projective change. Hence we have: When the connection of a space undergoes a projective change, the same is true of the induced connection of a sub- space, and the vector determining the latter is the derived vector of that determining the former. From (67.2) it is evident that m — » independent vectors Ve, exist such that there is no change in the induced connection. If we take we = iy, #3, where is not summed, then gj; == af (ef. § 65). Consequently when the vectors »(, can be chosen so that 5, = 0 (¢ not summed), there is no projective change in the induced connection. This BIBLIOGRAPHY bibliography contains only the books and memoirs which are referred to in the text. 1869. 1. Christoffel, K. B.: Uber die Transformation der homogenen Diffe- pt 1891. 1393. 1. 1901. 1. 1905. 1. 1908. 1. 1909. 1. 1918, 1. [\ 1922. 1. rentialausdriicke zweiten Grades. Journal fiir die reine und angew. Mathematik (Crelle), vol. 70, pp. 46-70. . Gowrsat, K.: Lecons sur l'intégration des équations aux dérivées partielles du premier ordre. Hermann, Paris. Lie, S.: Vorlesungen iiber kontinuierliche Gruppen. Teubner, Leipzig. Ricci, G. and Levi-Civita, T.: Méthodes de calcul différentiel absolu et leurs applications. Math. Annalen, vol. 54, pp. 125-201, 608. Fine, H. B.: A college algebra. Ginn and Company, Boston. Wright, J. E.: Invariants of quadratic differential forms. Cambridge Tract. No. 9. Fisenhart, L. P.: A treatise on the differential geometry of curves and surfaces. Ginn and Company, Boston. . Kowalewski, G.: Einfithrung in die Determinantentheorie. Veit und Comp., Leipzig. . Levi-Civita, T.: Nozione di parallelismo in una varieta qualunque e consequente specificazione geometrica della curvatura Riemanniana. Rendiconti di Palermo, vol. 42. pp. 173-205. Bianchi, L.: Lezioni sulla teoria dei gruppi continui finiti di trasformazioni. Spoerri, Pisa. . Kinsler, P.: Uber Kurven und Flichen in allgemeinen Raumen. Dissertation. Gottingen. . Weyl, H.: Space, time, matter. Translated by H. L. Brose. Methuen, London. . Weyl, H.: Zur Infinitesimalgeometrie; Einordnung der projektiven und der konformen Auffassung. Gottinger Nachrichten, 1921, pp 99-112. Eisenhart, L. P.: Fields of parallel vectors in the geometry of paths. Proceedings of the Nat. Acad. of Sciences, vol. 8, pp. 207-212. . Eisenhart, L. P.: Spaces with corresponding paths. Proceedings of the Nat. Acad. of Sciences, vol. 8, pp. 233-238. 181 182 1923. 1924. 1925. 1. DO Lo Ot C - 1 . Eisenhart, L. P.: Geometries of paths for which the equations no Lo i, 6. BIBLIOGRAPHY Veblen, O.: Projective and affine geometry of paths. Proceedings of Nat. Acad. of Sciences, vol. 8, pp. 347-350. . Eisenhart, L. P. and Veblen, O.: The Riemann geometry and its generalization. Proceedings of Nat. Acad. of Sciences, vol. 8, Pp. 19-23. . Veblen, O.: Normal coordinates for the geometry of paths. Pro- ceedings of Nat. Acad. of Sciences, vol. 8, pp. 192-197. . Weyl, H.: Zur Infinitesimalgeometrie; p-dimensionale Fliche im 7 -dimensionalen Raum. Math. Zeitschrift, vol. 12, pp. 154-160. . Fermi, K.: Sopra i fenomeni che avvengono in vicinanza di una linea oraria. Rendiconti dei Lincei, vol. 31!, pp. 21-23, 51-52. Veblen, O. and Thomas, T. Y.: The geometry of paths. Trans- actions of the Amer. Math. Soc., vol. 25, pp. 551-608. . Weitzenbick, R.: Invariantentheorie. Noordhoff, Groningen. . Hisenhart, L. P.: Symmetric tensors of the second order whose first covariant derivatives are zero. Transactions of the Amer. Math. Soc., vol. 25, pp. 297-306. . Kisenhart, L. P.: The geometry of paths and general relativity. Annals of Mathematics, ser. 2, vol. 24, pp. 367-392. . Cartan, E.: Sur les variétés & connexion affine et la théorie de la relativité généralisée. Annales de I'Ecole Norm. Super., ser. 3, vol. 40, pp. 325-412. Bianchi, L.: Lezioni di geometria differenziale, vol. 2. Zanichelli, Bologna. . Schouten, J. A.: Uber die Bianchische Identitit fiir symmetrische Ubertragungen. Math. Zeitschrift, vol. 17, pp. 111-115. . Veblen, O.: Equiaffine geometry of paths. Proceedings of Nat. Acad. of Sciences, vol. 9, pp. 3, 4. . Eisenhart, L. P.: Affine geometries of paths possessing an invariant integral. Proceedings of Nat. Acad. of Sciences, vol. 9, pp. 4-7. Schouten, J. A.: Der Ricci-Kalkiil. Springer, Berlin. of the paths admit a quadratic first integral. Transactions of the Amer. Math. Soc., vol. 26, pp. 378-384. . Cartan, E.: Sur les variétés a connexion projective. Bulletin de la Soc. Math. de France, vol. 52, pp. 205-241. Friesecke, H.: Vektoriibertragung, Richtungsiibertragung, Metrik. Math. Annalen, vol. 94, pp. 101-118. . Thomas, T. Y.: On the projective and equiprojective geometries of paths. Proceedings of Nat. Acad. of Sciences, vol. 11, pp. 199-203. . Thomas, .J. M.: Note on the projective geometry of paths. Pro- ceedings of Nat. Acad. of Sciences, vol. 11, pp. 207-209. . Veblen, O. and Thomas, J. M.: Projective normal coérdinates for the geometry of paths. Proceedings of Nat. Acad. of Sciences, vol 11., pp. 204-207. 1926. - < -1 10. 11. an 10. j= 11. BIBLIOGRAPHY > 183 . Levi-Cwita, T.: Lezioni di calcolo differenziale assoluto. Stock, Roma. 5. Schouten, J. A.: On the conditions of integrability of covariant differential equations. Transactions of the Amer. Math. Soc., vol. 27, pp. 441-473. . Veblen, O.: Remarks on the foundations of geometry. Bulletin of the Amer. Math. Soc., vol. 31, pp. 121-141. . Synge, J. L.: A generalization of the Riemannian line-element. Transactions of the Amer. Math. Soc., vol. 27, pp. 61-67. . Taylor, J. H.: A generalization of Levi-Civita's parallelism and the Frenet formulas. Transactions of the Amer. Math. Soc., vol. 27, pp. 246-264. Thomas, T. Y.: Note on the projective geometry of paths. Bulletin of the Amer. Math. Soc., vol. 31, pp. 318-322. FKinstein, A.: Einheitliche Feldtheorie von Gravitation und Elektri- zitit. Sitzungsber. der Preufi. Akad. der Wissensch. zu Berlin, pp. 414-419. . Eisenhart, L. P.: Riemannian Geometry. Princeton University Press. . Berwald, L.: Uber Paralleliibertragung in Rumen mit allgemeiner Mafibestimmung. Jahresber. Deut. Math. Vereinigung, vol. 24, pp- 213-220. 3. Thomas, J. M.: Asymmetric displacement of a vector. Trans- actions of the Amer. Math. Soc., vol. 28, pp. 658-670. . Levi-Civita, T.: Sur I'écart géodésique. Math. Annalen, vol. 97, pp. 291-320. . Levy, H.: Symmetric tensors of the second order whose covariant derivatives vanish. Annals of Mathematics, ser. 2, vol. 27, pp. 91-98. 5. Veblen, O. and Thomas, J. M.: Projective invariants of affine geo- metry of paths. Annals of Mathematics, ser. 2, vol. 27, pp. 279-296. . Thomas, J. M.: First integrals in the geometry of paths. Pro- ceedings of Nat. Acad. of Sciences, vol. 12, pp. 117-124. . Thomas, J. M.: On normal coordinates in the geometry of paths. Proceedings of Nat. Acad. of Sciences, vol. 12, pp. H8—63. 9. Hisenhart, L. P.: Geometries of paths for which the equations of the paths admit » (n+1)/2 independent linear first integrals. Transactions of the Amer. Math. Soc., vol. 28, pp. 330-338. Thomas, T. Y.: A projective theory of affinely connected manifolds. Math. Zeitschrift, vol. 25, pp. 723-733. Berwald, L.: Untersuchung der Kriimmung allgemeiner metrischer Raume auf Grund des in ihnen herrschenden Parallelismus. Math. Zeitschrift, vol. 25, pp. 40-73. . Cartan, E. and Schouten, J. A.: On Riemannian geometries admitting an absolute parallelism. Proceedings Kon. Akad. v. Wetenschaffen Amsterdam, vol 29, pp. 933-946. 184 14. 1927. 1. i 1998. 1. BIBLIOGRAPHY . Thomas, J. M.: On various geometries giving a unified electric and gravitational theory. Proceedings of Nat. Acad. of Sciences, vol. 12, pp. 187-191. hisenhart, L. P.: Congruences of parallelism of a field of vectors, Proceedings of Nat. Acad. of Sciences, vol. 12, pp. 757-760. Levy, H.: Congruences of curves in the geometry of paths. Rendi- conti di Palermo, vol. 51, pp. 304-311. . Bisenhart, L. P. and Knebelman, M. S.: Displacements in a geometry of paths which carry paths into paths. Proceedings of Nat. Acad. of Sciences, vol. 13, pp. 38-42. . Thomas, T.Y. and Michal, A. D.: Differential invariants of relative quadratic differential forms. Annals of Mathematics, ser. 2. vol. 28, pp. 631-688. . Knebelman, M. S.: Groups of collineations in a space of paths. Proceedings of Nat. Acad. of Sciences, vol. 13, pp. 396-400. Douglas, .J.: The general geometry of paths. Annals of Mathematics. ser. 2, vol. 29. 68% tL co C h PER aI ae ti