Geometry of differential equations

GEOMETRY

OF

DIFFERENTIAL

Vo I . B l i z n i k a s

and

Z. Yu.

EQUATIONS Lupeikis

UDC 513.7:517.9

Introduction G e o m e t r i c s t r u c t u r e s on differentiable manifolds and, in p a r t i c u l a r , on jet s p a c e s (or other fibered spaces) a r e defined by v a r i o u s methods and a r e studied not only in differential g e o m e t r y , beginning with the work of R i e m a n n and Christoffel, but also in differential topology. Along with t h e s e g e o m e t r i c s t r u c t u r e s , which lead to the notion of R i e m a n n s p a c e s or other m e t r i c s p a c e s , the r e c e n t y e a r s h a v e s e e n an intensive study m a d e of v a r i o u s s p a c e s of support e l e m e n t s (in L a p t e v ' s terminology), of a l g e b r a i c s t r u c t u r e s on diff e r e n t i a b l e manifolds and other tangent bundles of finite o r d e r of c e r t a i n f i b e r e d s p a c e s (these s t r u c t u r e s a r e defined by t e n s o r fields or s y s t e m s of t e n s o r fields), etc. Only those g e o m e t r i c s t r u c t u r e s which yield connections in fibered s p a c e s a r e defined not by t e n s o r fields but by the fields of d i f f e r e n t i a l - g e o m e t r i c obj e c t s of higher o r d e r . Among the g e o m e t r i c s t r u c t u r e s defined by fields of d i f f e r e n t i a l - g e o m e t r i c objects of h i g h e r - o r d e r , an i m p o r t a n t place is occupied by those g e o m e t r i c s t r u c t u r e s which a r e a s s o c i a t e d with specific s y s t e m s of differential equations. The study of manifolds or of fibered s p a c e s with s t r u c t u r e s g e n e r a t e d by a given differential equation s y s t e m is of i n t e r e s t botl~ f r o m the local as well as the global point of view. The development of the g e o m e t r y of differential equation s y s t e m s is c l o s e l y tied in with the developm e n t of the g e o m e t r y of Riemann, F i n s l e r , C a f t a n s p a c e s and with the t h e o r y of v a r i o u s connections in f i b e r e d s p a c e s (see the s u r v e y a r t i c l e s by Bliznikas [14], Laptev [60], Laptev [64], L u m i s t e [65], Shirokov I109], and the p a p e r s by Ku [25-29], Yen [276]). H i g h e r - o r d e r connections (see Bliznikas [5-7, 11, 13, 17]), t e n s o r connections, and,in p a r t i c u l a r , affinor connections (see Hombu [173-176]), and b i v e c t o r connections (see H o k a r i [167-171]) a l s o find application in the g e o m e t r y of differential equation s y s t e m s . The study of v a r i o u s manifolds and submanifolds, i m m e r s e d in a p r o j e c t i v e , affine, Euclidean, or other space, as well as the investigation of v a r i o u s g e o m e t r i c objects of specific s p a c e s lead to the study of differential equation s y s t e m s (for e x a m p l e , s e e Vuiichich [23], D e m a r i a [133], H a y a s h i [165], Matsumoto [210, 211], Miller [216], Obata [218], P r v a n o v i t c h [222], R u s c i o r [225], Sauer [226], Segre [227], T a m ~ s s y [245], T s a n g a s [253], Vala [255], V a r g a [256], Vincensini [264], Weise [271], Yen [276], and others). It should be noted that the initial r e s e a r c h e s on the g e o m e t r y of differential equation s y s t e m s w e r e connected with the t h e o r y of the i n v a r i a n t s of differential quadratic f o r m s (Christoffel, Lipschitz, Ricci, K r a m l e t , N o e t h e r , Weft, and others) and with the w o r k s of Lie touching on the i n v a r i a n t p r o p e r t i e s of differential equation s y s t e m s (see Lie [206, 207]). Lie e s t a b l i s h e d the connection between c e r t a i n a s p e c t s of the t h e o r y of differential equations and of line g e o m e t r y whose foundations w e r e c r e a t e d by PlUcker, as well as with the g e n e r a l t h e o r y of c u r v e s . He succeeded not only in significantly advancing integration methods for differential equations but also in investigating c e r t a i n c l a s s e s of equations r e l a t i v e to a specified group and in giving an effective g e o m e t r i c int e r p r e t a t i o n of the r e s u l t s obtained. It is n e c e s s a r y to r e m a r k that L i e ' s g e o m e t r i c a p p r o a c h to the t h e o r y of s y s t e m s of partial differential equations not only p e r m i t t e d h i m to develop and extend the r e s e a r c h e s , existing in his day, of Jacobi M a y e r , Monge, and o t h e r s on the t h e o r y of differential equations, but also to pose new p r o b l e m s , to open up new d i r e c t i o n s , and by this, to r e n d e r a c o n s i d e r a b l e influence on the development of g e o m e t r y . LieVs ideas w e r e developed in the r e s e a r c h e s of Vessio, Z u r a v s k i i , and o t h e r s ; m o r e o v e r , it was p r o p o s e d that the Vession s y s t e m s of differential equations, admitting of s i m p l e finite nondegenerate groups (Lie groups), be n a m e d Lie s y s t e m s . T r a n s l a t e d f r o m Itogi Nauki i Tekhniki (Algebra. Topologiya. G e o m e t r i y a ) , Vol. l l , ' p p . 209-259. 9 76 Plenum Pubhshmg Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f thts pubfieation may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, mierofilming, recording or otherwise, without written permission o f the publisher. A eopy o f this article is available from the publisher for $15.00.

591

The m o s t b r i l l i a n t a n d profound development of L i e ' s gifted ideas is due to Caftan. Caftan* e x a m i n e d the m o s t g e n e r a l c a s e s of differential equations, i . e . , the c a s e s when the c o r r e s p o n d i n g s i m p l e group is infinite. T h e s e groups, c o r r e s p o n d i n g to specific differential equation s y s t e m s , a d m i t e i t h e r of an i n t e g r a l i n v a r i a n t of m a x i m a l d e g r e e (theory of Jacobi m u l t i p l i e r s ) or of a r e l a t i v e l i n e a r i n t e g r a l i n v a r i a n t (theory of equations r e d u c i b l e to canonic form) or of an invariant P f a f f equation (equations r e d u c i b l e to f i r s t - o r d e r p a r t i a l differential equations. Cartan, in his investigations touching on the t h e o r y of continuous groups (Lie groups and infinite groups), on the P f a f f p r o b l e m , on the t h e o r y of g e n e r a l i z e d s p a c e s , on the t h e o r y of diff e r e n t i a l and i n t e g r a l i n v a r i a n t s , on the t h e o r y of s y s t e m s of e x t e r i o r differential equations, often dwelled on the m o s t i n t e r e s t i n g a s p e c t s of the g e o m e t r y of differential equations; m o r e o v e r , he a p p r o a c h e d the g e o m e t r y of differential equations e i t h e r f r o m the point of view of i n t e g r a l i n v a r i a n t s or f r o m the point of view of g e n e r a l i z e d s p a c e s . One of the m e r i t s of C a f t a n is p r e c i s e l y that his w o r k on the g e o m e t r y of differential equations w a s b a s e d on the g e o m e t r y of differential equations as an independent direction in t h e g e o m e t r y of g e n e r a l i z e d spaces. The g e o m e t r i c p a p e r s on the t h e o r y of connexes, whose initiator was Clebsch, found application in the t h e o r y of differential equations. The application of the methods of the t h e o r y of i n v a r i a n t s and of a l g e b r a i c g e o m e t r y to the t h e o r y of connexes proved to be useful both for g e o m e t r i c as well as for the analytic a s p e c t s of the t h e o r y of differential equations. The t h e o r y of only a t e r n a r y convex was worked out in C l e b s c h ' s p a p e r s . The subsequent development of this t h e o r y is due to Sintsov (see [94]) who c o n s t r u c t e d the t h e o r y of a q u a t e r n a r y connex (an e l e m e n t is a point-plane) and e s t a b l i s h e d the connection of its principal coincidence with the t h e o r y of f i r s t - o r d e r p a r t i a l differential equations, i.e., with the fundamental ideas of L i e ' s integration t h e o r y (see Lie [206, 207]), T h e s e fundamental r e s e a r c h e s on the t h e o r y of connexes led to the n e c e s s i t y of developing the g e o m e t r y of p a r t i c u l a r c l a s s e s of differential equation s y s t e m s , i.e., to the g e o m e t r y of P f a f f and Monge manifolds (see B e r z o l a r i , Bonsdorf, Veneroni, I s s a l i , K a s n e r , Kowle, L a z z e r i , Lilienthal [208, 209], Ogura, R o d g e r s , Sintsov [94], Voss [265,266], and others). T h e s e r e s e a r c h e s a r e connected with the m o d e r n t h e o r y of nonholonomic and s e m i n o n h o l o n o m i c r u l e s manifolds. In the investigation of differential equation s y s t e m s c o n s i d e r a b l e attention was paid to the study of t h e i r group p r o p e r t i e s . By a group p r o p e r t y of a differential equation s y s t e m S we m e a n a p r o p e r t y of this s y s t e m which r e m a i n s unchanged when the dependent and independent v a r i a b l e s o c c u r r i n g in s y s t e m S a r e subjected to t r a n s f o r m a t i o n s f r o m s o m e t r a n s f o r m a t i o n group G. If such a p r o p e r t y holds for a given differential equation s y s t e m S, then we say that s y s t e m S a d m i t s of group G. Under such t r a n s f o r m a t i o n s any solution of s y s t e m S is led once again to a solution of this Same s y s t e m . T h e s e group p r o p e r t i e s can be used not only for the c o n s t r u c t i o n of v a r i o u s c l a s s e s of p a r t i c u l a r solutions of the given s y s t e m S but also to effect a group c l a s s i f i c a t i o n of the differential equation s y s t e m s t h e m s e l v e s . Such p r o b l e m s , it a p p e a r s , w e r e f i r s t put forth at the end of the l a s t century by Lie (1885-1895). Lie and his students worked out the a n a l y t i c a l tools and studied a wide c i r c l e of applications of the t h e o r y of continuous t r a n s f o r m a t i o n groups, which a r o s e when solving the p r o b l e m s mentioned for differential equation s y s t e m s . An i m p e t u s for the subsequent development of this t h e o r y was given by the applied a r e a s of m a t h e m a t i c s dealing with differential equations or with the g e o m e t r i c p r o p e r t i e s of model s p a c e s . In applied p r o b l e m s , as a r u l e , we encounter differential equations of c o n c r e t e f o r m , admitting of nontrivial t r a n s f o r mation groups. T h e s e p r o p e r t i e s a r e p o s s e s s e d , for e x a m p l e , by the equations of h y d r o d y n a m i c s , the equations of the t h e o r y of e l a s t i c i t y and p l a s t i c i t y , the equations of combustion theory, the equations of detonation, the equations of m a g n e t o h y d r o d y n a m i c s and of other applied physical or m e c h a n i c a l t h e o r i e s . Only individual r e s u l t s exist in this direction (see Ovsyannikov [75-87], I b r a g i m o v [32-42]) and intensive investigations a r e being c a r r i e d out at the p r e s e n t time. B e s i d e s the a b o v e - m e n t i o n e d p a p e r s , to w o r k connected with the study of groups of analytic t r a n s f o r m a t i o n s of differential equation s y s t e m s we can r e f e r the p a p e r s by Evtushik [30], D u m i t r a s [137], Spencer [231], Hangan [161-162], and others. Thus, the g e o m e t r y of differential equation s y s t e m s can be i n t e r p r e t e d in m a n y ways, but t h e r e e x i s t profound r e l a t i o n s between all the t h e o r i e s even though they a r e not equivalent.

*E. Cartan, O e u v r e s C o m p l e t e s , P a r t i e II, Vols. 1,2; P a r t i e III, Vol. 2, G a u t h i e r - V i l l a r s , P a r i s (1953).

592

Let us note other features of the origin of the g e o m e t r y of differential equation s y s t e m s . Many a s pects of the local c l a s s i c a l theory of s u r f a c e s a r e connected with the study of. the s t r u c t u r e of geodesic lines (of the e x t r e m a l surface) whose differential equations have the form (Fik is the Christoffel object, n = 2) 9 d~s dxk d~x___~+ r~(x) ds 2

o (i,/ . . . . .

ds

The geodesic tines of any affinely-connected space also eral), i.e., to any affinely-connected space there always It is not difficult to note that if the differential equation manifold we can define affine connections for which the These connections have the form ~i

rik =

Fi

,

1,~ ..... n)

(0.1)

are d e t e r m i n e d by the same s y s t e m (n ~ 2, in genc o r r e s p o n d s the differential equation s y s t e m (0.1). s y s t e m (0.1) is given, then on the differentiable integral c u r v e s of s y s t e m (0.1) a r e geodesic lines.

i

'

jk ~- a j p k + a ~ pi,

(o.2)

where Pi is an a r b i t r a r y coveetor field. If we r e c k o n that s v a r i e s by the l a w ~ = as + b (a, b are a r b i t r a r y constants), then for the given differential equation s y s t e m (0.1) there exists a unique (torsion-free) affine connection whose geodesic lines coincide with the integral c u r v e s of the differential equation s y s t e m being examined. It is obvious that to any affinely-connected space t h e r e c o r r e s p o n d s a s e c o n d - o r d e r line-element space ~(2) n in which s u r f a c e (0.1) is given as the distinctive "absolute" of this space. Since the absolutes of a space a r e invariant r e l a t i v e to t r a n s f o r m a t i o n s (0.2), then, in general, the inverse c o r r e s p o n d e n c e is not unique. If linear t r a n s f o r m a t i o n s alone a r e admissible t r a n s f o r m a t i o n s for p a r a m e t e r s, then the abovementioned c o r r e s p o n d e n c e is unique. Affinely-eonnecte d spaces were studied f r o m this point of view by T h o m a s [248] and T h o m a s [249, 250], Veblen, and others. If we note that the quantities

9

dxk dxn ds

H i = V~h(x) ds

(0.3)

f o r m a d i f f e r e n t i a l - g e o m e t r i c object defined on a f i r s t - o r d e r l i n e - e l e m e n t space L ), then the g e o m e t r y of an affinely-connected space is equivalent to the g e o m e t r y of space L(n1) with the fundamental differentialg e o m e t r i c object H I. Cartan noted (see [48]) that the geodesic lines of a p r o j e c t i v e l y - c o n n e c t e d space have the form

d~ua_~ .~_ pZ ( f~, t, @-t ) - - d u dt Z~(ua,t,~t

(0.4)

where p a and ~ a r e a r b i t r a r y s e c o n d - d e g r e e polynomials in duff/dr. If the differential equation s y s t e m (0.4) is given, then the c u r v e s defined by the integrals of s y s t e m (0.4) can always be looked upon as the geodesic lines of a (torsion-free) p r o j e c t i v e l y - c o n n e c t e d space; m o r e o v e r , the projective connection of this space is defined nonuniquely (to within n (n + 1)/2 functions,of n arguments), i.e., a part of the components of the object of projective connection (of c e n t r o p r o j e c t i v e connection in general) can be chosen a r b i t r a r i l y . If the R i c c i t e n s o r of the c u r v a t u r e t e n s o r of a (torsion-free) p r o j e c t i v e l y - c o n n e c t e d space equals zero, then in C a r t a n ' s t e r m i n o l o g y (see [48]) such a p r o j e c t i v e l y - c o n n e c t e d space is called a space of n o r mal projective connection. It turns out that a space of n o r m a l projective connection always c o r r e s p o n d s uniquely to differential equation s y s t e m (0.4). Thus, a unique (local) c o r r e s p o n d e n c e exists between differential equation s y s t e m s (0.4) and spaces of n o r m a l projective connection (the p a r a m e t r i z a t i o n of the integral c u r v e s is a r b i t r a r y ) . In other words, spaces of n o r m a l projective connection play for the differential equation s y s t e m s (0.4) the v e r y same r o l e that Riemann spaces (with Levi-Civita parallelism) play in the theory of nondegenerate quadratic differential forms.

593

The differential invariants of the e l e m e n t a r y differential equation

dx~

w e r e examined by T r e s s e [252]. Cartan [121] proved that the integral c u r v e s of differential equation (0.5) can be looked upon as the geodesic lines of a l i n e - e l e m e n t space with a n o r m a l projective connection. We r e m a r k that the dual spaces p o s s e s s the same property. T h e s e investigations by T r e s s e and Caftan p r e cede the investigations of Koppisch [193] (Koppisch's papers predate the p a p e r s of T r e s s e , but Koppisch's dissertation [193] was published after T r e s s e ' s dissertation [252]). Koppisch studied, f r o m an analytic viewpoint, the c o r r e s p o n d e n c e between the solutions of two c l a s s e s of s e c o n d - o r d e r differential equations. The g e o m e t r i c interpretation of these r e s e a r c h e s is the following: the r e l a t i o n s existing between the families of geodesic lines of two mutually-dual l i n e - e l e m e n t spaces with n o r m a l projective connection w e r e studied. Analogous questions w e r e analyzed f r o m the analytic point of view by Yoshida [277], Zorawski [278], K a i s e r [183], P o d o l ' s k i i [91, 92], Neuman [217], and others. C a r t a n ' s g e o m e t r i c ideas w e r e c a r r i e d over to the multidimensional c a s e by Douglas [135, 136] who took it that the geodesic lines of a space a r e the solutions of the s y s t e m d~x~ L H ~ ( x k , t, dxk ~ = O, dt 2 dt ]

(0.6)

where H i a r e a r b i t r a r y homogeneous functions in dxk/dt. 9

t

,

The investigations of ]3ark [113], Berwald [114], Veblen [257-263], Weszely and Szflagyl [272]; Gaukhman [24], Kawaguchi and Hombu [185], Laptev [62], Slebodzinski [228-230], Stepanov [95-98, 100-102], Chern [122, 123], and others r e l a t e to the g e o m e t r y of such differential equation s y s t e m s . A cycle of papers by ]31iznikas [9-11, 17], Bompiani [116, 117], Kosambi [194-196], Moor [214, 215], Rzhekhina [93],Stepanov[99, 104], Takano [244], Udalov [105-107], Homann [172], Hombu [173-175], and others examines the g e o m e t r y of n o r m a l s y s t e m s of h i g h e r - o r d e r differential equations. Vasil'ev [22], t31iznikas [13, 16], Kil'p [54, 55], Lupeikis [66-73], P e t r u s h k e v i c h y u t e [90], and others devoted their papers to the g e o m e t r y of quasilinear s y s t e m s of differential equations. Many c l a s s i c a l p r o b l e m s in the t h e o r y of s u r f a c e s r e d u c e to the investigation of the p r o p e r t i e s of solutions of s e c o n d - o r d e r partial differential equations. If the s y s t e m ( a ~ /3) O~ x I Ou~ Ou~

Fv

O#

cz~ O-'--~-+

F ~ Oxk Oxn = 0 , kt~Ou-T Ou-T

where F~fl is the intrinsic object of affine connection of the surface, F~h is the object of affine connection of a Riemann space, has a solution of maximal dimension, then this solution defines a s u r f a c e in the R i e mann space, whose tangent coordinate lines a r e conjugate. This example is a special case of a s y s t e m of the f o r m (pk = 8xk/su~, etc.): r

X'

i k + no~l...o~p+ ` (Xk ,U "l~, p,Z...~J a) == 0 .

(0.7)

au~. . . . au%+ , (i,] . . . . . i

1,2..... n; a, 1~. . . . . 1,2, ..., m; a----l, 2 .... , p), k

where H~(_+l, a r e a r b i t r a r y functions (not n e c e s s a r i l y s e c o n d - o r d e r polynomials in p~ when p = 1). Such J d i f f e r e n t ~ equation s y s t e m s w e r e f i r s t studied f r o m a g e o m e t r i c viewpoint by Douglas. A generalization of these ideas of Douglas is due to Kawaguchi and Hombu [185]. The g e o m e t r y of such differential equation s y s t e m s has been examined by B i e b e r b a c h [115], Bortolotti [118], Ku [198-200], Ishihara and Fukami [180], Katzin and Levine [184], Mikami [212], Okubo [219, 220], Su [232-240], Suguri [241, 242], T a r i n a and Artin [246], Hashimoto [164], Hokari [167-171], Hua [178], Yano and H i r a m a t s u [274-275], and others. In connection with the development of the global t h e o r y of fibered spaces, of the t h e o r y of jets in the sense of E h r e s m a n n , and of other aspects of modern g e o m e t r y , t h e r e e m e r g e d investigations on the t h e o r y

594

of differential equations of differentiable manifolds, and also investigations on the global aspects of the g e o m e t r y of s y s t e m s of differential equations of various o r d e r s (Kumpera [201], Kuranishi [202-205], and others). We should r e m a r k that the p a p e r s on the g e o m e t r y of differential equations s y s t e m s , which a r e r e viewed in this survey, w e r e a c c o m p l i s h e d by various methods of d i f f e r e n t i a l - g e o m e t r i c investigations and a feature of the p r e s e n t s u r v e y is that the local r e s u l t s on the g e o m e t r y of differential equation s y s t e m s a r e presented by a single method of m o d e r n d i f f e r e n t i a l - g e o m e t r i c investigations - L a p t e v ' s method [61]. In this s u r v e y we s y s t e m a t i c a l l y examine the t h e m e s of only those papers which appeared in print f r o m 1953 on, while e a r l i e r p a p e r s a r e mentioned only under the n e c e s s i t y of a m o r e complete presentation of the development of the g e o m e t r y of differential equation s y s t e m s in its h i s t o r i c a l aspect. The s u r v a y ' s authors have made an attempt to unify to some extent the t e r m i n o l o g y and to p r e s e n t c e r t a i n r e s u l t s in a single invariant f o r m . The p a p e r s are concentrated around questions connected with the g e o m e t r y of differential equations. By the latter in this survey we mean a c o n c r e t e c i r c l e of p r o b l e m s which can be identified by the geometric concepts encountered in the t h e o r y of generalized spaces with fundamental-group connections and, in p a r ticular c a s e s , with the usual concepts of the t h e o r y of submanifolds of Klein spaces. We have left aside important sections of the g e o m e t r y of differential equations, which a r e connected with the fundamental p r o b l e m s of the general t h e o r y of r e l a t i v i t y (see P e t r o v [89]), the t h e o r y of differentialg e o m e t r i c methods in the calculus of v a r i a t i o n s (see Vagner [270], Winternitz [273], Kabanov [46], C a r a th6odory [120]), the t h e o r y of motions in generalized d i f f e r e n t i a l - g e o m e t r i c spaces (see E g o r o v [31]), the global aspects of the basic p r o b l e m s in the t h e o r y of equations on complex manifolds (Schwartz, Grothendieck, Dolbeau, Malgrange, H i r z e b r u c h , and others), and the a s p e c t s of differential topology, having a direct connection with the main p r o b l e m s of the g e o m e t r y of differential equations. The authors endeavored to pick out the fundamental directions of investigations in the g e o m e t r y of differential equations, which a r e being taken both in our own native as well as in the foreign schools of geometry. w

Fibered

Space

JP(Vm,

Vn)

Let Vn and V m be differentiable manifolds of c l a s s Cw (dim Vn = n, dim Vm = m) whose local c o o r dinates a r e the f i r s t integrals of the fully-integrable s y s t e m s r 1 = 0, 0 cx = 0(i, j , . . . = 1, 2, . . . , n; cL fi, . . . = 1, 2 , . . . , m). ~ 9 .fla a r e s y m m e t r i c relative to the The s t r u c t u r e equations for the 1 - f o r m s c0i a r e (wjI" .. Ja' 0ill. lower indices of the 1-form) n~o~ = cok /~ok, D(o} = o~1/~oJ k k~+ cok/\o)ik, f

(1.1)

a

O~}..j,

=

~! (~ - s)! ~...i~A~'(J~+~. (a =

The 1 - f o r m s 0 a, 0~, " ' "

1,2 .....

]a) k ,_,.,k ' w ,A,.,~ ~wir..lalr

p).

O~:t. " .fla are connected by analogous s t r u c t u r e equations.

Let us consider the set c P (Vm, Vn) of point functions ( f , y), where f is a mapping of c l a s s C p, i.e.,

f : (ya) _ (xi). Let u

(vm, v,,) = c; (vm,

YEV m

T w o e l e m e n t s (f, g), (g, g) ~Cp( V~, V~) a r e s a i d t o b e p - e q u i v a l e n t a t p o i n t (g~)~ V~, i f f ( y ) = g (y) and the partial derivatives of functions f and g coincide at point ( y ~ ) ~ V ~ up to o r d e r p (inclusive). This establishes a p-equivalence r e l a t i o n in the set of mappings f : Vm - - Vn taking (y~) into (xi). The c l a s s of p-equivalent elements is uniquely defined by any element of thos c l a s s .

595

The c l a s s of p-equivalent e l e m e n t s in c P ( V m , Vn) is called a p-th o r d e r jet (or a p - j e t in the s e n s e of E h r e s m a n n (see [138-144])). We denote the p - j e t g e n e r a t e d by mapping f at point (y~)~ V~ by jPyf, while we denote the set of all p - j e t s f r o m Vm into Vn by JP (Vm, Vn), i.e.,

J" (v,,,, v.) = U if, f. YeVm

This set is a f i b e r e d space. If 1 < m - n - 1 , then a r e g u l a r p - j e t is called an m - d i m e n s i o n a l s u r face e l e m e n t (see Kawaguchi [186, 189], C r a i g [124-131], Oliva [221]). I f m = 1, the p - j e t is called a c u r v i l i n e a r e l e m e n t of p-th o r d e r . We shall use the following notation:

Kn,m =J'(V,n, (

Vn) , Pr e < n ;

)L(nm

JP(V1, V,),

f.~P) = .P (r(, v.). Since the differential equations of mapping f : V m -~ Vn can be w r i t t e n as (o = f ~~O =,

(1.2)

by prolonging s y s t e m (1.2) s u c c e s s i v e l y , by virtue of s t r u c t u r e equations (1.1) and of the analogous equations for the 1 - f o r m s 0, we obtain dr& + ,~'~,,-k ''~ - - -t~ ~r o~ = f~, o6, ~k m i - U / : ~ n o ) i .

.

.

.

.

,

.

.

.

.

.

0v __t:iO v

__9~i .

.

.

.

.

.

.

.

.

.

.

-

-

(1.3) .

.

.

.

a

0%...% - - d~%...% +

'

='

9 Z s!a' C%z"" ks s=l

1

Z

- - 9 •]s [ ]l!..

(]l+.,.+/s=a)

a

• [(%"~1,)'"

f:;,+

o6(%

~ .(a--s)!

..+,,_~ . . . . , ) - -

.% f,%+r"%) t~ =Z %...%vr

S~I

(f~=,"'%l = 0

for any

a > 1).

If col = 0, 0 ~ = 0, then the differential equation s y s t e m (1.3) t a k e s the f o r m (a = 1, 2, . . . . p): 0 ~t -_-O ,

0 ~6 i = 0 .... ' 0 %.-% i =0.

(1.4)

It is fully i n t e g r a b l e and its f i r s t i n t e g r a l s f o r m a d i f f e r e n t i a l - g e o m e t r i c object r e l a t i v e to the Lie group GLP(n, R) x GL p (m, R) [GL p (n, R) is a differential group of o r d e r p for the differentiable manifold Vn] , i.e., the differential equation s y s t e m (1.4) defines the s p a c e of r e p r e s e n t a t i o n of the differential groups GLP (n, R) and GLP (m, R) (see L a p t e v [62, 63]). The components of this d i f f e r e n t i a l - g e o m e t r i c object a r e the local c o o r d i n a t e s of a p-jet, i.e.,

J~

%}.

'

The differential equation s y s t e m oi

0,0 ~

0, O~

0,

0~

=o

(1.5)

is fully i n t e g r a b l e and its f i r s t i n t e g r a l s a r e the local c o o r d i n a t e s of the f i b e r e d space JP (Vm, Vn). The s t r u c t u r e equations of s p a c e JP (Vm, Vn) have the f o r m (a = 1, 2 . . . . . p) D~o~ - - c ok ~/ \ O)ik, DO~-O~AO~, a l k " ... D 0 %..%= Z %,.. ~cA 0~f..~al + ~ c--=l

596

'

k + OVA el, ..%v,

(1.6)

where gi$r..fic

dO~r..% Of~ I "~ pr '

~%...%k = a

_

l

...ft.,

h

(il+-...+is~a)

0 i t~ 1

(1.7)

~___

9( a - - s ) t

aa ?

v(% -.% %+r"%) 13"

If n = 1, we i n t r o d u c e the following notation: [,

f (a)

1 , . . 1 :"== t a

and i ~a ] o)k~. ks

~(a) i _ df(a)~ +

(al-i-,..+a s=a)

s= 1

The s y s t e m (a = 1, 2 . . . . .

~

1 [(a~)k~...:(%) ~ alI...as!

(1.8)

p) o)i= O,

@(a) i = 0

is fully i n t e g r a b l e 9and I its~ f i r s t i n t e ~g r a l s a r e the l o c a l c o o r d i n a t e s of a p a r a m e t r i z e d line e l e m e n t of h i g h e r o r d e r ( o r d e r p) (x t, f ( )l . . . . . f ( p ) i ) , i.e., aor(ep ) point of the s p a c e L (p) (the s p a c e of h i g h e r - o r d e r line e l e ments). The s t r u c t u r e equations of s p a c e L~ have the f o r m (a = 0, 1 . . . . . p) D ~ (a)~

-~ ,~(c)k^ ~(a)i

~

(1.9)

/ \u(c)k,

where

w',

~(o)i

~(a) i (b) I =

u(o)i ~ r

o~(a) t O[(o) i '

'.,(o)k ~ ~(k

Otot

t

~

0/(0}k

(1.10)

Ok,

It turns out that tct) i • ( (r a ) ki --- 0 (c > a), .q, -(b) i

a!

b! (a -- b)!

a?~"- ~ ' (a >~ b).

(1.11)

The 1-forms o(a)~= u(b)

~(a) il

(b)k/ol~ 0, o(a)i=0

(1.12)

a r e the i n v a r i a n t 1 - f o r m s of the Lie g r o u p GL (n, p, R); m o r e o v e r , Do(a) i U(b)]

~-

a(c) k ^ a(a) i uib) j / \ U ( c ) k .

Group GL (n, p, R) is the t r a n s f o r m a t i o n g r o u p of the f r a m e s of a t a n g e n t s p a c e for ~ P ) . ing s u b g r o u p s : GL (n, R) C GL (n, 1, R) C " .

It h a s the follow-

~ GL (n, p - - 1, R).

597

The infinitesimal t r a n s f o r m a t i o n s of the f r a m e v e c t o r s {~((0)i,-~(1)i . . . . . -e(p)i} of the tangent s p a c e T(P) for finp) have the f o r m de(a)i

= 0 ( ~(a), ) k / (c)k .

The e l e m e n t s of the t e n s o r product (T*(P) is the dual v e c t o r tangent space) 4 T (p) 4 T *(p)

a r e said to be q t i m e s e x c o v a r i a n t and r t i m e s e x c o n t r a v a r i a n t , the concept of which was introduced by C r a i g (see [124-131]). The d i f f e r e n t i a l - g e o m e t r i c objects r e l a t i v e to h i g h e r - o ro( d e r differential groups GL q (n, p, R) a r e called exobjects o f q - t h o r d e r . . The connection objects of s p a c e L(np) a r e e x a m p l e s of such objects. The 1 - f o r m s (a = 1, 2 . . . . .

p)

a--I

9( )' = {}(~)~-F ~,~(a)~@(~) ~ ( c )k

(1.13)

C~0

define a linear d i f f e r e n t i a l - g e o m e t r i c connection of s p a c e L(P) if and only if the d i f f e r e n t i a l - g e o m e t r i c obn j e e t r ( a ) i h a s the following s t r u c t u r e : ~(b)k P

dp(a), ~-(O)a

,~(a), ~ - ,,(a) l ( b ) k l ~ l, - - J-(c) h ~.~(c) ( b ) ~ --

tqO ) ~ .~(a)

=

(.1.14.)

Z ~(0)~(~)t~ .~(a) ~ ~ .~(c) a

9

c~O

The 1 - f o r m s (m - 3) w e r e taken up in the p a p e r s by I z r a i l e v i c h [44], Kawaguchi and Hombu [185], Mirodan [213], Tonooka [251], and o t h e r s , in which objects of v a r i o u s connections w e r e adjoined to s y s t e m (3.29) (it is a s s u m e d that the t e n s o r field gaff is given). Kawaguchi and Hombu noticed that the c o n s t r u c t i o n of an object of affine connection F f l y , c o v e r e d only by the d i f f e r e n tial prolongations of the d i f f e r e n t i a l - g e o m e t r i c object H i~ , + ~ )=_/ - / i =1...%+1". P

~ H~I*""O~p-~-I

=I.*""=p'~-I

co~+H =1 ~ "=p"I-1TM0'1~ + ~~~H=1" ~'1~1" P'= OV o "~p~-I~r ~ l ' " P a ' =1 'O~p~-l/~

(3.30)

a~l

where p+l

~,=, "=.+.= ~] ' (o+o!s: ~'kl""ks ~, 1 f~l( ,=~ (;~+ +j,=,+~ is' -'%'"=i, " " " p@l

'' " f~]I"~''"J-]S--I-I-I'"i~p"{-I - Z

s~2

(p + 1)r 0]= "~ ,~..(p's'-~ 1')! ( z'"%/%+z'"% +z)?'

p o s e s a v e r y difficult p r o b l e m in the g e o m e t r y of differential equation s y s t e m s (3.29) (with a r b i t r a r y p), which is called the Kawaguchi p r o b l e m . In a l m o s t all the g e o m e t r i c r e s e a r c h e s on the g e o m e t r y of diff e r e n t i a l equation s y s t e m s (3.29), by the g e o m e t r y of t h e s e s y s t e m s is m e a n t the g e o m e t r y of the f i b e r e d

610

jet s p a c e s JP (Vm, Vn) rigged by the d i f f e r e n t i a l - g e o m e t r i c object H~(~.,). By prolonging the differential equation s y s t e m (3,30), we obtain a sequence of fundamental d i f f e r e n t i a l - g e o m e t r i c objects of the rigged f i b e r e d s p a c e JP (Vm, Vn):

H (') (n, m, p ) g H (2) (n, rn, p ) C " . C H (~) (n, m, P ) C ' " The d i f f e r e n t i a l - g e o m e t r i c object H 2 (n, m , p) h a s a subobject ized t e n s o r ,

H :'~(m ~(P) which is always a g e n e r a l Cc(p_]_l)kl~

If L~ is a c h a r a c t e r i s t i c o p e r a t o r defined by the equalities p--1

then the quantities Ng Cr

~ L

i [1~Jt]e]CC(P)

f o r m a t e n s o r c o v e r e d by the f i r s t differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c obi ject H~ If p = 2, 3, 4, then a t e n s o r a~fi exists which is an a l g e b r a i c c o v e r of the t e n s o r s N~(m~ 8 ~(P+D "

and ~i ~(mv(p) (see Bliznikas [17, 18]). JJ C~(p~_l)kh If p > 2, t h e r e e x i s t s a t e n s o r a a l - " ~ q c o v e r e d by those s a m e t e n s o r s (q = 2 ( p - 2 ) ) . With the aid of t e n s o r a ~ t...~q and of o p e r a t o r L T in the g e n e r a l c a s e we can c o n s t r u c t an object of affine connection --I~BT (it is c o v e r e d by the t h i r d differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object). This p r o b l e m still h a s not been solved for m > n and a r b i t r a r y p. w

On t h e

Global

Geometry

of Differential

Equation

Systems

In connection with the development of a global t h e o r y of fibered s p a c e s and of the t h e o r y of differentiable mappings on differentiable manifolds t h e r e e m e r g e s a s e r i e s of new questions and p r o b l e m s lying at the juncture of differential topology, a l g e b r a i c g e o m e t r y , functional a n a l y s i s , and the theory of differential equations. T h e s e m u l t i f a c e t e d connections a r e b r i e f l y explained by the c i r c u m s t a n c e that in the a n a l y s i s of differentiable mappings an i m p o r t a n t r o l e is played by the t h e o r y of s i n g u l a r i t i e s , in which t h r e e c o m ponent p a r t s can be delineated: differential, topological, and homological. The f i r s t p a r t is m o s t closely r e l a t e d with g e o m e t r i c p r o b l e m s not only in the t h e o r y of jets and fibered s p a c e s , but also in the g e o m e t r y of differential equation s y s t e m s . It is evident that all t h r e e p a r t s have c o n c r e t e applications also for the intrinsic theory of differential equation s y s t e m s (the w o r k s of Shwarz, Atiyah, Singer, and others) and for M o r s e theory. In v a r i o u s a r e a s of g e o m e t r y , in p a r t i c u l a r in differential g e o m e t r y , we encounter questions on the e x i s t e n c e of the solutions of the s y s t e m s obtained and often it is n e c e s s a r y to know as well the a r b i t r a r i n e s s of the solution. When solving such p r o b l e m s in the local s e n s e we e s s e n t i a l l y use C a r t a n ' s t h e o r y of f u l l y - i n t e g r a b l e differential equation s y s t e m s and, p a r t i c u l a r l y , C a r t a n ' s t h e o r y of s y s t e m s of differential equations in involution. T h e r e f o r e , when solving c e r t a i n g e o m e t r i c p r o b l e m s in .the global s e n s e we need the global t h e o r y of differential equations and e s p e c i a l l y those sections which have d i r e c t g e o m e t r i c applications. In this direction the first, most systematic, investigations are due to Kuranishi (see [202-205]), Kodaira [191], Spencer [192,231], and others. These investigations were based on the local results of other authors, on the theory of higher-order jets (see Ehresmann [138-144]), and on the theory of jets of local cross sections of a fibered space (in the local sense these spaces are Vagner's manifolds or are fibered

s p a c e s in the s e n s e of Vagner). Let N, N' be differentiable manifolds, (N', N, ~r) be a fibered space (v is a s u r j e c t i v e mapping, i.e., 7r: N' ~ N is a projection), JP ~- J P ( N ' , N, v) is the s p a c e o f p - t h o r d e r jets of local c r o s s sections for the f i b e r e d space (N' N, 7r), i.e., f :x EN---~y=f(x)E n-I(x). The s p a c e JP (N', N, ~r) is a fibered s p a c e as well (see Kuranishi [202-205]) and the c o o r d i n a t e s of the points of this s p a c e (in a s p e c i a l l y - s e l e c t e d a t l a s a r e the v a r i a b l e s x i, y a ~ v f / O x i l . . . ~ x i V (v = 1, 2 . . . . . p; i, j = 1, 2 . . . . . n; o~, fi = 1, 2 . . . . . m), w h e r e f is a

611

local c r o s s section. If U'is an open subset of s e t JP and A~ is the g e r m s h e a f of functions on U C l P , then a p-th o r d e r p a r t i a l differential equation on the f i b e r e d s p a c e (N T, N, ~r) is defined by the giving of a certain UCIP and of a s u b s h e a f Z of ideals of s h e a f A ~ locally g e n e r a t e d by s h e a f A ~ i.e., in s o m e a t l a s f o r m a b l e by 1 - f o r m s (v - p - l ) : d y ~"- - p~ dx t .... fOix...t~, ~ ~ d p l~1. .-i v ~ p tz 1 . .i v k d x k.

A c r o s s section f is a solution of Z if E: jP ( f ) --~ 0 for e v e r y JP the differential d (go pp+l), w h e r e (pp+l is the n a t u r a l projection)

zEN.

F o r any function g given on

~+1 : jp+1 -)- jn,

g e n e r a t e s a 1 - f o r m on jp+l. If we apply this operation to equation ~, we obtain a t r a n s i t i o n Z to a (P + D - s t o r d e r differential equation P "(Z) which is called the s t a n d a r d prolongation for ~. It t u r n s out that f is a solution for ~ if and only if it is a solution for P (Z). L e t T be a tangent bundle o v e r N and F (N f) be a bundle o v e r N', c o n s i s t i n g of v e c t o r s tangent to the f i b e r s . A f i b e r e d submanifold RPCJP is called a p - t h o r d e r partial differential equation on N ' (as on the bundle space). Then a solution of equation R p will be the c r o s s section whose j e t s lie in R p. We set Rp+q=]

q ( R p) (-] ] p + q ( N " ) .

The f a m i l y of s u b s p a c e s gp = F (Rp) ('] ~-1SP.T*@n~-I F (N')

(4.1)

in the bundle n-ISpT * | ~-t F(N') o v e r R p is called the symbol of equation R p, w h e r e sPT * is the s y m m e t r i c t e n s o r product (p-fold) of the cotangent s p a c e s T* and ~r0 ~ P~. If V and W a r e f i n i t e - d i m e n s i o n a l v e c t o r s p a c e s and, m o r e o v e r , (vl, . . . . Vn) is a b a s i s in V, while (v 1. . . . . v n) is a c o b a s i s in V*, then the h o m o m o r p h i s m 6(6 = 6p+l,j) 5: W@S 0+1V*@ A/V~-*-W|

p V* | Ai+1 V"

(4.2)

is defined by the r e l a t i o n n

6 ([

where

[EW|

|

..

. A v ' i ) = ~]~o, [ |

v " A . . "A o'i,

*, AIV * is the e x t e r i o r d e g r e e of s p a c e V* and 5v : W | S p+I V~

W | S p V'.

We a l w a y s h a v e 62-~-0,

i.e., ~p,j+l(Sp+l,3-~-0,

while the sequence 0_...~ ~7 @ SPV* ~ 1~7 @Sp-I V" | V* ~-~ W |174

A ~Y "~ -*-...---~ W|

SP-~V" |

| Am*-+ 0 is an e x a c t sequence ( s r v * = 0 for r < 0). If p :" k, then for the sequence {gP} of s u b s p a c e s g~ C W | SP V ~

612

(4.3)

we can define the sequence

8

6

6

0---~ gp_+gp-1 | V*-,.gp-~ | 6 gk | -~

6 V*-~W

|

9

k-1 V* | A p-k+1 V*

(4.4)

Sequences (4.3)and (4.4) a r e connected with differential equations in a natural way, b e c a u s e the family of subspaces (4~ looked upon as a sequence of spaces, is determined, f r o m the algebraic point of view, with the aid of two vector spaces. The groups of Spencer holonomies or the Spencer cohomologies H p - j ' j (see Kodaira and Spencer [192], Goldschmidt [146, 147], Guillemin and Kuranishi [15217 a r e determined with the aid of the equalities (HP-J,J = HP-J,J (gk), p > k):

Hv-I'l = Ker 6p_l,i/Im ~p__1_}_1,1.1.

(4.5)

The sequence of subspaces (4.1), generated by the prolongation of a given differential equation, is called involutive if sequence (4.4) is exact (it is always exact only in two t e r m s ) . The sequence of subspaces (4.1) is said to be q - c y c l i c if Hp,J=O

(p>~k, O