Geometry of nonlinear differential equations

39. 40. 41. 42. 43.

44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.

Shigeru Numata, nOn the c u r v a t u r e t e n s o r Shijk and the tensor Thijk of g e n e r a l i z e d R a n d e r s s p a c e s , n T e n s o r , 2_99, No. 1, 35-39 (1975). Shigeru Numata, nOn L a n d s b e r g s p a c e s of s c a l a r c u r v a t u r e , " J. K o r e a n Math. Soc., 12, No. 2, 97-100 (1975). Shigeru Numata, "On the t o r s i o n t e n s o r s Rhj k and Phjk of F i n s l e r s p a c e s with a m e t r i c ds = (gij (dx) x dxidxJ) 1/2 + bi(x)dxi, " T e n s o r , 3_22, No. 1, 27-31 (1978). Shigeru Numata, 'TOn C3-1ike F i n s l e r s p a c e s , " Rep. Math. P h y s . , 17, No. 1, 20-25 (1980). G. R a n d e r s , 'tOn an a s y m m e t r i c a l m e t r i c in the f o u r - s p a c e of g e n e r a l r e l a t i v i t y , " Phys. R e v . , 59, 195199 (1941). H. Rund, The Differential Geometry of Finsler Spaces, Springer, Berlin-G~ttingen-Heidelberg (1959). H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand, London (1966). Choko Shibata, "On Finsler spaces with Kropina metric, n Rep. Math. Phys., i_~3, No. i, 117-128 (1978). Choko Shibata, "On the curvature tensor Rhijk of Finsler spaces of scalar curvature," Tensor, 3_22, No. 3, 311-317 (1978). Choko Shibata, Hideo Shimada, Masayoshi Azuma, and Hiroshi Yasuda, nOn Finsler spaces with Randers' metric," Tensor, 31, No. 2, 219-226 (1977). Hideo Shimada, "On the Ricci tensors of particular Finsler spaces," J. Korean Math. Soc., i_~4, 41-63 (1977). G. Stephenson, "Affine field s t r u c t u r e of g r a v i t a t i o n and e l e c t r o m a g n e t i s m , Nuovo Cimento, 1 0 , 3 5 4 355 (1953). G. Stephenson, " L a g e o m e t r i e de E i n s l e r et les theories du c h a m p unifie," Ann. Inst. H. P o i n c a r e , 1__55, No. 3, 205-215 (1957). G. Stephenson and C. W. K i l m i s t e r , nA unified field theory of g r a v i t a t i o n and e l e c t r o m a g n e t i s m , " Nuovo Cimento, 10, 230-235 (1953). Hiroshi Yasuda, "On extended Lie s y s t e m s . III (Finsler s p a c e s ) , " T e n s o r , 2_33, No. 1, 115-130 (1972). H i r o s h i Yasuda, " F i n s l e r s p a c e s as distributions o n R i e m a n n i a n manifolds, n Hokkaido Math. J . , 1, No. 2, 280-297 (1972). Hiroshi Yasuda, "On F i n s l e r s p a c e s with absolute p a r a l l e l i s m of line e l e m e n t s , n J. K o r e a n Math. Soc., 1_33, No. 2, 1976-1992 (1976). Hiroshi Yasuda and Hideo Shimada, "On R a n d e r s s p a c e s of s c a l a r c u r v a t u r e , " Rep. Math. P h y s . , 11, No. 3, 347-360 (1977).

GEOMETRY A.

M.

OF NONLINEAR

DIFFERENTIAL

EQUATIONS UDC 514.763.8

Vinogradov

The p a p e r contains a s u r v e y of c e r t a i n c o n t e m p o r a r y concepts and r e s u l t s connected with the geom e t r i c foundations of the theory of nonlinear partial differential equations. At the base of the account is situated the g e o m e t r y and a n a l y s i s on jet s p a c e s , finite and infinite.

INTRODUC

TION

1. In this s u r v e y we c o n s i d e r c e r t a i n g e o m e t r i c ideas and r e s u l t s connected with the c l a r i f i c a t i o n of the foundations of the theory of nonlinear differential equations. Our a p p r o a c h to the f o r m u l a t i o n of the p r o b l e m as well as the method of its p r e s e n t a t i o n is based on the g e n e r a l o b s e r v a t i o n that any " g e o m e t r y " is the r e a l i z a tion of a c o r r e s p o n d i n g " a l g e b r a . " This kind of " a l g e b r a n in the c a s e c o n s i d e r e d is a s y s t e m of f u n c t o r s , s e r v ing as the theory of differential o p e r a t o r s in c o m m u t a t i v e r i n g s , which it would be a p p r o p r i a t e to call simply the differential calculus. F o r this r e a s o n , the p a p e r begins with the n e c e s s a r y d i g r e s s i o n into the so u n d e r stood differential calculus, in whose language the r e s t of the account is conducted. This is not only a question of convenience or s t y l e , but is of principal m o m e n t , and in p a r t i c u l a r , in counterweight to C o u r a n t ' s r e m a r k * *Questions connected with p a r t i a l differential equations of o r d e r higher than the f i r s t a r e so v a r i e d that the c o n s t r u c t i o n of a unified theory does not s e e m possible. Courant, P a r t i a l Differential Equations [Russian t r a n s lation], M i r , Moscow (1964), p. 159.

Translated from Itogi Nauki i Tekhniki, Seriya Problemy

1624

Geometrii,

Vol. ii, pp. 89-134, 1980.

0090-4104/81/1701-1624507.509 1981 Plenum Publishing C o r p o r a t i o n

to observe

the essential oneness

of differential mathematics.

2. The geometric theory of differential equations in the contemporary sense has its origins in the classical works of Lie (cf. [51, 52]), in which there is created a complete theory of first order equations and the foundation is laid for the classical theory of symmetry. Cartan indicated the importance of invariant methods in the general theory of differential equations and realized his program of invariantization of this theory into the geometric language of vector fields and differential forms. Coordinates, however, still play a noticeable role in Cartan's works, which, for example, is apparent in the fact that he never gave complete proofs of some of his deepest results (cf. [18, 32, 33, 29~ 46]). In the 50's and 60's Cartan's theory underwent further invariantization and refinement. A central place here is occupied by the papers of Kuranishi (el. [48-50, 45]). Approximately at this time the language of jets introduced by Ehresmann [35] was transformed from the language of "good form" into a useful working language for the theory of differential equations, which, for example, led Goldschmidt to a considerably- more satisfactory formulation of the Cartan-K~hler existence theorem and the Cartan-Kuranishi theorem on extensions (el. [37, 38]). An important moment here was the opening of the mechanism of Spencer cohomology (cf. [30]), which, as is now clear, is an important component part of the differential calculus in the sense indicated above. As a result of this development it became obvious that manifolds of jets are the natural geometric base of the theory of differential equations and this circumstance is one of our starting points. It is useful here to turn our attention to the space of tangent elements, which is the foundation of Lie's theory, and is nothing else than the manifold of jets of the first order. We emphasize, finally, the general impact of the papers of Spencer and Sternberg (e.g., [53]) on the questions we are considering. 3. The relation between the volume of material and the size of the paper defined its style, which is close to telegraphic: formulation of definitions and results plus minimal clarification. For this reason, in many places we could not mention the necessary motivations and examples, indicate the useful applications known at the time the theory developed (propagation of discontinuities, Cauchy's problem, Hamiltonian formalism in field theory) and describe its many and interesting connections with other domains of mathematics and mathematical physics. Implicitly the paper is divided into three parts. In Sees. 1 and 2 the language is described, Sees. 3-7 are essentially devoted to the geometry of submanifolds of manifolds of jets, and Sees. 8-10 to the geometry of infinitely extended equations. This last "nonclassical" part is the most important, since the objects of the category of nonlinear differential equations turn out to be precisely infinitely extended equations. Here the language of Sees. 1 and 2 is used essentially, while for the second part the ordinary language of the theory of smooth manifolds is completely sufficient. References to the literature and brief remarks of a priority character are given at the end of each section. Here we do not pretend to completeness and objectivity. In connection with this we refer to the survey [2], where the extensive literature is cited, and also to Forsyth's work [36], from which one can get not only a good representation of the development of the domain of interest to us in the last century, but also derive an inspiring formulation of a whole series of problems. The B~cklund transformation is by no means the on_iy example of the oblivion of beautiful geometric ideas left us by the classics. Finally, we note the papers [39-42, 31, 47], closely connected with the questions examined below. 4. The geometric foundation of the theory of nonlinear differential equations has been considered during the last several years in the seminar of workers on the mechanico-mathen~atical faculty of Moscow State University under the guidance of the author. In this paper, essentially, there is elaborated the point of view which results from these discussions, which explains a certain one-sidedness in our references. Now, when one, apparently, can assume that a return to foundations was justified, the author thallus the participants in this seminar, and in particular, V. V. Lychagin, B. A. Kupershmidt, and I. S. Krasil'shchik for the necessary optimism and enthusiasm. I.

Differential

Operators

in

Commutative

Rings

i.I. The hope of constructing differential operations on spaces of sufficiently general character, which arise in the successive constructions of the geometry of differential equations, as well as the general considerations of paragraph 0.2, leads to thenecessity of extending the boundaries of the classical differential calculus to smooth manifolds. Such an extension turns out to be possible if one follows the general algebraic point of view described below.

1625

1.2. Let K be a c o m m u t a t i v e ring with unit and A be a unital commutative K-algebra. F o r an element a ~ A and a K - h o m o m o r p h i s m A : P - - Q , w h e r e P and Q a r e A - m o d u l e s , one can c o n s t r u c t a K - h o m o m o r p h i s m 5a(A) : P - - Q , by setting ~ (A) (p) = A (up)-- aA (p),

p~P.

The o p e r a t o r 6a : HOmK(P , Q) --* HomK(P, Q) a r i s i n g in this way is a K - h o m o m o r p h i s m and 6 a o6 b = 5bO 6a. Let 6a0,at,...,as =6a0O6alO...O6as. Definition. A ~ HomK(P, Q) is called a K-differential o p e r a t o r (d. o.) of o r d e r _<s over the K-algebra A, if for any a0, . . . , a s ~A, 6 a o , . . . , a s ( A ) -- 0. 1.3. Let M be a s m o o t h manifold, F -- R or C, C~(M) be the F - a l g e b r a of F - v a l u e d C ~ over M. We denote by F(~)(Floc(~)) the C~(M)-module of all (all local) C ~ of the smooth v e c t o r bundle $: E~ M. In the c a s e when K = F, A = C~(M), P = F(~), Q = F(~), the definition of p a r a g r a p h 1.2 is equivalent with the usual definition of a linear differential o p e r a t o r acting f r o m the vector bundle ~ to the v e c t o r bundle 7. The situation d e s c r i b e d is later called the c l a s s i c a l case. 1.4. The collection of all d. o. of o r d e r --_s, acting f r o m P to Q, equipped with a left A - m o d u l e s t r u c t u r e [i.e., (azX)(p) = aA(p), a E A, p E P], wilI be denoted by Diffs(P, Q). This same collection of d . o . , equipped with a right A - m o d u l e s t r u c t u r e , ,is denoted by Diff~s(P, Q) [i.e., (a +A) (p) = A(ap)], and the A-bimodule which a r i s e s as a r e s u l t is denoted by Diffs~)(P, Q). Since Diff~+~ (P, Q)~Diff~ +> (P, Q) if s _< t, there is defined the A - b i m o d u l e Diff (+)(P, Q) of all K - d i f f e r e n tial o p e r a t o r s f r o m P to Q, filtered by the A-bimodules Diff(+) (P, Q). It is useful to keep in mind that the maps of A - m o d u l e s i+

i+

D~L (P, Q)~-D~ + (P, Q) ~d D~fff (P, Q)~-D~ (P, Q), generated by the identity maps of the supporting sets a r e d. o, of o r d e r -< s. 1.5. The d. o. introduced in p a r a g r a p h 1.2 a r e subject to the s a m e general rules as o r d i n a r y d. o. Here a r e some of them. 1) An o p e r a t o r of o r d e r z e r o is an A - h o m o m o r p h i s m and c o n v e r s e l y , i.e., Diff~+)- (P, Q) -- HomA(P, Q). 2) I f / h 6 Dills(P, Q), A2 E Difft(Q, R), then z~2 oA1 ~ Diffs+t(P , R). [This follows directly f r o m the rule 6a(A2 oA1) -- 6a(A2)oA 1 + a 2 O6a(A1) , A 1 ~ HomK(P , Q), A2 ~ HomK(Q , R).] In p a r t i c u l a r , the c o m p o s i tion operation turns Diff(P, Q) into a right Diff(P, P)-module and a left Diff(Q, Q)-module, and Diff(P, P) into an a s s o c i a t i v e A - a l g e b r a . 3) Let A' be some localization of the K - a l g e b r a A, A ~ Diffs(p , Q). Then there exists a unique extension A' : P ' ~ Q ' of the o p e r a t o r A to the c o r r e s p o n d i n g localizations P ' , Q' of the A - m o d u l e s P and Q. In other w o r d s , if the o p e r a t o r A is u n d e r s t o o d as an o p e r a t o r on the "manifold" SpecA, then it can be r e s t r i c t e d to any open set of this "manifold." 4) Let @:A 1 - - A 2 be a h o m o m o r p h i s m of K - a l g e b r a s , P and Q be A2-modules and A e Diffs(P , Q). Then is a d.o. over A 1 if P and Q a r e understood as A l - m o d u l e s by means of @. In other w o r d s , the concept of d. o. is invariant with r e s p e c t to the operation of change of rings. 1.6. We set DiffsQ = Diffs(A, Q), DiffQ = Diff(A, Q), and analogously, for Diff+Q, Diff+Q, Diff~+)Q, Diff (+) Q. Let further f~ = J1 (Q): Diff Q ~ Q be defined by the equation ~ (A) ~ A (1), ~ - ~ID~ff,Q , and ~+ = fl[oi+, 3I~ ~fbi+]mf~+ Q. Then ~ . ( ~ ) is an A - h o m o m o r p h i s m , while j1 + is a d. o. of o r d e r < s. This is evident f r o m the last r e m a r k of P a r a g r a p h 1.4. With any ~ E Diffs(P , Q) one can a s s o c i a t e Ah 6 HomA(P, Diff~Q), by setting zxh(P)(a) = A(ap), a ~ A, p ~ P. Proposition. The c o r r e s p o n d e n c e s A -- Ah and ~h~il, oAh a r e mutually inverse i s o m o r p h i s m s of the A modules Diff +(p, Q) and HomA(P, Diff + Q), p r e s e r v i n g filtrations. Let A~Diif~(P, Q), A~/=Ao~+(P), h o m o m o r p h i s m of right D i f f A - m o d u l e s .

~ht~z~(~)h:Di[L+P~Diff+~Q . Then~h*~limah~:Di[~+P~Diff+Q is a ~

In the special case when A ~ + ( P ) , we denote the h o m o m o r p h i s m .~h~:DiW~(DiII+P)~Di~++~P by Cs, t and we call it the gluing o p e r a t o r . Here s and t can be a s s u m e d to be equal to ~ in the obvious sense.

1626

We

identify Diff 0 P = Diff~ P = HomA(A,

A E Diff P, leads to a representation Analogously, Di~f~P=P|

P) with P. Then

of Diff P as a direct sum

the decomposition of A-modules:

A = (A-

P|

P,

A(1)) + A(1), where

where

Diff P = ker f~ (P).

Diff~P~ ker ]1~ (P).

1.7. The A - m o d u l e D (P)=Difi~P consists of all K-differentiations c~of the a l g e b r a A with values in P. In the case when A = C~(M), D(A), as is known, can be identified with the CF(M)-module D(M) of all s m o o t h vector fields on M, which we shall do in what follows. We define, f u r t h e r , the A - m o d u l e Ds (P) as the eollection of P - v a l u e d K - s e m i l i n e a r functions q)(a~. . . . , as) , a i E A, which a r e s k e w - s y m m e t r i c and have the p r o p e r t y that the operation a ~-~ q~(a, a 2 , . . . , a s) is a K-differentiation for any fixed collection a2, . . . , a s e A. It will be a s s u m e d that D0(P) = P and D(P) = D1 (P). There is another useful definition of the modules Ds (P) clarifying their value. Let D (@~_Q)={v~D (Q) lirnv~@}, where Q1 is a Subset of the A - m o d u l e Q, and (Diff+) (P) = D i f f ' ( . . . D i f f ' ( P ) . . . ) (i times). Then the modules Di(P) a r e defined inductively: D ~ ( P ) - D (D~_~(P)~(Diif~+) z-' (P)). Here the inclusion D~ (P)~(Dtff+)~(P) (which is not a h o m o m o r p h i s m of A - m o d u l e s ! ) is the following composition of natural inclusions: D; (P)-- D (D,_, (P)~(Diif+) '-~ (P))~D (DiH~+)'c~ (P)~(DiH~y (P). It is obvious f r o m the f i r s t definition that the inclusion Di (P)crzD~_~(D) (P)) together with the inclusion Dr (D (P))~..D~_~(Dill+P} r e d u c e s t o t h e inclusion De (P)~D~_~ (Diff~P). Diff is the Spencer o p e r a t o r s Si, j for the module P defined as the c o m p o s i t i o n .4-

Thus,

there arises the sequence Si,

0

§

-[-

D~-l(cl,

(SIP): 9

§

3i--i, I

O~ D~ (P)--~D~_~ (Dlff~ P) . . . . .

+

~ D~_k (Diffk P)

Si--#

'

k_~

. . . - + Diff+P-~ P-~ 0.

Proposition. I) The sequence SiP is a complex (Diff is the Spencer complex). 2) Si,j are d. o. of order _ l _> O. .

T H E O R E M . Let f : U ~ U ~ be a Lie t r a n s f o r m a t i o n and ~ = 0 o r l a c c o r d i n g as m > l o r m = l o Then i f U is c o n n e c t e d and the f i b e r s of the p r o j e c t i o n s a r e c o n n e c t e d , then f = g ( k - a ) w h e r e g : ,~k,e(U) ~ ~k,a(U ~) is a Lie t r a n s f o r m a t i o n . We note that a Lie t r a n s f o r m a t i o n of d o m a i n s f r o m N ~ = N is s i m p l y a d i f f e o m o r p h i s m (the c a s e e = 0), and o f d o m a i n s f r o m N1 is a c o n t a c t t r a n s f o r m a t i o n (the c a s e ~ = 1). A Lie t r a n s f o r m a t i o n p r e s e r v e s the f i b e r s of the p r o j e c t i o n ~k,k-1 in the n o n e x c e p t i o n a l e a s e s i n c e they a r e i n t e g r a l manifolds of the C a f t a n d i s t r i b u t i o n of m a x i m a l d i m e n s i o n ( C o r o l l a r y 5.6), and h e n c e g e n e r a t e s T~ a Lie t r a n s f o r m a t i o n lrk,k_l(U) ~ 7rk,k_l(U'). R e p e a t i n g this a r r a n g e m e n t , one c a n " d e s c e n d " to N m. This i s , r o u g h l y speakir~g, the r e a s o n f o r the validity of the t h e o r e m . The e a s e n = m = 1 r e q u i r e s d i f f e r e n t a r g u m e n t s . 5.9. A n i n f i n i t e s i m a l v a r i a n t of P a r a g r a p h 5.8 c o n s i s t s of the following. Definition. We call a v e c t o r field XoD(U), UcN~ ~ a Lie field if the (local) t r a n s l a t i o n o p e r a t o r s along t r a j e c t o r i e s of this field a r e Lie t r a n s f o r m a t i o n s . If A t a r e the t r a n s l a t i o n o p e r a t o r s m e n t i o n e d , then one c a n c o n s i d e r the Lie t r a n s f o r m a t i o n A~s) and s e t X (~)= d (A ~) )* It=0. Then X (s) is a Lie field on '~:~+s,k (u), c o v e r i n g the Lie field X. THEOREM.. Let X be a Lie field on UcN,~ h and let the h y p o t h e s e s of T h e o r e m 5.8 be s a t i s f i e d . X = y ( k - a ) , w h e r e Y is a Lie field on ~k,~(U).

Then

As in P a r a g r a p h 5.8, it is useful to keep in mind that an a r b i t r a r y field on N ~ = N is a Lie field, and c o n t a c t fields on N~ and Lie fields a r e the s a m e . In p a r t i c u l a r , the natural, lifts of H a m i l t o n i a n fields on T*(M) to Ji(l M) are Lie fields. The commutator of two Lie fields is again a Lie field, so that the Lie fields on 5/~N,,k form a Lie algebra, isomorphic under the conditions of Theorem 5.9 to the Lie algebra of all vector (contact) fields on ,~k,0(U) 0rk,i(U)). 5.10. In the case when U~J~(n) and T is linear, it is convenient to characterize Lie fields by their "generating functions." The generating function for a Lie field X is the section h=X~UI (n)~F(~I* (,~)). The field X is uniquely determined by this section h. If m = I, then any section from F(~*(~)) ~C~(]:(~) =:Y-~(n) can figure as a generating function. If m > i, this is not so. Proposition_:. A section h ~ F(~(,~)) is a "generating function" for some Lie field if and only if for any 01 and 02 ~ ji(~) such that ,xl,0(8i) = ,~i,0(02) = 0, one can find a vector ~ e Tx(M) , x = n(0), depending only on 0, such that h,(O1)--h(O2) = ~ J ( 0 1 - - 0 2 } , w h e r e 01 - 02 is u n d e r s t o o d as a n e l e m e n t f r o m kerr1, 0[~=Al(M)|

On the c o l l e c t i o n G ( n ) ~ r (.~ (n)) of all g e n e r a t i n g functions, t h e r e is a Lie a l g e b r a s t r u c t u r e (th, h 2) ~-{hi, h 2 }. H e r e {h~, h 2 } d e n o t e s the g e n e r a t i n g function of the c o m m u t a t o r [Xt, X2], w h e r e hi is the g e n e r a t i n g function of X i. 5.11. Any Lie X on U~Y~(rO d e t e r m i n e s a in time along t r a j e c t o r i e s of the field X, then f o r and h e n c e has the f o r m (locally) slk)(M) f o r s o m e m a k e s s e n s e . The following p r o p o s i t i o n indicates

" o n e - p a r a m e t e r g r o u p ~ in gloc(Tr). If A t : U ~ U ~s t r a n s l a t i o n any s e Floe (=), the manifold At(s (k)(M)) is an R - m a n i f o l d s t e Floc0r). If 7r is l i n e a r , then the d e r i v a t i v e ~ = d s t / d t l t - 0 the m e a n i n g of the g e n e r a t i n g function.

1635

Proposition. If the bundle ~ is l i n e a r , then s ~ j~ (s)* ( X ~ U ~ (~)). More g e n e r a l l y , Jk-~ (s)

=

7~ (s)* (X_~U~ (n)).

COROLLARY. The Cauchy p r o b l e m for the evolution equation st = A(s), w h e r e s ~ F(v), A E dif(~, ,~), has a unique solution for any initial data if and only if q~A is a g e n e r a t i n g function. F o r any g e n e r a t i n g function h t F(~(~)) the c o r r e s p o n d i n g Lie field X, c o n s i d e r e d on jl(~), is tangent to the equation {h=O}~]~(~). H e r e the t r a j e c t o r i e s of the field X a r e the c h a r a c t e r i s t i c s of this equation. Syst e m s of f i r s t o r d e r equations of the f o r m {h = 0}, w h e r e h is a generating function, f o r m p r e c i s e l y the c l a s s of equations d i r e c t l y i n t e g r a b l e by "the method of c h a r a c t e r i s t i c s . " The s o - c a l l e d method of R i e m a n n i n v a r i ants in extended i n v a r i a n t f o r m r e c e n t l y s e t f o r t h by Lychagin [12] allows one to extend this c l a s s considerably. The theory of generating functions for Lie fields X on N k is c o n s t r u c t e d analogously: a generating function for the field X is a s e c t i o n U~(X) E F(/t). In this c a s e the Lie field X on U ~ N m k in a s i m i l a r way d e t e r mines a " o n e - p a r a m e t e r group" on the s p a c e of all n - d i m e n s i o n a l submanifolds in N and the e l e m e n t jk(L)* (U~(X)) (cf. 5.4) should be i n t e r p r e t e d as the speed of change of the ( k - 1)-jet of the submanifold L along the flow c o r r e s p o n d i n g to this o n e - p a r a m e t e r group. 5.12. It follows f r o m what was said above that U - t r a n s f o r m a t i o n s (U-flows) a r e Lie t r a n s f o r m a t i o n s (Lie fields). In p a r t i c u l a r , the r e s u l t s of P a r a g r a p h s 5.10 and 5.11 a r e also valid for U-fields. The generating functions of all possible U-fields f o r m a s u b a l g e b r a G U (,~) in the a l g e b r a G (~). These generating functions can be d e s c r i b e d c o m p l e t e l y analogously to P r o p o s i t i o n 5.10, in which the words "depending only on 0" m u s t be r e p l a c e d by "depending only on x." 5.13. C o m m e n t a r y . The f o r m s dp~ - Zp~+ldXi, oz and e s p e c i a l l y d u - pidxi, defining the C a r t a n d i s t r i b u tion, still figure in v a r i o u s s p e c i a l situations as useful working tools in the c l a s s i c a l w o r k s on differential g e o m e t r y . E. C a r t a n used t h e m s y s t e m a t i c a l l y in his p r o c e d u r e of reducing s y s t e m s of differential equations to Pfaffian s y s t e m s . The C a f t a n point of view, which was e x t r e m e l y i m p o r t a n t and fruitful in his t i m e , is now, however, psychologically awkward, in view of the fact that the functorial a s p e c t s of the theory, c l e a r l y viewed in the language of jet s p a c e s , a r e lost in the throng of equivalent Pfaffian equations. The t e r m i n o l o g y and g e n e r a l a p p r o a c h adopted in this section a r e due to the author (cf. [5, 6, 11, 12]). Operations of the type U~ in v a r i o u s s p e c i a l c a s e s o c c u r r e d e a r l i e r in p a p e r s on the theory of connections and G - s t r u c t u r e s (cf. also [43]). T h e i r g e n e r a l definitions and i n t e r p r e t a t i o n s as "nonlinear Uk(,~)" a r e due to K u p e r s h m i d t [21]. V. V. Lychagin found the r o l e of the bundles lk. To him, a p p a r e n t l y , is also due the d e s c r i p tion of the m a x i m a l involutive s u b s p a c e s in C 0 (cf. [11, 12]). F o r m o r e complete i n f o r m a t i o n in connection with P a r a g r a p h 5.4, cf. [11, Chap. 3], and in connection with P a r a g r a p h s 5.5 and 5.6, Chap. 4 there. T h e o r e m s on the s t r u c t u r e of Lie fields and t r a n s f o r m a t i o n s in a local v e r s i o n w e r e known a p p a r e n t l y to Beckiund a l r e a d y (this is t r i v i a l l y so for the c a s e k = 2, m = 1). B. A. K u p e r s h m i d t , not knowing this, proved them by d i r e c t calculations in 1972 for k = 2 and m a r b i t r a r y (unpublished). I m m e d i a t e l y a f t e r this the author found a s i m p l e g e n e r a l proof for a r b i t r a r y manifolds m, n, and k, whose idea is indicated in P a r a g r a p h 5.8 (cf. [5, 6, 11, 21]). T h r e e y e a r s a f t e r this the local v a r i a n t of these t h e o r e m s was again given in I b r a g i m o v and A n d e r s o n [17]. The p r o p o s i t i o n and c o r o l l a r y of 5.11 a r e due to the author [6, 11]. We s t r e s s , finally, that Lie fields and t r a n s f o r mations w e r e called c l a s s i f y i n g in [5, 6, 11, 12, 21]. 6.

Characteristics,

Deviations,

Singularities

6.1, In the fibers of the projection ~k,k-I one can define in an invariant way with respect to Lie transformations certain geometric structures. In this section we describe some of them. We consider, firstly, a group G of Lie transformations of the manifold N~ I, having 0 E N~ I as a fixed point and_preserving the fiber ~r , 0 = 7rk_i,k_2(0). Let G -- {g(01g ~ G} be the lift of G to N k (cf. 5.8) andG O=GI

~!k_1(~),

Proposition. In ~ik_ I(0)(~1~l) one can introduce in a unique w a y an affine structure such that G 0 becomes a subgroup of the full affine group (with respect to this structure) containing all translations. COROLLARY. The fibers of the projection ~k,k-1 : N k - Nk~ I have a natural affine structure if a) k > I, m > I or ~ k > 2 , m = i. The fibers of the projection ~k,k-1 :jk(~) _ jk-1(~) have an affine structure invariant with respect to U-transformations if k m ~ i. 6.2. With any subspace P~ T~(N~ ~-~ ) one can associate submanifolds S~ (P) ~ {@~N~ I.~k,~-~ (O)~ 8, dim P N L0 i}, i = 0, i, .... n, forming a stratification of the fiber ~l,k_1(~). In particular, the submanifolds Si(0) = Si(L0),

1636

0 ~ N k called characteristic cones play an important role. Analogously, if V C N,n is some subrnanis set S z (V)= {0EN~+~]nk+~.~ (0)GV, dim Lo ~ T-~(V) ~ i, -0= =k+~,k (O)} . Let further OGS~(V), O~nk+~. k (O) and L~=

(dnk+~, k I o)-:(LonT~(V)) 9 We

set S~j(V) ~ (O'ENk, +2 IO ~ nk+2, k+~ (O')CS~(V), dim L01NL~ = Jl. Analogously

we

one can

define "manifolds" Si,..z,c=~, (V). If on N k there is given some foliation ~', then if the corresponding conditions on .• k+s are satisfied, there arises a stratification consisting of strata S~ ......~(S)-- U S~ ......is (V), where V is a leaf of the foliation ~.

Here any submanifold

L~N

decomposes

into pieces j~+~ (L) -~ (L( ~+~I (] S~ ......~ (f)),

which can be called Thom-Boardman singularities relative to ~'. In fact, the ordinary Thom-Boardman singularities are the special case of the construction described when N~n = N = Nt x N 2 and f consists of the fibers of the projection N~ • N 2 ~ N 2. We note in conclusion that if E~N~ is some equation, then Sn,n," . .,n(E) (s times) is its s-th extension. 6.3. Now we consider the singular points sing ~ of the R-manifold .~N~ or jk(,~) for mk ~ i, i.e., the points 0@,R, where the rank of the mapping ~n~.~_~ i~ is less than n, which in the ease when 5~Y ~ (~n) is equivalent with the fact that the rank of the mapping

~]~

is less than n, or that no neighborhood

of the point

0 in ~ can be represented in the form s(k)(M), s ~ Floc(,n) (cf. 5.1, 5.7). The case m = k = 1 in view of Theorem 5.8 is exceptional in this respect and was considered in [i, 16]. We denote by sing, ~ the collection of those 0@~ at which the rank of the map .~,k-t L~. drops by I. Then if O~sing~ , then in view of 5.6, T~(~)~V , where V is a maximal involutive subspace in C 0 of type n- I. Here d~,~_~ (To (.~))~ ~n~.~_~ (V). Hence, we call an n-dimensional subspace P~V satisfying the latter condition an /-deviation. Proposition. I) Any/-deviation P is realizable, i.e., we can represent it in the form T~(~) for some Rmanifold ~. 2) The space of orbits of l-deviations with respect to the action of the group of Lie transformations preserving the point 0 coincides with the space of orbits of the group GL (l, R) | GL (n~, R), acting naturally on S ~ (l~)|

~

6.4. One can pose the question of the structure of the germ of an R-manifold close to a singular point 0. A. P. Krishchenko described germs ~ such that 0~sing~, and indicated a certain procedure for studying the general case. It follows in particular from his results that the germ of any closed set without interior points in R can be realized as the germ of a set sing~. 6.5. Commentary. The subjects of this section, it is clear, represent only the initial fragment of a theory which could be developed to a very high degree. In particular, it would be quite useful to reconsider the theory of singularities of smooth mappings, placing in it the foundations of the theory of the Caftan distribution. The treatment given in 6.2 of the Thorn-Boardman singularities and their generalizations illustrates the simplifications which can be achieved along this path. The problem of realization and classification of l-deviations is an example of the new interesting problems. The property of affineness of the leaves ~l,k_ 1 (0) was noted long ago (e.g., [38]). In 6.1 a more precise formulation of it is given, cf. [Ii] for details. The construction of "cones ~' given in 6.2 is due to Krasil'shehik (cf. [ii]). The results of Paragraphs 6.3, 6.4 are dueto Krishchenko; el. [19, 20], where one can find further details. 7. Elementary Intrinsic

and

Geometry Extrinsic

of

Nonlinear

Equations.

Geometry

7.1. In this section there are first tion E~Nm h [or jk(~)] from the extrinsic question of which of them are "intrinsic" we mean the geometry of the distribution emphasize that this point of view, if one equivalent with the representation of the

considered the simplest geometric characteristics of a nonlinear equapoint of view, i.e., E is considered as a submanifold of Nkl. Then the invariants is discussed. By the intrinsic geometry of the equation E CE on E, which is the restriction of the Cartan distribution to E. We does not consider the psychological preferences mentioned in 5.3, is equation in the form of a system of Pfaffian equations.

7.2. We recall that L~N (s~F1oc(n)) is called a solution (in the usual sense) of the equation E if L(k) (s(k)(M)) lies on E. It is useful, however, to extend the concept of solution, just as in the theory of linear equations. A geometrically obvious method for doing this consists of the following. E

De____finit__ ion__z~ We call a map c~: L ~ N [or J~ if for any x~L one can find a neighborhood ~x,

where dimL = n, a (generalized) solution of the equation a number s -> 0, and an imbedding ~U: U ~ Nkm+S(jk+s(~)),

1637

such that ira/3 U is an R - m a n i f o l d lying on E s. R i e m a n n s u r f a c e s , which a r e muitivalued solutions of the C a u c h y - R i e m a n n s y s t e m , can s e r v e as illustrations of this definition. R e m a r k 1. If E ~ ] k ( ~ ) , then s i m i l a r l y one can understand as a g e n e r a l i z e d solution a partially defined multivalued section of the bundle ~. Hence, in the case of a linear equation E with such a solution one can a s sociate an o r d i n a r y generalized solution by combining (if possible) its s e p a r a t e branches. This constitutes a naive connection with the linear theory. Here to a singular c a r r i e r in the linear theory there c o r r e s p o n d singular points of the R - m a n i f o l d i m ~ U. We note that the number s, figuring in the definition, shows that in a neighborhood U ~ x the derivatives of order k + s + 1 are discontinuous if singim~ U ~ (7). W e recall that the submanifold IE'cNlm, l 1) is quasilinear. It is a p p r o p r i a t e to call the orbit of the subspace g(0) with r e s p e c t to the natural action of the group of Lie t r a n s f o r m a t i o n s in the fiber 7~,k_l -~ ( ~ ) ( ~ S k (R~),| the type (typ0E) of the equation E at the point 0. This kind of space of orbits is, as a rule, multidimensional, although for sufficiently small k there occur situations with a finite number of types, e.g., second order equations in one independent variable. 7.4. The way in which g(0) intersects with the cones Si(0) (cf. 6.2) is an invariant of the type in its own right. A n y point 0'Cg (0)~ Si (0) determines an i-dimensional subspaee L0, n L0 , which means also an (n - i)dimensional subspace Ann (L0, (]L0) of L$. The collection of all such subspaces forms a suhmanifold charn_i(E , 0) of the G r a s s m a n manifold Gn,n_i(L~). If E<J~(=), then L0* can be identified naturally with Tx(M) , x = 7rk(0). The homology class of the submanifold chariE = U chah (E, 0)c [J On,z (Lo) is a topological invariant of the equation E. ~ ~ The direction vectors of the lines ~ ~ char t (E, 0) are the characteristic eoveetors (characteristics) of the equation E at the point 0. Proposition:charl(E~, 01)=(d=k,k_~ [0)* chah(E,0)

if ,Tk+l,k(0') = O.

7.5. We illustrate the connection between the manifold ehari E and the c h a r a c t e r of the solution of the equation E. We call the equation E elliptic if char 1E = 0. Proposition. If an R - m a n i f o l d is a g e n e r a l i z e d solution of an elliptic equation, then points of 1-deviations a r e m i s s i n g f r o m it. An elliptic equation can have as intermediate integrals only elliptic equations. These a s s e r t i o n s follow d i r e c t l y f r o m Propositions 5.6, 7.4 and the definitions. It is useful to c o m p a r e the f i r s t of them with the c l a s s i c a l r e s u l t a s s e r t i n g that a g e n e r a l i z e d solution of a linear elliptic equation is actually o r d i n a r y , in connection with R e m a r k 1 of P a r a g r a p h 7.2. The example of Riemann s u r f a c e s shows that there exist g e n e r a l i z e d solutions of elliptic equations, singular points of which r e a l i z e /-deviations, l > 1. 7.6. We t u r n now to the intrinsic g e o m e t r y of the equation E. Let O~.Co(E)=C~(~ F~(E), O~E, be the r e s t r i c t i o n of the C a f t a n distribution to E. The question d i s c u s s e d below consists of whether, knowing only the manifold E and the distribution C (E) : 0 ~-- C0(E), one can r e c o n s t r u c t the inclusion E c N ~ (jk (~)). In other w o r d s , when does the intrinsic g e o m e t r y of the equation E uniquely determine the extrinsic ? Equations for which the answer to this question is positive we call rigid. F i r s t we give the definitions needed. F i r s t l y , we call E C - g e n e r a l if ~k,k-ll E is s u r j e c t i v e and the fibers of the projections ~k,k-ll E a r e connected and only they, as before, a r e integral manifolds of maximal dimension in C(E). F u r t h e r , we say that E is C - c o m p l e t e , if for a l m o s t all 0 E N ~ t ( J k - t ( ; ) ) the linear span of the subspaces d~,~_, (Co (E)), 0 ~ , ~ - , (0) ~ E, g e n e r a t e s C ~. Finally, we call the equation E 1-solvable, if Et -- E is sur]ective. It is easy to see that the collection of these three conditions, called n o r m a l i t y , can be f o r m u lated in purely intrinsic t e r m s .

1638

THEOREM. A normal equation is rigid. The r01e of the conditions constituting the normality consists in the proof of this theorem, of the fact that C-generality allows one to reconstruct N~ I (jk-i (,~)), and C-completeness allows one to reconstruct the Cartan distribution on N~ 1(Jk-l(;)), while the 1-solvability guarantees the compatibility of C (E) and Ck_ 1 (N) (Ck_ I (~)). 7.7. We discuss the normality conditions so as to convince ourselves of their harmlessness. First of all we note that C-generality essentially means that the equation E is not too strongly overdetermined. In view of -< mCk_~l_ 2~.~` under the condition of the connectedness of the 5.6, for it to be satisfied it suffices that eodimE fibers of the projections ~k,k-ll E. For the sake of simplicity we formulate the condition guaranteeing C-completeness, assuming that E jk(;), where ,~ is linear. Let g0(0) denote the affine hull of =k,k-1(0)rlE -i in ,Xk,k_l(0) , understood as a subspace in S k (T*x(M))| ~ (x), x=~k_~ ~). We also set V 0 = { k e T~(M) I the map (smbl0~)(k~| (x)-~ -~ (x) is not epimorphic}, where 0 ~ E, x = ~k(E), and A e difk(~ , ,7') is such that E = {~A = 0}. Proposition. i) If for all 0e jk-l(~) and Y ~ Tx(M) , Y ~ 0, x = ,~k_l(0) one has K~(g0(0))=Sk-~(Tx(M))| 7r-l(x)~ then the equation E is C-complete. 27 If for all0~E

Vo~Y~(/d).

x==k(0),

then the equation E is 1-solvable.

We note in conclusion that among "ordinarily occurring" equations, only equations with one independent variable or first order equations with one dependent variable are rigid (cf. Theorem 5.8). 7.8. Now we consider the classification problem. We call equations E and E' equivalent if there exists a diffeomorphism f:E --E', carrying C(E) isomorphieally into C(E'). Analogously one defines local equivalence of equations near given points. Is the question of (local) classification with respect to this equivalence relation understood? If the equations E and E' are rigid, then the transformation f realizing their equivalence extends to a Lie transformation of their ambient jet spaces 9 Hence, for example, the extrinsic invariants of the equation E, constructed in Paragraphs 7.3, 7.4, are also invariants with respect to the equivalence relation considered. Even as crude an invariant as the function 0 ~ typ0 E (cf. 7.3) shows how "inconceivably" large a number of equivalence classes one gets here. Moreover, equations which have identical functions 0~-~ typ0E can be distinguished with the help of an invariant of the type of the structural functions in the theory of G-structures. Thus, the formulated classification hardly has practical value. However certain invariants arising in this connection can be useful for the qualitative characteristics of equations (cf. the curvature tensor). Thus, a reasonable classification can be expected only in classes of highly nonrigid equations. For nonrigid equations there may be many intrinsic (but not extrinsic) symmetries, guaranteeing structural simplicity (cf. end of Paragraph 7.7). 7.9. Commentary. The definition of generalized solutions in Paragraph 7.2 was given by the author Proposition 7.3 was noted by Goldschmidt [38], and the construction of the manifolds chari(E ~ 0) is due to shchik [ii]. Proposition 7.4 is a nonlinear analogue (cf. [1117 of Proposition 6.4 of [37]. The first part of sition 7.5 is taken from [20] and the second from [12]. Theorem 7.6 is due to the author, and Proposition to the author and V. V. Lychagin. We note that it would be useful to consider the problems of the theory of G-structures the equivalence problem) from the point of view of Paragraph 7.8. 8.

Infinite

Nonlinear

Extensions Differential

of

Equations.

Category

[5~ ii]. Krasil'Propo7.7

(and, in particular,

of

Equations

8.19 Later we shall turn to the consideration of spaces E~o, which are inverse limits of sequences 9 . .--E s-. . .--E l-E 0=E of extensions of an equationE~Nh,,~ [or Jk(Tr)]. The need for this is connected with the fact that the fundamental operations and constructions of the theory "jump" from jet spaces of order k to jet spaces of higher order 9 Example: Ps, Us, A. Thus, a closed calculus can be constructed only in the space of infinite jets. For any L~_N the inverse limit of the maps jk(L) [or jk(s), s e Floc(,'r)] as k ~ ~ reduces to the map j(L):L--N~n [or j(s):M--J~176 We setimj(L) = L ~~ imj(s) = s~~ It is convenient for/F~N~(Yk(rQ) to introduce the not ation : E s = E s_ k for s -> k, E s = N s (js (~r)for k > s -> 0. Let is.t = is,t (E) = (~, t Is~)* :C~ (E') -+ C~ (E ~) We denote the direct limit of the sequence of homomorphisms under the natural inclusion in S(F) by ~s(E) . Thus, ~(E)

is+l# by ~(E), and the image of the ring C~(ES) is filtered by the subrings Ss(E). N% [or J~(r0]

t639

l a t e r denotes the i n v e r s e l i m i t of the sequence of mappings 7rk+l,k, and WIn(N) [or W (~) ] denotes the d i r e c t h m l t of the sequence of h o m o m o r p h i s m s ~k+l,k. O b v l o u s l y , E ~ N m (J (7)) , so that one can c o n s i d e r the ideals 2/(E) = {es [or $(~)]lq~Jz~=0} c F , ~ (N) [or W (n)]}, 2/= (E) = 2/(E) (] ~'~n(N) [or ~= (n) 1. The defined ideals •i (E) of the " m a n i f o l d " E = ~ E s , g e n e r a l l y speaking, differ f r o m E s, but on the other hand f o r m a f o r m a l l y integrable sequence of equations, having the s a m e supply of solutions as E. If E is f o r m a l l y i n t e g r a b l e , then is,t(E) a r e m o n o m o r p h i s m s , ~= (E) = C~ (E =) , and E s = ES. 9

9

9

~

'

ao

co

co

Despite the fact that E is a manifold, ES as well as ~ s can have s i n g u l a r i t i e s . Hence in o r d e r to cons t r u c t in c o m p l e t e g e n e r a l i t y a theory of differential o p e r a t o r s o v e r a l g e b r a s of the f o r m ~ (E) it is n e c e s s a r y to use the g e n e r a l a p p r o a c h developed in Sees. i and 2. 8.2. A differential o p e r a t o r A~Diff (P, Q), w h e r e P = { P i ) , Q = t Q i } a r e filtered modules, is called filtered if there exists an s such that ,A(P~)~Q~+=, i ~ 0 . Our c o n s i d e r a t i o n s l a t e r will be developed in the domains of the theory of filtered differential o p e r a t o r s in the c a t e g o r y of g e o m e t r i c filtered modules o v e r a filtered F - a l g e b r a ~ ( E ) . The s t a n d a r d notations for functors of the differential calculus and the objects r e p r e senting t h e m (cf. Sec. 2) a r e applied l a t e r only for objects of this theory. F o r e x a m p l e , Diff(P, Q) denotes the collection of all filtered o p e r a t o r s f r o m P to Q. We note that the c a t e g o r y we c o n s i d e r of 2~-(E)-modules is differentially closed (cf. 2.4), and r e p r e senting objects exist and a r e d i r e c t limits of the c o r r e s p o n d i n g objects o v e r the a l g e b r a s 9-~(E) as s - - ~r 8.3. The bundle ~s (ef. 3.2), r e s t r i c t e d to E s , we denote by ,Ts(E) and we s e t W= (E, ~)~=r(ns(E)). The h o m o m o r p h i s m s ~s,t g e n e r a t e , as above, maps i s , t = is,t(E, ~7): Wt (E, ri)-~W= (E, ~), whose d i r e c t l i m i t we denote by Y (E, ~1). It is obvious that S~(E,~) is an ~ (E)-module and these modules f o r m a filtration in F (E, ~). In the c a s e when E~Jk(.~) and ( : E ( -~ M is s o m e bundle, we shall w r i t e N=(E, ~) instead of ~ ( E , .n* (D). In this e a s e it is convenient also to s e t ~_~ (E, t) = r (1) and W-~ ( E ) = C~ (N). Also let N (~, g) ~--lira dir F(.~; (~)). A pair U ~ N , ~:U-+R", c o n s i d e r e d in P a r a g r a p h 3.1 d e t e r m i n e s a s y s t e m of affine c h a r t s J ~ ( ~ ) ~ N ~ , k -> 0, obviously c o m p a t i b l e with the maps '~s,t- Hence, passing to the i n v e r s e limit as k - - ~, we get an affine c h a r t J~ ( D ~ N ~ , , within whose bounds a r e defined coordinates xt . . . . . Xn, p~, w h e r e c~ = 1, . . . . m and runs through all possible collections (it . . . . . i s) (cf. 3.1). By an affine c h a r t on E :r we shall m e a n E~NJ~(~), w h e r e d~ (s is an affine c h a r t on N m. ~o

8.4. To w o r k with the C a r t a n distribution on the " m a n i f o l d s " E ~ in the g e o m e t r i c a l l y d e s c r i p t i v e manner adopted in Sec. 5 would be v e r y awkward. In this connection we introduce the operation ~ in the following way. Let 9 be a functor of the differential calculus (cf. 2.1) and @ = ~(E ~) be its r e p r e s e n t i n g object in the theory c o n s i d e r e d on E ~. A s s u m i n g that E ~ N ~ [or joo(,~)], we set: ~O)--{~p6fl~ i [j (L)* (~p)]( x ) = 0

whenever,

j (L) (x)6.E~}.

In p a r t i c u l a r , if L is a solution of the equation E, then c ~ ]c~=0. The modules ~I) a r e obviously i n v a r i a n t with r e s p e c t to the natural o p e r a t o r s of the differential calculus, connecting the r e p r e s e n t i n g o b j e c t s , the s c h e m e of whose a p p e a r a n c e is given in P a r a g r a p h 2.3. Hence, on the quotient objects ~ q ~ / ~ t h e r e a r e defined c o r r e s p o n d i n g quotient o p e r a t o r s , which will be denoted by adding a line above the s y m b o l of the original o p e r a t o r . F o r e x a m p l e , d ( ~ A ~ ) ~ A ~§ and d:X~-~Sj+L We call a submanifold V ~ E ~ i n t e g r a l if ~O)lv=0 for all r e p r e s e n t i n g objects ft. F r o m T h e o r e m 5.7 follows the following proposition. Proposition. A submanifold I Z ~ E ~~ is i n t e g r a l if ~ M l v ~ O . Any n - d i m e n s i o n a l integral submanifold has the f o r m V = L "~ w h e r e L is a solution of the equation E. T h e r e do not exist i n t e g r a l manifolds of dimension g r e a t e r than n. 8.5. The p r e s e n c e of the o p e r a t i o n ~ allows us to c o n s i d e r in the differential calculus on E ~ the following " s u b t h e o r y . " If, for e x a m p l e , 9 is a o n e - p l a c e functor of the differential calculus, then we s e t % ~ A n n % q b , w h e r e ff is a r e p r e s e n t i n g object of r More p r e c i s e l y , this means that for any y ( E ) - m o d u l e P, ~ ( P ) = {/z~Hom~-(e) (~, P) ] / z ( ~ ) ~ 0 } . F o r e x a m p l e , if 9 :Q ~-~ Diffs(P, Q), w h e r e P is a fixed and Q is a " v a r i a b l e " .~(E)module, then ~ ( Q ) - ~ Dill= (P, Q) c o n s i s t s of o p e r a t o r s which we shall call ~ -differential. ~ - d i f f e r e n t i a l o p e r a t o r s , like, however, other objects of ~ - t h e o r y too, a r e c h a r a c t e r i z e d by the fact that for t h e m one can define the r e s t r i c t i o n to any submanifold of the f o r m L ~~ Namely, if we s e t (AIz~)(Plc~)~ A (p)[c~, A6~ Diff (P, Q), p ~ p , then this definition is p r o p e r .

1640

~.

Dually there is defined a quotient theory, the representing objects in which, by definition, are modules Notation: ~ (P)~ Homs-(~ ) (~q), P)(~O (P)/~(P) if 9 is projective). For example, ~ ~ Diff s (P, Q) is a co-

set mod ~ Diffs (P, Q) of differential operators if P is projective. 8.6. Now we can informally describe what, in our idea, should be called the category of nonlinear differential equations (NDE). The objects of NDE will be filtered algebras of the form g-(E) ~ the differential calculus in which is equipped with the operation ~. Ring homomorphisms ~(E~)-+Sr(E2) , preserving the operation ~ , will be considered morphisms of this category. Clarifying the nature of these morphisms, we note that intuitively they can be considered as smooth mappings of the space of solutions of the equation E 2 into the space of solutions of the equation E l. Remark i. Although finally NDE is described algebraically, we stress the role of the geometric ations motivating the present definition and also playing an indispensable role in work with NDE.

consider-

Remark 2. As a matter of fact, the definition above is only the "kernel" of the true category of nonlinear differential equations. Iris clear immediately that it is useful to add sub- and quotient-objects, which appear automatically in the course of things, for example, the ~manifold of Cauchy data," etc. It is not possible for us to discuss these aspects in more detail here. 8.7. In the case when E~Yk(~) , which means E~J~(~) also, the presence of the projection ~i~:L "~-+ M, .~limproj ~k, allows us to furnish the differential calculus over ~(E ~) with additional structure - a horizontalization operation ~I, which is invariant under morphisms manifold M. By a horizontal module

on E ~ we mean

over M, filtered by the submodules

a module

of NDE,

of the form ~ (E)|

(n~), where

,~' is a linear bundle

I~ (E)Oc~(M)F (~). With any representing object 9 of a functor of the dif-

ferential calculus 9 on E one can associate its "horizontal part" ~0c~: representing

preserving the projection onto the base

~)o~(E)|

where

~(iM) is a

object for 9 on M; we identify ~(IV[) with (~IE~)*q)(M).

Proposition.

There exists a unique operation ~I~ 2~: ~)-+ (P0, having the property that (s~) * (H~p) ~ (s~) * (~),

COROLLARY. Example

~I2~ ~i.

I. q

(dfl~)-- ~ p~+~dx~; ~I(~0~Ao2)

~ ~I (~O~)Al (0~2); "I (A~)-- 0, / > n. The role of the operation q is

i

that it, in view of the following theorem, THEOREM.

splits sequences

~q> ~ kerH~ ~ im(l--~I~).

Consequently,

of the form ~(P+~0

0-+ ~(D-+ q)-+ ~-+ 0, and[ ~ =

go.

is generated by the image of the operator Js Ps. AnalExample 2. P~--~I -o Y~, " and the ~(E) -module ~ ogously, ~:~iod and ~A ~+~ is generated by the image of the operator d--d:A~-+A~*L Moreover, (d-d)I~(z)~U~. 8.8. In the situation of the previous paragraph we describe the module ~ Diff~ (P, Q). To this end, with any A ~ Diffs(P , Q) we associate an operator ~ e Diffs(P, Q) by setting: A(p) = (A, Ps(P)), where (., .) denotes the natural pairing of ~ (P) and Diff s(P, Q). Proposition. Diff~ (P, Q)}.

The operator A is uniquely determined

by the restriction A1P-i and ~ Diff (P, Q)=~IA@

This assertion follows directly from the results of the previous paragraph. COROLLARY. /~ = A and (A- A)(P_I) = 0. The identity A = ~ + (A- ~) shows, in view of what was said above, that there is a direct decomposition Diff~ (P, Q):~ Diff~ (P, Q)eDifff (P, Q), where [~Dif~ ~p( , Q)v~E(P_~)-0. The operators A introduced earlier have the property that ~as an ~(E) -module is generated by operators of the form A.

~ and A I~_~Q_~.

Moreover,

~ Diff~ (P, Q)

8.9. We consider, finally, the operation Ui< on E ~ which is introduced locally geometrically in Paragraph 5.4. For this, we consider the ~'(E) -module D(k)(p) of all differentiations of the ring ~(E) in the ~'(E) -~(~> (P)~{X~D (~) (P)I X - ~ = O, module P, which is considered as an ~-~ (E) -module too. We set, further, ~/~ v(o@~A ~ ]~} and D(~)(P):D (~) (P)/~D (~)(p). The image of the vector field X under the composition D(P) --D(k) (p) ~ ~(k) (p) we denote by U~(X). Here D(P) ~-~ D(k) (P) is the operation of restriction

X~+X !~_ (~). 1641

In the situation of 8.7 we s e t D(vk)(P) = {X E D(k) (P) IX IC ~ (M) = 0 }. Then a r g u m e n t s analogous to those used in P a r a g r a p h 8.8 prove the following proposition. Proposition: The o p e r a t i o n U k induces an i s o m o r p h i s m of modules D(vk)(P) and T)(k)(P) and a d i r e c t dec o m p o s i t i o n D (~) (P) = ~ D (k) (p)| ~) (P). In the case when

~ is linear, D(vk) (P) can be identified naturally with 5k (E, ak_0|

8.10. To conclude this section we shall briefly discuss the question of the structure of morphisms in oO NDE. Let EF~(Ni)~p i= l, 2, dimNi=nQ-m~. We call a morphism E:EF ~ E 2 nondegenerate if the homomorphism F* :Hn(E~)~'An(EF) has no kernel. Proposition. We emphasize

A nondegenerate

that degenerate

morphism

morphisms

F is uniquely determined

exist and the nondegeneracy

by giving the restriction F* l~ro(~,).

condition here is essential.

We call isomorphisms in NDE diffeomorphisms. If there,.,is given a diffeomorphism F s :E~ _~s s -> max k i preserving the Cartan distribution, where E i c (Ni)~i , then it extends naturally to a diffeomorphism F : EF ~ E~ ~ We call diffeomorphisms of this kind Lie transformations. Algebraically Lie transformations are characterized by the fact that the homomorphism F* preserves filtration. ao ~o THEOREM. If m i i, then any diffeomorphism F : (N1)ml ~ (N 2)m2 is a Lie transformation. For ml > 1 there exist diffeomorphisms which are not Lie transformations. Let ~i :E~ i ~ M be a vector bundle, Pi = F(~i) and A ~ Diff(Pl, P2) be o p e r a t o r s such that A -I ~ Diff(P2, Pl). If dim~ 1 = dim~ 2 > 1, then t h e r e exist s i m i l a r 9 + + o p e r a t o r s of d e g r e e g r e a t e r than z e r o . F o r e x a m p l e , let P1 = DiffkQ, P2 = DfffkQ and A = i+ : DiffkQ - - DiffkQ (cf. P a r a g r a p h 1.4). We c o n s i d e r , f u r t h e r , the map FA:J~~ --J=~ , uniquely d e t e r m i n e d by the condition that F A oj = j o A. Then FA is a d i f f e o m o r p h i s m which is not a Lie t r a n s f o r m a t i o n if the o r d e r of the o p e r a t o r A is g r e a t e r than z e r o . The natural nonlinear g e n e r a l i z a t i o n of this e x a m p l e gives the d e s c r i p t i o n of all diffeom o r p h i s m s in NDE.

R e m a r k . The r e s u l t s of this p a r a g r a p h and the next s e c t i o n show that v e c t o r fields on Nm, p r e s e r v i n g the C a r t a n distribution, a s a r u l e do not have local t r a n s l a t i o n o p e r a t o r s . Hence the " o r d i n a r y " a p p r o a c h to the c o n s t r u c t i o n of the theory of these fields (cf., e.g., [171]) is i m p r o p e r . 8.11. C o m m e n t a r y . The theory of this s e c t i o n is due to the author. F o r a m o r e detailed account, cf. [11, 12]. The o p e r a t i o n ~I on J~(~) for ,~ = Ai is s y s t e m a t i c a l l y used by B. A. K u p e r s h m i d t (cf. Chap. 1 of the s u r vey [26], containing an account of his r e s u l t s ) . He also gave for J~(,v) the d i r e c t decompositions A~=~A~| D : D (~I = ~ D |

~).

The space g~(J, not speaking now of E ~, apparently was not an object of independent study but was used loosely for solving various kinds of concrete questions, sometimes under a dressing of differential algebra. As characteristic examples, we cite [3, 15, 26, 54]. 9. Infinitesimal Evolutions,

Symmetries and

Higher

of Nonlinear

Equations,

Characteristics

9.1. Geometrically speaking, by an infinitesimal symmetry of a nonlinear equation E, if one considers this from the point of view of the category NDE, one should mean a vector field on E ~ preserving the Cartan distribution on E ~, i.e., the operation ~. If what was said is translated to algebraic language, then we get, in view of Proposition 8.4, the following. Definition. By an infinitesimal (intrinsic) symmetry of the equation E or a p-field is meant a differentiation X~D(Sr(E)) such that X(~AI)cIFA I. IX(A) denotes the Lie derivative of the object A.] The collection of all algebra D (5 (E)).

~-fields on E ~ we denote by

D~(E).

Obviously

Dc~(E)

is a subalgebra of the Lie

Proposition. a) Any ~-field X is uniquely determined by the restriction X]j-o(m, b) ~D (5 (E)) is an ideal of the Lie algebra D~ (E), c) D~ (E) coincides with the normalizer of the subalgebra ~D (5 (E)) in the Lie algebra D (y (E)). Remark. Condition b) signifies that the algebraic conditions of the Frobenius theorem are satisfied for the Caftan distribution on E ~. In this case, the Frobenius theorem itself is as a rule untrue, although there are important special cases, say Cauehy-Riemann systems, when it is valid.

1642

9.2. We introduce the s u b a l g e b r a sym E ~ D~ (Y (E))/~D (S (E)). A s s e r t i o n b) of P r o p o s i t i o n 9.1 signifies that the t r a n s l a t i o n o p e r a t o r s along t r a j e c t o r i e s of the field XE~D (9" (E)), if they exist, would leave i n v a r i a n t any integral manifold of the C a r t a n d i s t r i b u t i o n on E ~~ and hence would define the z e r o v e c t o r field on the " s p a c e " of i n t e g r a l manifolds, i . e . , the " s p a c e of solutions of the equation E." Thus, s y m E can be i n t e r preted as the Lie a l g e b r a of v e c t o r fields on this " s p a c e of solutions," and the ~ -fields f r o m ~}9(F(E)) c o n ~O aO s i d e r e d as trivial. We shall w r i t e ~ n, one has TIi(E) = 0 for i > n. C o n s e r v a t i o n laws which usually appear in physics and mechanics c o r r e s p o n d to the group TIn-I(E). The groups TIi(E), as will be shown below, for "good" equations for i < n - 1 coincide with Hi(E) and hence have a purely topological c h a r a c t e r . Finally, the group TIn(E) should be i n t e r p r e t e d (el. below) as the collection of all L a g r a n g i a n s inthe Lagrangian f o r m a l i s m with c o n s t r a i n t s , e x p r e s s e d by" the equation E. Hence l a t e r by c o n s e r v a tion laws one understands elements of the group TIn-I(E). If X ~ D ~ (E), pE~A ~, then by definition X (p)~A ~, and if d9 = 0 mod~A* and X ~ D (Y (s then X (,o) : d(X_~p)mod~A*~ Thus, the Lie derivative is well defined: %(h)= {X (p)}G/7~(E), h~N~(E), ~CsymE, where X =

X + ~ D (~ (E)), h =~)+~h ~. Proposition. servation law.

If )/ is a symmetry

and h is a conservation

law for the equation E, then •

is again a con-

10.3. We need the concept of conjugate operator in the class of ~ -differential operators. As a preliminary, we define the action of the operators A~Dilfy(s on forms co e-~n, which is denoted by Alto] ff~n and is uniquely determined by the following properties of it: i) c Icol:cco, c'GY(E), 2) X [co]~ --X (0~), XC~D (f (E)), 3)

(a~o~) [~1 =A~ [~ (~)]. We note, f u r t h e r , that ~ n is a locally free o n e - d i m e n s i o n a l ~- (E) - m o d u l e , since in an affine c h a r t ~ n = A 0- If co is a local g e n e r a t o r in ~ n , we define the o p e r a t o r 5~E~ Diif ~ (E) (in the domain of definition of ~o) f r o m the condition 5(T)=5~(~).~o and we set A* (f)-~5o[fm]. The right side of the last equation is independent of the choice of ~z and we call the o p e r a t o r 5 " ~ Diff A~ so defined the conjugate of A. Let Q ==Hem~- (~) (Q, A~). With any AE~ Diff (P, (~) we can a s s o c i a t e a family of o p e r a t o r s 5 (p, q)~Diff A~, w h e r e A(p, q)(f) = A(fp)(q), p ~ P, q ~Q. It is c l e a r that the family {~(p, q)} d e t e r m i n e s A uniquely. As the conjugate o p e r a t o r A* :Q ~ P in this case we define the family A*(q, p) = ~(p, q)*. It is obvious that (zx*)* = A and that ~ Diff A" ~ ~ Diff+A~:> ~ Diff A'~ a r e i s o m o r p h i s m s of ~ ( E ) - m o d u l e s . Thanks to the latter, there a r i s e is omorphis ms -- ~Di&) ~D~ (g Di*{~A") , "A gD~ (g Diff K~)-+ ~, Di~i gD,A'~-+~g DifI X=-~,

w h e r e the last a r r o w is a consequence of P o i n c a r e duality Di~ n = 7~n-i (substitutionof an i - v e c t o r in an n - f o r m ) , the composition with which we denote by ~. The o p e r a t o r s r

Di[f X*-+g Dif* A~+*, r

(a)=//oh, turn the collection {g DiffT~*} into a complex.

THEOREM. a) voS~q-g~ov=O, w h e r e Si a r e the Spencer o p e r a t o r s for P = A n in g - t h e o r y on E ~ (el. 1.7, 8.5). 1645

b) The Spencer sequence in

~-theory on E ~~ is exact. L------a

~D

Since im S~:~ Dif.fA~, and the latter module is projective, there exists a homomorphism ~:~ Difi An-+ (~ Diif ~n) such that S I o ~ = i. From a) of the theorem cited and the fact that ~~ it follows that

(A (p), q)+(p, A* (q))~d)Kz(A (p, q)), Afi~Dlfi(P, Q), w h e r e )K~(A)=(~ovok)(A--~(1)) , and we w r i t e (a, a) instead of ~(a) for a e ~, a ~A. The equation cited is G r e e n ' s f o r m u l a in ~ - t h e o r y on E ~. 10.4. We shall show now that the basic facts of the v a r i a t i o n a l calculus a r e consequences of G r e e n ' s f o r mula. We note, f i r s t l y , that the density of the L a g r a n g i a n , defined on sections of the bundle ~ : E ~ - M, is a differential o p e r a t o r Z:F (~)-+A" (M) (the L a g r a n g i a n itself h e r e can be d e s c r i b e d as o = S ~ ) , which means /

~ o is a horizontal f o r m on s o m e jk(~) and hence on j~o(,~). In view of P a r a g r a p h 8.7, this gives the basis for c o n s i d e r i n g the density of the L a g r a n g i a n in the g e n e r a l c a s e as an e l e m e n t of A n and the L a g r a n g i a n itself S Z as the d - c o h o m o l o g y c l a s s fl = {~} of the f o r m ~0, i.e., an e l e m e n t of the group TIn(N)(TIn(~)). If one conM s i d e r s the p r o b l e m with c o n s t r a i n t s , i.e., the varying quantity is subject to s o m e equation E, the density of the L a g r a n g i a n and the L a g r a n g i a n i t s e l f by the s a m e c o n s i d e r a t i o n s should be understood as e l e m e n t s of ~ n on E ~ and Hn(E), r e s p e c t i v e l y . A v a r i a t i o n of a section s ~ Floc(~) is, e x p r e s s e d intuitively, the speed of change of s in the s p a c e of sections along the trajectories of some ~veetor field." In view of 9.2, such a field, strictly speaking, is ~( ~(TF), while the speed of change of the Lagrangian at the "point" s, denoted by ~ Is, is here equal to (s~~ Identifying X with the corresponding vertical field (cf. 9.2), it makes sense to write X (w) and then % (~q)~ Z ({0)}) (% (c0)}= {l~ (~)} (cf. 9.8). Green's formula for the operator lw shows that {l~ (%)} = {(%, l~ (1))}. Thus, ~ I~:(s~) * {(%, l~ (1))}:{((s~) * ()C),(s~) * (l* (I))}. In viewofthe arbitrariness ofX anddu Bois Raymond's lemma, it follows from this that for extremals (s ~) (l~ (i))~0, so that l~ (I)I---0 ~s the Euler-Lagrange equatmn corresponding to the density of the Lagrangian co. We

introduce the Euler operator @:~/~ (~)-,~, @ ({~0})=l~ (I).

10.5. Let ~ ~ Hn(,T). We call a "field" • ~ ~(~r) a symmetry of the Lagrangian ~ if ~((~) = 0 on E ~, where E~{8(9.)~0}. Considering, as above, X as a vertical ~ -field, we have 0~((~)~{%(co)}~=~%(o~)=~p on E ~. But since %(c0)~/~(~)~(%, l~* (1)) & dkK~ ^~ (l~ 0(, 1)), we get the following a s s e r t i o n . THEOREM. If • is a s y m m e t r y of the L a g r a n g i a n fl, then the d-cohomology c l a s s {)K~(/~(~, 1))--p} is a c o n s e r v a t i o n law for the equation ~ (~) ~ 0. The c l a s s i c a l t h e o r e m of Noether gives the c o n s t r u c t i o n of a c o n s e r v a t i o n law under the condition~IX(0~)= 0, w h e r e X is a Lie f i e l d . , R e p r e s e n t i n g X in the f o r m ~( + Y, w h e r e ~ is v e r t i c a l , and Y ~ D (~ (E)), we find that %(c0)-- - - d ( F _ ~ ) ~ - - d (X_~0). Thus, N o e t h e r ' s t h e o r e m is a v e r y s p e c i a l c a s e of the t h e o r e m formulated, w h e r e it is n e c e s s a r y to s e t p ~ --X.~0~. We s t r e s s the effectiveness of the definition given of s y m m e t r i e s of the Lagrangian. It follows f r o m the r e s u l t s of P a r a g r a p h 10.6 that it is equivalent to the conditions: ~ (~((~))~0, 0 ~ * (7~(~))~ H~ (E~), w h e r e ~: E ~r ~ J~(~) is any s e c t i o n of the bundle J~(~) - - j0(~) = E n. 10.6. Withthis, in o r d e r to get a functional valued f o r m u l a t i o n of the calculus of v a r i a t i o n s in the s i t u a tion with c o n s t r a i n t s as well as on Nm,~ we c o n s i d e r the following s p e c t r a l sequence on E ~, w h e r e E ~ N ~ [or jk(~)] is the equation of the c o n s t r a i n t [E ~ = N~n or J~(~) if there is no constraint]. Let ~ A * be the k - t h power of the ideal ~A*. It is obvious that the ideal ~ A * is stable with r e s p e c t to d, so that we a r r i v e at a complex with f i l t r a t i o n A* ~ ~ 9~ A * ~ .... The s p e c t r a l sequence which a r i s e s as a r e s u l t {EP,q, dP,q} we call the ~ - s p e c t r a l sequence of the equation E. As usual, the index p denotes the filtration, and p + q the d e g r e e . L e t ~(~)DiIf~P 9 be t h e k - t h e x t e r i o r power of the ~ ' ( E ) - m o d u l e ~ D i [ f + p ,

~.(E)~Hom~-~)(~.(E), A~) and

E = {~0 = 0 } , w h e r e q~ ~ P i n t e r s e c t s the b a s e t r a n s v e r s e l y . Taking the t e n s o r product of the ~ - c o m p l e x of Spencer SP (cf. P a r a g r a p h 1.8) for the ~ ( E ) - m o d u l e P, o v e r ~ ( E ) , by ~(~)Diff+~(E) we get the complex Ak(p) = {Ak(p), sk}, sik:Ak(p) ---Ak_~(P). The o p e r a t o r l ~ t ~ g e n e r a t e s a map of Spencer c o m p l e x e s S (/~):S ( ~ ) ~ (~ (E)) (cf. P a r a g r a p h 1.8), which m e a n s also a map of c o m p l e x e s a~ (l~):A ~ (/5)_+ A ~ (x (E)). The " i - d i m e n s i o n a l

1646

chains" of the complex coker ak(,lq~ we denote by B k, and its i-dimensional homology by ikE). THEOREM. i) E~,q = 7kq, d~,q = ~ 2) in an affine chart E0p'q = BP-~, p > 0; 3) E ~ fine chart Ep,q~p-1 (E) and, in particular, E~q~//n_q (l~IE~) (cf. Paragraph 1.8). 1 n--q

= ~lq(E)~ 4) in an af-

We call an equation E S - r e g u l a r if ker S (l*wIE~)=0 (cf. P a r a g r a p h 1.8). The S - r e g u l a r i t y condition is not stringent, and as a rule, it can be v e r i f i e d effectively on the symbol of the o p e r a t o r l~ I~= in ~ - t h e o r y . P r o p o sition 1.8 and Theorem

10.6 lead directly to the following result.

COROLLARY. Let the equationE be S-regular, so thatEl%q--0ifq >n; Elp'q=0 for p >0 ifq ~n-l~ n (q r n for E ~~ = N~n or j~o(~)). Thus, E ,q = EP, q, Hq(E) = Hq(E) for q < n- 1 [q < n if E ~~ = Nm or j~o(~)] if E is formally integrable. Remark.

The complex

g~ equal to the composition

{Tk i, d} can be united with the complex

{E n,i, d n'i} with the help of the operator

A~-+/~n (E)-+ E~.~:

O-+ Ao-+ ...

-+ ~ " - +

EI,"-+

E~,"-+

. . .

It is easy to calculate the cohomology of this complex using Theorem 10.6. For example, if E ~~ = J~(~), the groups coincide with Hi(j~ Thanks to this one can find the "inversion formula," i.e., represent w 67~ n in the form dp (in an appropriate neighborhood), if ~' ((~)- ~ (~q)~ 0 10.7. The value of the cited spectral sequence for the circle of questions considered includes the fact that El~ according to Corollary 10.6, is the group of conservation laws, and E~ ,n is the group of all Lagrangians. Moreover, we get the following: a solution L~N (sCP (~)) of the constraint equation is an extremal of the Lagrangian Z if and only if j (L)* (d~,~(5r i.e., d~,n(Z)=0 should be considered the Euler-Lagrange equation. The restriction of the operator d ~ operator ~ (Paragraph 10.4).

to an affine chart, in the case of no constraints,

coincides with the

We emphasize the naturality of the cited formulation of the calculus of variations in the category NDE. Actually, the morphisms inthis category preserve the operation ~, and hence, generate homomorp~sms of the ~ -spectral sequences, one of whose differentials is the Euler operator. This generalizes a classical result: the Euler- Lagrange equation is invariant under arbitrary changes of dependent and independent variables. We

note yet another curious consequence

Proposition. p is a conservation

of Theorem

10.6.

Let E = {~0 = 0} (cf. above) and l~IEoo = _+le!E~

(i.e., E is self- or skew-adjoint).

Then if

law for E, then d~'n-l(p) E symE.

This follows from the fact that, by virtue of Theorem kerl~le~, and symE=ker/~I~ (Theorem 9.8}.

10.63) and Proposition 1.8, E~,~-I=H~ (l~!eo~)~

10.8. Commentary. In this section~ results of the author are cited (cf. [7, 8], where one can find further details). An interesting invariant formulation of the Lagrangian formalism on j~o(~) was given earlier by B. A. Knpershmidt (of. the account of his work in the survey [26, Chap. I]), starting from the axiomatized first variation formula. Theorem 10.5 is close to a theorem of Kupershmidt, who also discovered in geometric terms the ~resolvent ~ for operators g on J~(~) (cf. Remark 10.6). The calculation of the term E2l,n is equivalent to the solution of the "inverse problem of the variational calculus" (cf. Remark 10.6), to which Takens [54] is devoted (its solution also follows fromthe cited work of Kupershrnidt). A construction of the calculus of variations on a special class of Lagrangians from an unexpected point of view (theory of Hodge-Lepage and contact geometry) is given by Lychagin [25]. Interesting in connection with the theme of this section are the work of Goldschmidt and Sternberg [43] and the important work of Dedecker [34]. The use of the language of the Hamiltonian formalism in the theory of fields is considered in [9]. LITERATURE i.

2.

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