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Symmetries and Recursion
Operators for Classical and
Supersymmetric
Di¬erential Equations

by

I.S. Krasil™shchik
Independent University of Moscow
and Moscow Institute of Municipal Economy,
Moscow, Russia

and

P.H.M. Kersten
Faculty of Mathematical Sciences,
University of Twente,
Enschede, The Netherlands




KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Contents

Preface xi

Chapter 1. Classical symmetries 1
1. Jet spaces 1
1.1. Finite jets 1
1.2. Nonlinear di¬erential operators 5
1.3. In¬nite jets 7
2. Nonlinear PDE 12
2.1. Equations and solutions 12
2.2. The Cartan distributions 16
2.3. Symmetries 21
2.4. Prolongations 28
3. Symmetries of the Burgers equation 30
4. Symmetries of the nonlinear di¬usion equation 34
4.1. Case 1: p = 0, k = 0 35
4.2. Case 2: p = 0, k = 0, q = 1 35
4.3. Case 3: p = 0, k = 0, q = 1 36
4.4. Case 4: p = ’4/5, k = 0 36
4.5. Case 5: p = ’4/5, p = 0, k = 0 36
4.6. Case 6: p = ’4/5, k = 0, q = 1 36
4.7. Case 7: p = 0, p = ’4/5, k = 0, q = 1 37
4.8. Case 8: p = 0, p = ’4/5, q = p + 1 37
4.9. Case 9: p = 0, p = ’4/5, q = 1, q = p + 1 37
5. The nonlinear Dirac equations 37
5.1. Case 1: = 0, »’1 = 0 39
5.2. Case 2: = 0, »’1 = 0 43
5.3. Case 3: = 0, »’1 = 0 43
5.4. Case 4: = 0, »’1 = 0 43
6. Symmetries of the self-dual SU (2) Yang“Mills equations 43
6.1. Self-dual SU (2) Yang“Mills equations 43
6.2. Classical symmetries of self-dual Yang“Mills equations 46
6.3. Instanton solutions 49
6.4. Classical symmetries for static gauge ¬elds 51
6.5. Monopole solution 52

Chapter 2. Higher symmetries and conservation laws 57
1. Basic structures 57
v
vi CONTENTS

1.1. Calculus 57
1.2. Cartan distribution 59
1.3. Cartan connection 61
1.4. C-di¬erential operators 63
2. Higher symmetries and conservation laws 67
2.1. Symmetries 67
2.2. Conservation laws 72
3. The Burgers equation 80
3.1. De¬ning equations 80
3.2. Higher order terms 81
3.3. Estimating Jacobi brackets 82
3.4. Low order symmetries 83
3.5. Action of low order symmetries 83
3.6. Final description 83
4. The Hilbert“Cartan equation 84
4.1. Classical symmetries 85
4.2. Higher symmetries 87
4.3. Special cases 91
5. The classical Boussinesq equation 93

Chapter 3. Nonlocal theory 99
1. Coverings 99
2. Nonlocal symmetries and shadows 103
3. Reconstruction theorems 105
4. Nonlocal symmetries of the Burgers equation 109
5. Nonlocal symmetries of the KDV equation 111
6. Symmetries of the massive Thirring model 115
6.1. Higher symmetries 116
6.2. Nonlocal symmetries 120
6.2.1. Construction of nonlocal symmetries 121
6.2.2. Action of nonlocal symmetries 124
7. Symmetries of the Federbush model 129
7.1. Classical symmetries 129
7.2. First and second order higher symmetries 130
7.3. Recursion symmetries 135
7.4. Discrete symmetries 138
7.5. Towards in¬nite number of hierarchies of symmetries 138
7.5.1. Construction of Y + (2, 0) and Y + (2, 0) 139
7.5.2. Hamiltonian structures 140
7.5.3. The in¬nity of the hierarchies 144
7.6. Nonlocal symmetries 146
8. B¨cklund transformations and recursion operators
a 149

Chapter 4. Brackets 155
1. Di¬erential calculus over commutative algebras 155
1.1. Linear di¬erential operators 155
CONTENTS vii

1.2. Jets 159
1.3. Derivations 160
1.4. Forms 164
1.5. Smooth algebras 168
2. Fr¨licher“Nijenhuis bracket
o 171
2.1. Calculus in form-valued derivations 171
2.2. Algebras with ¬‚at connections and cohomology 176
3. Structure of symmetry algebras 181
3.1. Recursion operators and structure of symmetry algebras 182
3.2. Concluding remarks 184

Chapter 5. Deformations and recursion operators 187
1. C-cohomologies of partial di¬erential equations 187
2. Spectral sequences and graded evolutionary derivations 196
3. C-cohomologies of evolution equations 208
4. From deformations to recursion operators 217
5. Deformations of the Burgers equation 221
6. Deformations of the KdV equation 227
7. Deformations of the nonlinear Schr¨dinger equation
o 231
8. Deformations of the classical Boussinesq equation 233
9. Symmetries and recursion for the Sym equation 235
9.1. Symmetries 235
9.2. Conservation laws and nonlocal symmetries 239
9.3. Recursion operator for symmetries 241

Chapter 6. Super and graded theories 243
1. Graded calculus 243
1.1. Graded polyderivations and forms 243
1.2. Wedge products 245
1.3. Contractions and graded Richardson“Nijenhuis bracket 246
1.4. De Rham complex and Lie derivatives 248
1.5. Graded Fr¨licher“Nijenhuis bracket
o 249
2. Graded extensions 251
2.1. General construction 251
2.2. Connections 252
2.3. Graded extensions of di¬erential equations 253
2.4. The structural element and C-cohomologies 253
2.5. Vertical subtheory 255
2.6. Symmetries and deformations 256
2.7. Recursion operators 257
2.8. Commutativity theorem 260
3. Nonlocal theory and the case of evolution equations 261
3.1. The GDE(M ) category 262
3.2. Local representation 262
3.3. Evolution equations 264
3.4. Nonlocal setting and shadows 265
viii CONTENTS

3.5. The functors K and T 267
3.6. Reconstructing shadows 268
4. The Kupershmidt super KdV equation 270
4.1. Higher symmetries 271
4.2. A nonlocal symmetry 273
5. The Kupershmidt super mKdV equation 275
5.1. Higher symmetries 276
5.2. A nonlocal symmetry 278
6. Supersymmetric KdV equation 280
6.1. Higher symmetries 281
6.2. Nonlocal symmetries and conserved quantities 282
7. Supersymmetric mKdV equation 290
8. Supersymmetric extensions of the NLS 293
8.1. Construction of supersymmetric extensions 293
8.2. Symmetries and conserved quantities 297
8.2.1. Case A 297
8.2.2. Case B 303
9. Concluding remarks 307

Chapter 7. Deformations of supersymmetric equations 309
1. Supersymmetric KdV equation 309
1.1. Nonlocal variables 309
1.2. Symmetries 310
1.3. Deformations 312
1.4. Passing from deformations to “classical” recursion operators 313
2. Supersymmetric extensions of the NLS equation 315
2.1. Case A 316
2.2. Case B 318
3. Supersymmetric Boussinesq equation 320
3.1. Construction of supersymmetric extensions 320
3.2. Construction of conserved quantities and nonlocal variables 321
3.3. Symmetries 322
3.4. Deformation and recursion operator 323
4. Supersymmetric extensions of the KdV equation, N = 2 324
4.1. Case a = ’2 325
4.1.1. Conservation laws 326
4.1.2. Higher and nonlocal symmetries 328
4.1.3. Recursion operator 330
4.2. Case a = 4 331
4.2.1. Conservation laws 331
4.2.2. Higher and nonlocal symmetries 334
4.2.3. Recursion operator 335
4.3. Case a = 1 337
4.3.1. Conservation laws 337
4.3.2. Higher and nonlocal symmetries 341
CONTENTS ix

4.3.3. Recursion operator 347
Chapter 8. Symbolic computations in di¬erential geometry 349
1. Super (graded) calculus 350
2. Classical di¬erential geometry 355
3. Overdetermined systems of PDE 356
3.1. General case 357
3.2. The Burgers equation 360
3.3. Polynomial and graded cases 371
Bibliography 373
Index 379
x CONTENTS
Preface

To our wives, Masha and Marian



Interest to the so-called completely integrable systems with in¬nite num-
ber of degrees of freedom aroused immediately after publication of the fa-
mous series of papers by Gardner, Greene, Kruskal, Miura, and Zabusky
[75, 77, 96, 18, 66, 19] (see also [76]) on striking properties of the
Korteweg“de Vries (KdV) equation. It soon became clear that systems of
such a kind possess a number of characteristic properties, such as in¬nite
series of symmetries and/or conservation laws, inverse scattering problem
formulation, L ’ A pair representation, existence of prolongation structures,
etc. And though no satisfactory de¬nition of complete integrability was yet
invented, a need of testing a particular system for these properties appeared.
Probably, one of the most e¬cient tests of this kind was ¬rst proposed
by Lenard [19] who constructed a recursion operator for symmetries of the
KdV equation. It was a strange operator, in a sense: being formally integro-
di¬erential, its action on the ¬rst classical symmetry (x-translation) is well-
de¬ned and produces the entire series of higher KdV equations. But applied
to the scaling symmetry, it gave expressions containing terms of the type
u dx which had no adequate interpretation in the framework of the existing
theories. And it is not surprising that P. Olver wrote “The deduction of the
form of the recursion operator (if it exists) requires a certain amount of in-
spired guesswork...” [80, p. 315]: one can hardly expect e¬cient algorithms
in the world of rather fuzzy de¬nitions, if any.
In some sense, our book deals with the problem of how to construct
a well-de¬ned concept of a recursion operator and use this de¬nition for
particular computations. As it happened, a ¬nal solution can be explicated
in the framework of the following conceptual scheme.
We start with a smooth manifold M (a space of independent variables)
and a smooth locally trivial vector bundle π : E ’ M whose sections play
the role of dependent variables (unknown functions). A partial di¬erential
equation in the bundle π is a smooth submanifold E in the space J k (π) of k-
jets of π. Any such a submanifold is canonically endowed with a distribution,
the Cartan distribution. Being in general nonintegrable, this distribution
possesses di¬erent types of maximal integral manifolds a particular case of
which are (generalized) solutions of E. Thus we can de¬ne geometry of the
xi
xii PREFACE

equation E as geometry related to the corresponding Cartan distribution.
Automorphisms of this geometry are classical symmetries of E.
Dealing with geometry of di¬erential equations in the above sense, one
soon ¬nds that a number of natural constructions arising in this context is
in fact a ¬nite part of more general objects existing on di¬erential conse-
quences of the initial equation. This leads to introduction of prolongations
E l of E and, in the limit, of the in¬nite prolongation E ∞ as a submanifold
of the manifold J ∞ (π) of in¬nite jets. Using algebraic language mainly,
all ¬nite-dimensional constructions are carried over both to J ∞ (π) and E ∞
and, surprisingly at ¬rst glance, become there even more simple and elegant.
In particular, the Cartan distribution on E ∞ becomes completely integrable
(i.e., satis¬es the conditions of the Frobenius theorem). Nontrivial symme-
tries of this distribution are called higher symmetries of E.
Moreover, the Cartan distribution on E ∞ is in fact the horizontal dis-
tribution of a certain ¬‚at connection C in the bundle E ∞ ’ M (the Cartan
connection) and the connection form of C contains all vital geometrical in-
formation about the equation E. We call this form the structural element of
E and it is a form-valued derivation of the smooth function algebra on E ∞ .
A natural thing to ask is what are deformations of the structural element
(or, of the equation structure on E). At least two interesting things are
found when one answers this question.
The ¬rst one is that the deformation theory of equation structures is
closely related to a cohomological theory based on the Fr¨licher“Nijenhuis
o
bracket construction in the module of form-valued derivations. Namely, if
we denote by D1 Λi (E) the module of derivations with values in i-forms, the
Fr¨licher“Nijenhuis bracket acts in the following way:
o
[[·, ·]]fn : D1 Λi (E) — D1 Λj (E) ’ D1 Λi+j (E).
In particular, for any element „¦ ∈ D1 Λ1 (E) we obtain an operator
‚„¦ : D1 Λi (E) ’ D1 Λi+1 (E)
de¬ned by the formula ‚„¦ (˜) = [[„¦, ˜]]fn for any ˜ ∈ D1 Λi (E). Since

D1 Λ— (E) = i
i=1 D1 Λ (E) is a graded Lie algebra with respect to the
Fr¨licher“Nijenhuis bracket and due to the graded Jacobi identity, one can
o
see that the equality ‚„¦ —¦ ‚„¦ = 0 is equivalent to [[„¦, „¦]]fn = 0. The last
equality holds, if „¦ is a connection form of a ¬‚at connection. Thus, any
¬‚at connection generates a cohomology theory. In particular, natural co-
homology groups are related to the Cartan connection and we call them
i
C-cohomology and denote by HC (E).
We restrict ourselves to the vertical subtheory of this cohomological the-
0
ory. Within this restriction, it can be proved that the group HC (E) coincides
1
with the Lie algebra of higher symmetries of the equation E while HC (E)
consists of the equivalence classes of in¬nitesimal deformations of the equa-
tion structure on E. It is also a common fact in cohomological deformation
2
theory [20] that the group HC (E) contains obstructions to continuation of
PREFACE xiii

in¬nitesimal deformations up to formal ones. For partial di¬erential equa-
tions, triviality of this group is, roughly speaking, the reason for existence
of commuting series of higher symmetries.
The second interesting and even more important thing in our context
is that the contraction operation de¬ned in D1 Λ— (E) is inherited by the
i 1
groups HC (E). In particular, the group HC (E) is an associative algebra
0
with respect to this operation while contraction with elements of HC (E)
is a representation of this algebra. In e¬ect, having a nontrivial element
1 0
R ∈ HC (E) and a symmetry s0 ∈ HC (E) we are able to obtain a whole
in¬nite series sn = Rn s0 of new higher symmetries. This is just what is
expected of recursion operators!
Unfortunately (or, perhaps, luckily) a straightforward computation of
the ¬rst C-cohomology groups for known completely integrable equations
(the KdV equation, for example) leads to trivial results only, which is not
surprising at all. In fact, normally recursion operators for nonlinear inte-
grable systems contain integral (nonlocal) terms which cannot appear when
one works using the language of in¬nite jets and in¬nite prolongations only.
The setting can be extended by introduction of new entities ” nonlocal
variables. Geometrically, this is being done by means of the concept of a
covering. A covering over E ∞ is a ¬ber bundle „ : W ’ E ∞ such that the
˜
total space W is endowed with and integrable distribution C and the dif-
˜
ferential „— isomorphically projects any plane of the distribution C to the
corresponding plane of the Cartan distribution C on E ∞ . Coordinates along
the ¬bers of „ depend on coordinates in E ∞ in an integro-di¬erential way
and are called nonlocal.
Geometry of coverings is described in the same terms as geometry of
in¬nite prolongations, and we can introduce the notions of symmetries of
W (called nonlocal symmetries of E), the structural element, C-cohomology,
etc. For a given equation E, we can choose an appropriate covering and may
1
be lucky to extend the group HC (E). For example, for the KdV equation it
su¬ces to add the nonlocal variable u’1 = u dx, where u is the unknown
function, and to obtain the classical Lenard recursion operator as an ele-
ment of the extended C-cohomology group. The same e¬ect one sees for the
Burgers equation. For other integrable systems such coverings may be (and
usually are) more complicated.
To ¬nish this short review, let us make some comments on how recursion
operators can be e¬ciently computed. To this end, note that the module
D(E) of vector ¬elds on E ∞ splits into the direct sum D(E) = D v (E)•CD(E),
where D v (E) are π-vertical ¬elds and CD(E) consists of vector ¬elds lying in
the Cartan distribution. This splitting induces the dual one: Λ(E) = Λ1 (E)•
h
1 (E). Elements of Λ1 (E) are called horizontal forms while elements of
CΛ h
CΛ1 (E) are called Cartan forms (they vanish on the Cartan distribution).
By consequence, we have the splitting Λi (E) = p q
p+q=i C Λ(E) — Λ (E),
xiv PREFACE

where
Λq (E) = Λ1 (E) § · · · § Λ1 (E) .
C p Λ(E) = CΛ1 (E) § · · · § CΛ1 (E), h h
h
p times q times

This splitting generates the corresponding splitting in the groups of C-
p,q
i
cohomologies: HC (E) = p+q=i HC (E) and nontrivial recursion operators
1,0
are elements of the group HC (E).
The graded algebra C — Λ(E) = p
p≥0 C Λ(E) may be considered as the
algebra of functions on a super di¬erential equation related to the initial
equation E in a functorial way. This equation is called the Cartan (odd )
covering of E. An amazing fact is that the symmetry algebra of this covering
—,0 —,0
is isomorphic to the direct sum HC (E) • HC (E). Thus, due to the general
p,0
theory, to ¬nd an element of HC (E) we have just to take a system of forms
„¦ = (ω 1 , . . . , ω m ), where ω j ∈ C p Λ(E) and m = dim π, and to solve the
equation E ω = 0, where E is the linearization of E restricted to E ∞ . In
particular, for p = 1 we shall obtain recursion operators, and the action
of the corresponding solutions on symmetries of E is just contraction of a
symmetry with the Cartan vector-form „¦.


This scheme is exposed in details below. Though some topics can be
found in other books (see, e.g., [60, 12, 80, 5, 81, 101]; the collections [39]
and [103] also may be recommended), we included them in the text to make
the book self-contained. We also decided to include a lot of applications in
the text to make it interesting not only to those ones who deal with pure
theory.
The material of the book is arranged as follows.
In Chapter 1 we deal with spaces of ¬nite jets and partial di¬erential
equations as their submanifold. The Cartan distribution on J k (π) is intro-
duced and it maximal integral manifolds are described. We describe auto-
morphisms of this distribution (Lie“B¨cklund transformations) and derive
a
de¬ning relations for classical symmetries. As applications, we consider clas-
sical symmetries of the Burgers equation, of the nonlinear di¬usion equation
(and obtain the so-called group classi¬cation in this case), of the nonlinear
Dirac equation, and of the self-dual Yang“Mills equations. For the latter,
we get monopole and instanton solutions as invariant solutions with respect
to the symmetries obtained.
Chapter 2 is dedicated to higher symmetries and conservation laws. Ba-
sic structures on in¬nite prolongations are described, including the Cartan
connection and the structural element of a nonlinear equation. In the con-
text of conservation laws, we brie¬‚y expose the results of A. Vinogradov
on the C-spectral sequence [102]. We give here a complete description for
higher symmetries of the Burgers equation, the Hilbert“Cartan equation,
and the classical Boussinesq equation.
PREFACE xv

In Chapter 3 we describe the nonlocal theory. The notion of a covering
is introduced, the relation between coverings and conservation laws is dis-
cussed. We reproduce here quite important results by N. Khor kova [43] on
the reconstruction of nonlocal symmetries by their shadows. Several appli-
cations are considered in this chapter: nonlocal symmetries of the Burgers
and KdV equation, symmetries of the massive Thirring model and symme-
tries of the Federbush model. In the last case, we also discuss Hamiltonian
structures for this model and demonstrate the existence of in¬nite number
of hierarchies of symmetries. We ¬nish this chapter with an interpretation
of B¨cklund transformations in terms of coverings and discuss a de¬nition
a
of recursion operators as B¨cklund transformations belonging to M. Marvan
a
[73].
Chapter 4 starts the central topic of the book: algebraic calculus of form-
valued derivations. After introduction of some general concepts (linear dif-
ferential operators over commutative algebras, algebraic jets and di¬erential
forms), we de¬ne basic constructions of Fr¨licher“Nijenhuis and Richardson“
o
Nijenhuis brackets [17, 78] and analyze their properties. We show that to
any integrable derivation X with values in one-forms, i.e., satisfying the
condition [[X, X]]fn = 0, a complex can be associated and investigate main
properties of the corresponding cohomology group. A source of examples
for integrable elements is provided by algebras with ¬‚at connections. These
algebras can be considered as a model for in¬nitely prolonged di¬erential
equation. Within this model, we introduce algebraic counterparts for the
notions of a symmetry and a recursion operator and prove some results
describing the symmetry algebra structure in the case when the second co-
homology group vanishes. In particular, we show that in this case in¬nite
series of commuting symmetries arise provided the model possesses a non-
trivial recursion operator.
Chapter 5 can be considered as a speci¬cation of the results obtained
in Chapter 4 to the case of partial di¬erential equations, i.e., the algebra
in question is the smooth function algebra on E ∞ while the ¬‚at connection
is the Cartan connection. The cohomology groups arising in this case are
C-cohomology of E. Using spectral sequence techniques, we give a com-
plete description of the C-cohomology for the “empty” equation, that is for
the spaces J ∞ (π) and show that elements of the corresponding cohomol-
ogy groups can be understood as graded evolutionary derivations (or vector
¬elds) on J ∞ (π). We also establish relations between C-cohomology and
deformations of the equation structure and show that in¬nitesimal defor-
1,0
mations of a certain kind (elements of HC (E), see above) are identi¬ed
with recursion operators for symmetries. After deriving de¬ning equations
for these operators, we demonstrate that in the case of several classical
systems (the Burgers equation, KdV, the nonlinear Schr¨dinger and Boussi-
o
nesq equations) the results obtained coincide with the well-known recursion
xvi PREFACE

operators. We also investigate the equation of isometric immersions of two-
dimensional Riemannian surfaces into R3 (a particular case of the Gauss“
Mainardi“Codazzi equations, which we call the Sym equation) and prove its
complete integrability, i.e., construct a recursion operator and in¬nite series
of symmetries.
Chapter 6 is a generalization of the preceding material to the graded
case (or, in physical terms, to the supersymmetric case). We rede¬ne all
necessary algebraic construction for graded commutative algebras and in-
troduce the notion of a graded extension of a partial di¬erential equation. It
is shown that all geometrical constructions valid for classical equations can
be applied, with natural modi¬cations, to graded extensions as well. We
describe an approach to the construction of graded extensions and consider
several illustrative examples (graded extensions of the KdV and modi¬ed
KdV equations and supersymmetric extensions of the nonlinear Schr¨dinger
o
equation).
Chapter 7 continues the topics started in the preceding chapter. We
consider here two supersymmetric extensions of the KdV equations (one-
and two-dimensional), new extensions of the nonlinear Schr¨dinger equation,
o
and the supersymmetric Boussinesq equation. In all applications, recursion
operators are constructed and new in¬nite series of symmetries, both local
and nonlocal, are described.
Finally, in Chapter 8 we brie¬‚y describe the software used for
computations described in the book and without which no serious ap-
plication could be obtained.


Our collaboration started in 1991. It could not be successful without
support of several organizations among which:
• the University of Twente,
• NWO (Nederlandse Organisatie voor Wetenschappelijk Onderzoek),
• FOM (Fundamenteel Onderzoek der Materie / Samenwerkingsver-
band Mathematische Fysica),
• INTAS (International Association for the promotion of co-operation
with scientists from the New Independent States of the former Soviet
Union).
We are also grateful to Kluwer Academic Publishers and especially to Pro-
fessor Michiel Hazewinkel for the opportunity to publish this book.

Joseph Krasil shchik and Paul Kersten,
Moscow“Enschede
CHAPTER 1


Classical symmetries

This chapter is concerned with the basic notions needed for our exposi-
tion ” those of jet spaces and of nonlinear di¬erential equations. Our main
purpose is to put the concept of a nonlinear partial di¬erential equation
(PDE) into the framework of smooth manifolds and then to apply powerful
techniques of di¬erential geometry and commutative algebra. We completely
abandon analytical language, maybe good enough for theorems of existence,
but not too useful in search for main underlying structures.
We describe the geometry of jet spaces and di¬erential equations (its
geometry is determined by the Cartan distribution) and introduce classical
symmetries of PDE. Our exposition is based on the books [60, 12]. We also
discuss several examples of symmetry computations for some equations of
mathematical physics.

1. Jet spaces
We expose here main facts concerning the geometrical approach to jets
(¬nite and in¬nite) and to nonlinear di¬erential operators.

1.1. Finite jets. Traditional approach to di¬erential equations consists
in treating them as expressions of the form
‚u ‚u
F x1 , . . . , xn , ,..., , . . . = 0, (1.1)
‚x1 ‚xn
where x1 , . . . , xn are independent variables, while u = u(x1 , . . . , xn ) is an
unknown function (dependent variable). Such an equation is called scalar,
but one can consider equations of the form (1.1) with F = (F 1 , . . . , F r )
and u = (u1 , . . . , um ) being vector-functions. Then we speak of systems of
PDE. What makes expression (1.1) a di¬erential equation is the presence of
partial derivatives ‚u/‚x1 , . . . in it, and our ¬rst step is to clarify this fact
in geometrical terms.
To do it, we shall restrict ourselves to the situation when all func-
tions are smooth (i.e., of the C ∞ -class) and note that a vector-function
u = (u1 , . . . , um ) can be considered as a section of the trivial bundle
1m : Rm — Rn = Rn+m ’ Rn . Denote Rm — Rn by J 0 (n, m) and con-
n
sider the graph of this section, i.e., the set “u ‚ J 0 (n, m) consisting of the
points
(x1 , . . . , xn , u1 (x1 , . . . , xn ), . . . , um (x1 , . . . , xn ) ,
1
2 1. CLASSICAL SYMMETRIES

which is an n-dimensional submanifold in Rn+m .
Let x = (x1 , . . . , xn ) be a point of Rn and θ = (x, u(x)) be the corre-
sponding point lying on “u . Then the tangent plane to “u passing through
the point θ is completely determined by x and by partial derivatives of u at
the point x. It is easy to see that the set of such planes forms an mn-di-
mensional space Rmn with coordinates, say, uj , i = 1, . . . , n, j = 1, . . . , m,
i
j
where ui “corresponds” to the partial derivative of the function uj with
respect to xi at x.
Maintaining this construction at every point θ ∈ J 0 (n, m), we obtain
def
the bundle J 1 (n, m) = Rmn — J 0 (n, m) ’ J 0 (n, m). Consider a point
θ1 ∈ J 1 (n, m). By doing this, we, in fact, ¬x the following data: values
of independent variables, x, values of dependent ones, uj , and values of
all their partial derivatives at x. Assume now that a smooth submanifold
E ‚ J 1 (n, m) is given. This submanifold determines “relations between
points” of J 1 (n, m). Taking into account the above given interpretation of
these points, we see that E may be understood as a system of relations on
unknowns uj and their partial derivatives. Thus, E is a ¬rst-order di¬erential
equation! (Or a system of such equations.)
With this example at hand, we pass now to a general construction.
Let M be an n-dimensional smooth manifold and π : E ’ M be a
smooth m-dimensional vector bundle1 over M . Denote by “(π) the C ∞ (M )-
module of sections of the bundle π. For any point x ∈ M we shall also
consider the module “loc (π; x) of all local sections at x.

Remark 1.1. We say that • is a local section of π at x, if it is de¬ned on
a neighborhood U of x (the domain of •). To be exact, • is a section of the
pull-back — π = π |U , where : U ’ M is the natural embedding. If •, • ∈
“loc (π; x) are two local sections with the domains U and U respectively,
then their sum • + • is de¬ned over U © U . For any function f ∈ C ∞ (M )
we can also de¬ne the local section f • over U.

For a section • ∈ “loc (π; x), •(x) = θ ∈ E, consider its graph “• ‚ E
and all sections • ∈ “loc (π; x) such that
(a) •(x) = • (x);
(b) the graph “• is tangent to “• with order k at θ.
It is easy to see that conditions (a) and (b) determine an equivalence relation
∼k on “loc (π; x) and we denote the equivalence class of • by [•]k . The
x x
k becomes an R-vector space, if we put
quotient set “loc (π; x)/ ∼x

[•]k + [ψ]k = [• + ψ]k , a[•]k = [a•]k , •, ψ ∈ “loc (π; x), a ∈ R, (1.2)
x x x x x


1
In fact, all constructions below can be carried out ” with natural modi¬cations
” for an arbitrary locally trivial bundle π (and even in more general settings). But we
restrict ourselves to the vector case for clearness of exposition.
1. JET SPACES 3

while the natural projection “loc (π; x) ’ “loc (π; x)/ ∼k becomes a linear
x
k (π). Obviously, J 0 (π) coincides with
map. We denote this space by Jx x
Ex = π ’1 (x), the ¬ber of the bundle π over the point x ∈ M .
Remark 1.2. The tangency class [•]k is completely determined by the
x
point x and partial derivatives up to order k at x of the section •. From
k
here it follows that Jx (π) is ¬nite-dimensional. It is easy to compute the
dimension of this space: the number of di¬erent partial derivatives of order
i equals n+i’1 and thus
n’1
k
n+i’1 n+k
k
dim Jx (π) =m =m . (1.3)
n’1 k
i=0

Definition 1.1. The element [•]k ∈ Jx (π) is called the k-jet of the
k
x
section • ∈ “loc (π; x) at the point x.
The k-jet of • can be identi¬ed with the k-th order Taylor expansion of
the section •. From the de¬nition it follows that it is independent of coor-
dinate choice (in contrast to the notion of partial derivative, which depends
on local coordinates).
Let us consider now the set
J k (π) = k
Jx (π) (1.4)
x∈M

and introduce a smooth manifold structure on J k (π) in the following way.
Let {U± }± be an atlas in M such that the bundle π becomes trivial over each
U± , i.e., π ’1 (U± ) U± — V , where V is the “typical ¬ber”. Choose a ba-
± , . . . , e± of local sections of π over U . Then any section of π |
sis e1 U±
±
m
1 e± + · · · + um e± and the functions
is representable in the form • = u 1 m
1 , . . . , um , where x , . . . , x are local coordinates in U , con-
x1 , . . . , xn , u 1 n ±
’1 (U ). Let us de¬ne the functions
stitute a local coordinate system in π ±
m: k (π) ’ R, where σ = (σ , . . . σ ), |σ| = σ + · · · + σ ¤ k, by
uσ x∈U± Jx 1 n 1 n

‚ |σ| uj
def
uj [•]k = , (1.5)
σ x
‚xσ
x
def
‚xσ = (‚x1 )σ1 . . . (‚xn )σn . Then these functions, together with local coor-
dinates x1 , . . . , xn , de¬ne the mapping f± : x∈U± Jx (π) ’ U± — RN , where
k

N is the number de¬ned by (1.3). Due to computation rules for partial
derivatives under coordinate transformations, the mapping
’1
: (U± © Uβ ) — RN ’ (U± © Uβ ) — RN
(f± —¦ fβ ) U± ©Uβ

is a di¬eomorphism preserving the natural projection (U± © Uβ ) — Rn ’
(U± © Uβ ). Thus we have proved the following result:
Proposition 1.1. The set J k (π) de¬ned by (1.4) is a smooth manifold
while the projection πk : J k (π) ’ M , πk : [•]k ’ x, is a smooth vector
x
bundle.
4 1. CLASSICAL SYMMETRIES

Note that linear structure in the ¬bers of πk is given by (1.2).
Definition 1.2. Let π : E ’ M be a smooth vector bundle, dim M =
n, dim E = n + m.
(i) The manifold J k (π) is called the manifold of k-jets for π;
(ii) The bundle πk : J k (π) ’ M is called the bundle of k-jets for π;
(iii) The above constructed coordinates {xi , uj }, where i = 1, . . . , n, j =
σ
1, . . . , m, |σ| ¤ k, are called the special (or adapted ) coordinate system
on J k (π) associated to the trivialization {U± }± of the bundle π.
Obviously, the bundle π0 coincides with π.
Note that tangency of two manifolds with order k implies tangency with
less order, i.e., there exists a mapping πk,l : J k (π) ’ J l (π), [•]k ’ [•]l , k ≥
x x
l. From this remark and from the de¬nitions we obtain the commutative
diagram
πk,l
J k (π) ’ J l (π)
πk



l,s
π
,s







J s (π)
πk




πl




πs







M
where k ≥ l ≥ s and all arrows are smooth ¬ber bundles. In other words,
we have
πl,s —¦ πk,l = πk,s , πl —¦ πk,l = πk , k ≥ l ≥ s. (1.6)
On the other hand, for any section • ∈ “(π) (or ∈ “loc (π; x)) we can de¬ne
the mapping jk (•) : M ’ J k (π) by setting jk (•) : x ’ [•]k . Obviously,
x
jk (•) ∈ “(πk ) (respectively, jk (•) ∈ “loc (πk ; x)).
Definition 1.3. The section jk (•) is called the k-jet of the section •.
The correspondence jk : “(π) ’ “(πk ) is called the operator of k-jet.
From the de¬nition it follows that
πk,l —¦ jk (•) = jl (•), k ≥ l,
j0 (•) = •, (1.7)
for any • ∈ “(π).
Let •, ψ ∈ “(π) be two sections, x ∈ M and •(x) = ψ(x) = θ ∈ E. It is
a tautology to say that the manifolds “• and “ψ are tangent to each other
with order k + l at θ or that the manifolds “jk (•) , “jk (ψ) ‚ J k (π) are tangent
with order l at the point θk = jk (•)(x) = jk (ψ)(x).
Definition 1.4. Let θk ∈ J k (π). An R-plane at θk is an n-dimensional
plane tangent to some manifold of the form “jk (•) such that [•]k = θk .
x
1. JET SPACES 5

Immediately from de¬nitions we obtain the following result.
Proposition 1.2. Let θk ∈ J k (π) be a point in a jet space. Then the
¬ber of the bundle πk+1,k : J k+1 (π) ’ J k (π) over θk coincides with the set
of all R-planes at θk .
For a point θk+1 ∈ J k+1 (π), we shall denote the corresponding R-plane
at θk = πk+1,k (θk+1 ) by Lθk+1 ‚ Tθk (J k (π)).

1.2. Nonlinear di¬erential operators. Since J k (π) is a smooth
manifold, we can consider the algebra of smooth functions on J k (π). De-
note this algebra by Fk (π). Take another vector bundle π : E ’ M and
— —
consider the pull-back πk (π ). Then the set of sections of πk (π ) is a mod-
ule over Fk (π) and we denote this module by Fk (π, π ). In particular,
Fk (π) = Fk (π, 1M ), where 1M is the trivial one-dimensional bundle over
M.
def
The surjections πk,l and πk generate the natural embeddings νk,l =
def
— —
πk,l : Fl (π, π ) ’ Fk (π, π ) and νk = πk : “(π ) ’ Fk (π, π ). Due to (1.6),
we have the equalities
νk,l —¦ νl,s = νk,s , νk,l —¦ νl = νk , k ≥ l ≥ s. (1.8)
Identifying Fl (π, π ) with its image in Fk (π, π ) under νk,l , we can consider
Fk (π, π ) as a ¬ltered module,
“(π ) ’ F0 (π, π ) ’ . . . ’ Fk’1 (π, π ) ’ Fk (π, π ), (1.9)
over the ¬ltered algebra
C ∞ (M ) ’ F0 (π) ’ . . . ’ Fk’1 (π) ’ Fk (π). (1.10)
Let F ∈ Fk (π, π ). Then we have the correspondence
def
∆(•) = jk (•)— (F ),
∆ = ∆F : “(π) ’ “(π ), • ∈ “(π). (1.11)
Definition 1.5. A correspondence ∆ of the form (1.11) is called a (non-
linear ) di¬erential operator of order2 ¤ k acting from the bundle π to
the bundle π . In particular, when ∆(f • + gψ) = f ∆(•) + g∆(ψ) for all
•, ψ ∈ “(π) and f, g ∈ C ∞ (M ), the operator ∆ is said to be linear.
From (1.9) it follows that operators ∆ of order k are also operators of
all orders k ≥ k, while (1.8) shows that the action of ∆ does not depend on
the order assigned to this operator.
Example 1.1. Let us show that the k-jet operator jk : “(π) ’ “(πk )
(see De¬nition 1.3) is di¬erential. To do this, recall that the total space of

the pull-back πk (πk ) consists of points (θk , θk ) ∈ J k (π) — J k (π) such that

2
For the sake of briefness, we shall use the words operator of order k below as a
synonym of the expression operator of order ¤ k.
6 1. CLASSICAL SYMMETRIES

πk (θk ) = πk (θk ). Consequently, we may de¬ne the diagonal section ρk of
def

the bundle πk (πk ) by setting ρk (θk ) = (θk , θk ). Obviously, jk = ∆ρk , i.e.,
jk (•)— (ρk ) = jk (•), • ∈ “(π).
The operator jk is linear.
Example 1.2. Let „ — : T — M ’ M be the cotangent bundle of M and
„p : p T — M ’ M be its p-th external power. Then the de Rham di¬er-

— —
ential d is a ¬rst order linear di¬erential operator acting from „ p to „p+1 ,
p ≥ 0.
Example 1.3. Consider a pseudo-Riemannian manifold M with a non-
degenerate metric g ∈ “(S 2 „ — ) (by S q ξ we denote the q-th symmetric power
of the vector bundle ξ). Let g — ∈ “(S 2 „ ) be its dual, „ : T M ’ M be-
ing the tangent bundle. Then the correspondence ∆g : f ’ g — (df, df ) is a
(nonlinear) ¬rst order di¬erential operator from C ∞ (M ) to C ∞ (M ).
Let ∆ : “(π) ’ “(π ) and ∆ : “(π ) ’ “(π ) be two di¬erential opera-
tors. It is natural to expect that their composition ∆ —¦ ∆ : “(π) ’ “(π )
is a di¬erential operator as well. However to prove this fact is not quite
simple. To do it, we need two new and important constructions.
Let ∆ : “(π) ’ “(π ) be a di¬erential operator of order k. For any
θk = [•]k ∈ J k (π), let us set
x
def
¦∆ (θk ) = [∆(•)]0 = (∆(•))(x). (1.12)
x

Evidently, the mapping ¦∆ is a morphism of ¬ber bundles3 , i.e., the diagram
¦∆
J k (π) ’E
πk




π







M
is commutative.
Definition 1.6. The map ¦∆ is called the representative morphism of
the operator ∆.
For example, for ∆ = jk we have ¦jk = idJ k (π) . Note that there exists a
one-to-one correspondence between nonlinear di¬erential operators and their
representative morphisms: one can easily see it just by inverting equality
(1.12). In fact, if ¦ : J k (π) ’ E is a morphism of the bundle π to π ,
a section • ∈ F(π, π ) can be de¬ned by setting •(θk ) = (θk , ¦(θk )) ∈
J k (π) — E . Then, obviously, ¦ is the representative morphism for ∆ = ∆• .
Definition 1.7. Let ∆ : “(π) ’ “(π ) be a k-th order di¬erential oper-
def
ator. Its l-th prolongation is the composition ∆(l) = jl —¦ ∆ : “(π) ’ “(πl ).
3
But not of vector bundles!
1. JET SPACES 7

Lemma 1.3. For any k-th order di¬erential operator ∆, its l-th prolon-
gation is a (k + l)-th order operator.
(l) def
Proof. In fact, for any point θk+l = [•]x ∈ J k+l (π) let us set ¦∆ =
k+l
(l)
[∆(•)]l ∈ J l (π). Then the operator , for which the morphism ¦∆ is
x
representative, coincides with ∆(l) .
Corollary 1.4. The composition ∆ —¦ ∆ of two di¬erential operators
∆ : “(π) ’ “(π ) and ∆ : “(π ) ’ “(π ) of order k and k respectively is a
(k + k )-th order di¬erential operator.
(k )
Proof. Let ¦∆ : J k+k (π) ’ J k (π ) be the representative morphism
(k )
for ∆(k ) . Then the operator , for which the composition ¦∆ —¦ ¦∆ is the
representative morphism, coincides with ∆ —¦ ∆.
To ¬nish this subsection, we shall list main properties of prolongations
and representative morphisms trivially following from the de¬nitions.
Proposition 1.5. Let ∆ : “(π) ’ “(π ), ∆ : “(π ) ’ “(π ) be two
di¬erential operators of orders k and k respectively. Then:
(k )
(i) ¦∆ —¦∆ = ¦∆ —¦ ¦∆ ,
(l)
(ii) ¦∆ —¦ jk+l (•) = ∆(l) (•) for any • ∈ “(π), l ≥ 0,
(l) (l )
(iii) πl,l —¦ ¦∆ = ¦∆ —¦ πk+l,k+l , i.e., the diagram
(l)
¦∆
J k+l (π) ’ J l (π )


πk+l,k+l πl,l (1.13)
“ “
(l )
¦∆
J k+l (π) ’ J l (π )
is commutative for all l ≥ l ≥ 0.
1.3. In¬nite jets. We now pass to in¬nite limit in all previous con-
structions.
Definition 1.8. The space of in¬nite jets J ∞ (π) of the ¬ber bundle
π : E ’ M is the inverse limit of the sequence
πk+1,k π1,0 π
· · · ’ J k+1 (π) ’ ’ ’ J k (π) ’ · · · ’ J 1 (π) ’ ’ E ’’ M,
’’ ’ ’
i.e., J ∞ (π) = proj lim{πk,l ,k≥l} J k (π).
Though J ∞ (π) is an in¬nite-dimensional manifold, no topological or
analytical problems arise, if one bears in mind the genesis of this manifold
(i.e., the system of maps πk,l ) when maintaining all constructions. Below
we demonstrate how this should be done, giving de¬nitions for all necessary
concepts over J ∞ (π).
8 1. CLASSICAL SYMMETRIES

A point θ of J ∞ (π) is a sequence of points {x, θk }k≥0 , x ∈ M, θk ∈
J k (π), such that πk (θk ) = x and πk,l (θk ) = θl , k ≥ l. Let us represent
any θk in the form θk = [•k ]k . Then the Taylor expansions of any two
x
sections, •k and •l , k ≥ l, coincide up to the l-th term. It means that the
points of J ∞ (π) can be understood as m-dimensional formal series. But
by the Whitney theorem on extensions of smooth functions [71], for any
such a series there exists a section • ∈ “(π) such that its Taylor expansion
coincides with this series. Hence, any point θ ∈ J ∞ (π) can be represented
in the form θ = [•]∞ . x
A special coordinate system can be chosen in J ∞ (π) due to the fact
that if a trivialization {U± }± gives special coordinates for some J k (π), then
these coordinates can be used for all jet spaces J k (π) simultaneously. Thus,
the functions x1 , . . . , xn , . . . , uj , . . . can be taken for local coordinates in
σ
∞ (π), where j = 1, . . . , m and σ is an arbitrary multi-index of the form
J
(σ1 , . . . , σn ).
A tangent vector to J ∞ (π) at a point θ is de¬ned as follows. Let
θ = {x, θk } and w ∈ Tx M , vk ∈ Tθk J k (π). Then the system of vectors
{w, vk }k≥0 determines a tangent vector to J ∞ (π) if and only if (πk )— vk = w,
(πk,l )— vk = vl for all k ≥ l ≥ 0.
A smooth bundle ξ over J ∞ (π) is a system of bundles · : Q ’ M ,
ξk : Pk ’ J k (π) together with smooth mappings Ψk : Pk ’ Q, Ψk,l : Pk ’
Pl , k ≥ l ≥ 0, such that

Ψl —¦ Ψk,l = Ψk , Ψk,l —¦ Ψl,s = Ψk,s , k ≥ l ≥ s ≥ 0,

and all the diagrams

Ψk,l Ψl
’ Pl ’Q
Pk


ξk ξl ·
“ “ “
πk,l πl
J k (π) ’ J l (π) ’M

are commutative. For example, if · : Q ’ M is a bundle, then the pull-backs
πk (·) : πk (Q) ’ J k (π) together with the natural projections πk (·) ’ πl— (·),
— — —

πk (·) ’ Q form a bundle over J ∞ (π). We say that ξ is a vector bundle


over J ∞ (π), if · and all ξk are vector bundles and the mappings Ψk , Ψk,l
are ¬ber-wise linear.
A smooth mapping of J ∞ (π) to J ∞ (π ), where π : E ’ M , π : E ’
M , is de¬ned as a system F of mappings F’∞ : M ’ M , Fk : J k (π) ’
J k’s (π ), k ≥ s, where s ∈ Z is a ¬xed integer called the degree of F , such
that

πk’r,k’s’1 —¦ Fk = Fk’1 —¦ πk,k’1 , k ≥ s + 1.
1. JET SPACES 9

For example, if ∆ : “(π) ’ “(π ) is a di¬erential operator of order s, then
(k’s)
the system of mappings F’∞ = idM , Fk = ¦∆ , k ≥ s (see the previous
subsection), is a smooth mapping of J ∞ (π) to J ∞ (π ).
We say that two smooth mappings F = {Fk }, G = {Gk } : J ∞ (π) ’
J ∞ (π ) of degrees s and l respectively, l ≥ s, are equivalent, if the diagrams
πk’s,k’l
J k’s (π ) ’ J k’l (π )








Fk




k
G
J k (π)
are commutative for all admissible k ≥ 0. When working with smooth
mappings, one can always choose the representative of maximal degree in
any class of equivalent mappings. In particular, it can be easily seen that
mappings with negative degrees reduce to zero degree ones in such a way.
Remark 1.3. The construction above can be literally generalized to the
following situation. Consider the category M∞ , whose objects are chains
mk+1,k
m1,0
m
M’∞ ← M0 ← ’ M1 ← · · · ← Mk ← ’ ’ Mk+1 ← · · · ,
’ ’’ ’ ’’
where M’∞ and all Mk , k ≥ 0, are ¬nite-dimensional smooth manifolds
while m and mk+1,k are smooth mappings. Let us set
def def
mk = m —¦ m1,0 —¦ · · · —¦ mk,k’1 , mk,l = ml+1,l —¦ · · · —¦ mk,k’1 , k ≥ l.
De¬ne a morphism of two objects, {Mk }, {Nk }, as a system F of mappings
{F’∞ , Fk } such that the diagram
mk,l
’ Ml
Mk

Fk Fl
“ “
nk,l
’ Nl’s
Nk’s
is commutative for all admissible k and a ¬xed s (degree of F ).
Example 1.4. Let M and N be two smooth manifolds, F : N ’ M be
a smooth mapping, and π : E ’ M a be vector bundle. Consider the pull-
def
backs F — (πk ) = πF,k : JF (π) ’ N , where JF (π) denotes the corresponding
k k

total space. Thus {N, JF (π)}k≥0 is an object of M∞ .
k

To any section φ ∈ “(πk ), there corresponds the section φF ∈ “(πF,k )
def def
de¬ned by φF (x) = (x, φF (x)), x ∈ N (for any x ∈ N , we set φF (x) =
(x, φ(F (x))). In particular, for φ = jk (•), • ∈ “(π) we obtain the section
10 1. CLASSICAL SYMMETRIES

jk (•)F . Let ξ : H ’ N be another vector bundle and ψ be a section of the

pull-back πF,k (ξ). Then the correspondence
• ’ jk (•)— (ψ),
∆ = ∆ψ : “(π) ’ “(ξ), F
is called a (nonlinear ) di¬erential operator of order ¤ k over the mapping
F . As before, we can de¬ne prolongations ∆(l) : “(πk+l ) ’ “(ξl ) and these
(l) k+l
prolongations would determine smooth mappings ¦∆ : JF (π) ’ J l (ξ).
(l)
The system {¦∆ }l≥0 is a morphism of {JF (π)} to {J k (ξ)}.
k

Note that if F : N ’ M , G : O ’ N are two smooth maps and ∆, are
two nonlinear operators over F and G respectively, then their composition
is de¬ned and is a nonlinear operator over F —¦ G.
Example 1.5. The category M of smooth manifolds is embedded into
M∞ , if for any smooth manifold M one sets M∞ = {Mk , mk,k’1 } with
Mk = M and mk,k’1 = idM . For any smooth mapping f : M ’ N we also
set f∞ = {fk } with fk = f . We say that F is a smooth mapping of J ∞ (π)
to a smooth manifold N , if F = {Fk } is a morphism of {J k (π), πk,k’1 } to
N∞ . In accordance to previous constructions, such a mapping is completely
determined by some f : J k (π) ’ N .
Taking R for the manifold N in the previous example, we obtain a de¬-
nition of a smooth function on J ∞ (π). Thus, a smooth function on J ∞ (π)
is a function on J k (π) for some ¬nite but an arbitrary k. The set F(π) of
such functions is identi¬ed with ∞ Fk (π) and forms a commutative ¬l-
k=0
tered algebra. Using the well-known duality between smooth manifolds and
algebras of smooth functions on these manifolds, we deal in what follows
with the algebra F(π) rather than with the manifold J ∞ (π) itself.
From this point of view, a vector ¬eld on J ∞ (π) is a ¬ltered derivation
of F(π), i.e., an R-linear map X : F(π) ’ F(π) such that
f, g ∈ F(π), X(Fk (π)) ‚ Fk+l (π),
X(f g) = f X(g) + gX(f ),
for all k and some l = l(X). The latter is called the ¬ltration of the ¬eld
X. The set of all vector ¬elds is a ¬ltered Lie algebra over R with respect
to commutator [X, Y ] and is denoted by D(π) = l≥0 D(l) (π).
Di¬erential forms of degree i on J ∞ (π) are de¬ned as elements of the
def def
¬ltered F(π)-module Λi (π) = k≥0 Λi (πk ), where Λi (πk ) = Λi (J k (π)) and

the module Λi (πk ) is considered to be embedded into Λi (πk+1 ) by πk+1,k .
De¬ned in such a way, these forms possess all basic properties4 of di¬erential
forms on ¬nite-dimensional manifolds. Let us mention most important ones:
(i) The module Λi (π) is the i-th external power of the module Λ1 (π),
Λi (π) = i Λ1 (π). Respectively, the operation of wedge product
§ : Λp (π) — Λq (π) ’ Λp+q (π) is de¬ned and Λ— (π) = i
i≥0 Λ (π)
becomes a commutative graded algebra.
4
In fact, as we shall see in Section 1 of Chapter 2, Λi (π) is structurally much richer
than forms on a ¬nite-dimensional manifold.
1. JET SPACES 11

(ii) The module D(π) is dual to Λ1 (π), i.e.,
D(π) = homφ (π) (Λ1 (π), F(π)), (1.14)
F

where homφ (π) (·, ·) denotes the module of all ¬ltered homomorphisms
F
over F(π). Moreover, equality (1.14) is established in the following
way: there is a derivation d : F(π) ’ Λ1 (π) such that for any vector
¬eld X there exists a uniquely de¬ned ¬ltered homomorphism fX for
which the diagram
d
’ Λ1 (π)
F(π)
X




fX



F(π)
is commutative.
(iii) The operator d is extended up to maps d : Λi (π) ’ Λi+1 (π) in such
a way that the sequence
d d
0 ’ F(π) ’’ Λ1 (π) ’ · · · ’ Λi (π) ’’ Λi+1 (π) ’ · · ·
’ ’
becomes a complex, i.e., d —¦ d = 0. This complex is called the de
Rham complex on J ∞ (π) while d is called the de Rham di¬erential.
The latter is a derivation of the superalgebra Λ— (π).
Using the identi¬cation (1.14), we can de¬ne the inner product (or con-
traction) of a ¬eld X ∈ D(π) with a 1-form ω ∈ Λ1 (π):
def
iX ω = fX (ω). (1.15)
We shall also use the notation X ω for the contraction of X to ω. This
operation extends onto Λ— (π), if we set
iX (ω § θ) = iX (ω) § θ + (’1)ω ω § iX (θ)
iX f = 0,
for all f ∈ F(π) and ω, θ ∈ Λ— (π) (here and below we always write (’1)ω
instead of (’1)deg ω ).
With the de Rham di¬erential and interior product de¬ned, we can
introduce the Lie derivative of a form ω ∈ Λ— (π) along a ¬eld X by setting
def
LX ω = iX (dω) + d(iX ω)
(the in¬nitesimal Stokes formula). We shall also denote the Lie derivative by
X(ω). Other constructions related to di¬erential calculus over J ∞ (π) (and
over in¬nite-dimensional objects of a more general nature) will be described
in Chapter 4.
Linear di¬erential operators over J ∞ (π) generalize the notion of
derivations and are de¬ned as follows. Let P and Q be two ¬ltered F(π)-
modules and ∆ ∈ homφ (P, Q). Then ∆ is called a linear di¬erential operator
R
12 1. CLASSICAL SYMMETRIES

of order k acting from P to Q, if
(δf0 —¦ δf1 —¦ · · · —¦ δfk )∆ = 0
def
for all f0 , . . . , fk ∈ F(π), where (δf ∆)p = f ∆(p) ’ ∆(f p). We write k =
ord(∆).
Due to existence of ¬ltrations in F(π), P and Q, one can de¬ne di¬er-
ential operators of in¬nite order acting from P to Q, [51]. Namely, let
P = {Pl }l , Q = {Ql }l , Pl ‚ Pl+1 , Ql ‚ Ql+1 , Pl , Ql being Fl (π)-modules.
Let ∆ ∈ homφ (P, Q) and s be ¬ltration of ∆, i.e., ∆(Pl ) ‚ Ql+s . We can
R
def
always assume that s ≥ 0. Suppose now that ∆l = ∆ |Pl : Pl ’ Ql is a
linear di¬erential operator of order ol over Fl (π). Then we say that ∆ is a
linear di¬erential operator of order growth ol . In particular, if ol = ±l + β,
±, β ∈ R, we say that ∆ is of constant growth ±.
Distributions. Let θ ∈ J ∞ (π). The tangent plane to J ∞ (π) at the
point θ is the set of all tangent vectors to J ∞ (π) at this point (see above).
Denote such a plane by Tθ = Tθ (J ∞ (π)). Let θ = {x, θk }, x ∈ M , θk ∈ J k (π)
and v = {w, vk }, v = {w , vk } ∈ Tθ . Then the linear combination »v +µv =
{»w +µw , »vk +µvk } is again an element of Tθ and thus Tθ is a vector space.
A correspondence T : θ ’ Tθ ‚ Tθ , where Tθ is a linear subspace, is called a
distribution on J ∞ (π). Denote by T D(π) ‚ D(π) the submodule of vector
¬elds lying in T , i.e., a ¬eld X belongs to T D(π) if and only if Xθ ∈ Tθ for all
θ ∈ J ∞ (π). We say that the distribution T is integrable, if it satis¬es formal
Frobenius condition: for any vector ¬elds X, Y ∈ T D(π) their commutator
lies in T D(π) as well, or [T D(π), T D(π)] ‚ T D(π).
This condition can expressed in a dual way as follows. Let us set
T 1 Λ(π) = {ω ∈ Λ1 (π) | iX ω = 0, X ∈ T D(π)}
and consider the ideal T Λ— (π) generated in Λ— (π) by T 1 Λ(π). Then the
distribution T is integrable if and only if the ideal T Λ— (π) is di¬erentially
closed: d(T Λ— (π)) ‚ T Λ— (π).
Finally, we say that a submanifold N ‚ J ∞ (π) is an integral manifold
of T , if Tθ N ‚ Tθ for any point θ ∈ N . An integral manifold N is called
locally maximal at a point θ ∈ N , if there exist no other integral manifold
N such that N ‚ N .

2. Nonlinear PDE
In this section we introduce the notion of a nonlinear di¬erential equa-
tion and discuss some important concepts related to this notion: solutions,
symmetries, and prolongations.

2.1. Equations and solutions. Let π : E ’ M be a vector bundle.
Definition 1.9. A submanifold E ‚ J k (π) is called a (nonlinear ) dif-
ferential equation of order k in the bundle π. We say that E is a linear
’1 ’1
equation, if E © πx (x) is a linear subspace in πx (x) for all x ∈ M .
2. NONLINEAR PDE 13

We say that the equation E is determined, if codim E = dim π, that it
is overdetermined, if codim E > dim π, and that it is underdetermined, if
codim E < dim π.
We shall always assume that E is projected surjectively onto E under
πk,0 .
Definition 1.10. A (local) section f of the bundle π is called a (local)
solution of the equation E, if its graph lies in E: jk (f )(M ) ‚ E.
Let us show that these de¬nitions are in agreement with the traditional
ones. Choose in a neighborhood U of a point θ ∈ E a special coordinate sys-
tem x1 , . . . , xn , u1 , . . . , um , . . . , uj , . . . , where |σ| ¤ k, j = 1, . . . , m. Then,
σ
in this coordinate system, E will be given by a system of equations
±
F 1 (x1 , . . . , xn , u1 , . . . , um , . . . , u1 , . . . , um , . . . ) = 0,
 σ σ
(1.16)
.................................................
r

F (x1 , . . . , xn , u1 , . . . , um , . . . , u1 , . . . , um , . . . ) = 0,
σ σ

where the functions F 1 , . . . , F r are functionally independent. Now, let
f ∈ “loc (π) be a section locally expressed in the form of relations u1 =
f 1 (x1 , . . . , xn ), . . . , um = f m (x1 , . . . , xn ). Then its k-jet is given by the
equalities
‚ |σ| f j
uj = ,
σ
‚xσ
¯¯
where j = 1, . . . , m, 0 ¤ |σ ¤ k, and jk (f )(U ), U = πk (U) ‚ M , lies in E if
and only if the equations
± |σ| 1 |σ| m
F (x1 , . . . , xn , f 1 , . . . , f m , . . . , ‚ f , . . . , ‚ f , . . . ) = 0,
1


 ‚xσ ‚xσ
..........................................................

 |σ| 1 |σ| m
r
F (x , . . . , x , f 1 , . . . , f m , . . . , ‚ f , . . . , ‚ f , . . . ) = 0.
 1 n
‚xσ ‚xσ
are satis¬ed. Thus we are in a complete correspondence with the analytical
de¬nition of a di¬erential equation.
Remark 1.4. There exists another way to represent di¬erential equa-
tions. Namely, let π : Rr — U ’ U be the trivial r-dimensional bundle.
Then the set of functions F 1 , . . . , F r can be understood as a section • of
the pull-back (πk |U )— (π ), or as a nonlinear operator ∆ = ∆• de¬ned in U,
while the equation E is characterized by the condition
E © U = {θk ∈ U | •(θk ) = 0}. (1.17)
More general, any equation E ‚ J k (π) can be represented in the form similar
to (1.17). Namely, for any equation E there exists a ¬ber bundle π : E ’ M
and a section • ∈ Fk (π, π) such that E coincides with the set of zeroes for
• : E = {• = 0}. In this case we say that E is associated to the operator
∆ = ∆• : “(π) ’ “(π ) and use the notation E = E∆ .
14 1. CLASSICAL SYMMETRIES

Example 1.6. Consider the bundles π = „p : p T — M ’ M , π =


„p+1 : p+1 T — M ’ M and let d : “(π) = Λp (M ) ’ “(π ) = Λp+1 (M )


be the de Rham di¬erential (see Example 1.2). Thus we obtain a ¬rst-order

equation Ed in the bundle „p . Consider the case p = 1, n ≥ 2 and choose
local coordinates x1 , . . . , xn in M . Then any form ω ∈ Λ1 (M ) is represented
as ω = u1 dx1 + · · · + un dxn and we have
Ed = {uj i = ui j | i < j},
1
1

where 1i denotes the multi-index (0, . . . , 1, . . . , 0) with zeroes at all positions
except for the i-th one. This equation is underdetermined when n = 2,
determined for n = 3 and overdetermined for n > 3.
Example 1.7 (see [69]). Consider an arbitrary vector bundle π : E ’
M and a di¬erential form ω ∈ Λp (J k (π)), p ¤ dim M . The condition
jk (•)— (ω) = 0, • ∈ “(π), determines a (k + 1)-st order equation Eω in
the bundle π. Consider the case p = dim M = 2, k = 1 and choose a special
coordinate system x, y, u, ux , uy in J k (π). Let • = •(x, y) be a local section
and

ω = A dux § duy + (B1 dux + B2 duy ) § du
+ dux § (B11 dx + B12 dy) + duy § (B21 dx + B22 dy)
+ du § (C1 dx + C2 dy) + D dx § dy,
where A, Bi , Bij , Ci , D are functions of x, y, u, ux , uy . Then we have

j1 (•)— ω = A• (•xx dx + •xy dy) § (•yx dx + •yy dy)
• •
+ B1 (•xx dx + •xy dy) + B2 (•yx dx + •yy dy) § (•x dx + •y dy)
• • • •
+(•xx dx+•xy dy)§(B11 dx+B12 dy)+(•yx dx+•yy dy)§(B21 dx+B22 dy)
• •
+ (•x dx + •y dy) § (C1 dx + C2 dy) + D • dx § dy,
def
where F • = j1 (•)— F for any F ∈ F1 (π). Simplifying the last expression,
we obtain
• • • •
j1 (•)— ω = A• (•xx •yy ’ •2 ) + (•y B1 + B12 )•xx ’ (•x B2 + B12 )•yy
xy
• • • • • •
+ (•y B2 ’ •x B1 + B22 ’ B11 )•xy + •x C2 ’ •y C1 + D• ) dx § dy.

Hence, the equation Eω is of the form
a(uxx uyy ’ u2 ) + b11 uxx + b12 uxy + b22 uyy + c = 0, (1.18)
xy

where a = A, b11 = uy B1 + B12 , b12 = uy B2 ’ ux B1 + B22 ’ B11 , b22 =
ux B2 + B12 , c = ux C2 ’ uy C1 + D are functions on J 1 (π). Equation (1.18)
is the so-called two-dimensional Monge“Ampere equation and obviously any
such an equation can be represented as Eω for some ω ∈ Λ1 (J 1 (π)).
2. NONLINEAR PDE 15

Note that we have constructed a correspondence between p-forms on
J k (π)
and (p + 1)-order operators. This correspondence will be described
di¬erently in Subsection 1.4 of Chapter 2
Example 1.8. Consider again a ¬ber bundle π : E ’ M and a section
: E ’ J 1 (π) of the bundle π1,0 : J 1 (π) ’ E. Then the graph E =
(E) ‚ J 1 (π) is a ¬rst-order equation in the bundle π. Let θ1 ∈ E . Then,
due to Proposition 1.2 on page 5, θ1 is identi¬ed with the pair (θ0 , Lθ1 ), where
θ0 = π1,0 (θ1 ) ∈ E, while Lθ1 is the R-plane at θ0 corresponding to θ1 . Hence,
(or the equation E ) may be understood as a distribution
the section
5 n-dimensional planes on E : T : E θ ’ θ1 = L (θ) . A
of horizontal
solution of the equation E , by de¬nition, is a section • ∈ “(π) such that
j1 (•)(M ) ‚ (E). It means that at any point θ = •(x) ∈ •(M ) the plane
T (θ) is tangent to the graph of the section •. Thus, solutions of E coincide
with integral manifolds of T .
def
In local coordinates (x1 , . . . , xn , u1 , . . . , um , . . . , uj , . . . ), where uj = uj i ,
1
i i
i = 1, . . . , n, j = 1, . . . , m, the equation E is represented as
uj = j 1 m
i (x1 , . . . , xn , u , . . . , u ), i = 1, . . . , n, j = 1, . . . , m, (1.19)
i
j
being smooth functions.
i

Example 1.9. As we saw in the previous example, to solve the equation
E is the same as to ¬nd integral n-dimensional manifolds of the distribution
T . Hence, the former to be solvable, the latter is to satisfy the Frobenius
theorem conditions. Thus, for solvable E , we obtain conditions on the
section ∈ “(π1,0 ). Let us write down these conditions in local coordinates.
Using representation (1.19), note that T is given by the 1-forms
j
ω j = duj ’ dxi , j = 1, . . . , m.
i
i=1n
Hence, the integrability conditions may be expressed as
m
ρj § ω i ,
j
dω = j = 1, . . . , m,
i
i=1

for some 1-forms ρi . After elementary computations, we obtain that the
i
j
functions i must satisfy the following relations:
j j
m m
‚j j
‚ ‚ γ‚ ±
± β β
γ
+ = + (1.20)
± β ‚uγ
‚uγ
‚xβ ‚x±
γ=1 γ=1

for all j = 1, . . . , m, 1 ¤ ± < β ¤ m. Thus we got a naturally constructed
¬rst-order equation I(π) ‚ J 1 (π1,0 ) whose solutions are horizontal n-dimen-
sional distributions in E = J 1 (π).
5
An n-dimensional plane L ‚ Tθk (J k (π)) is called horizontal, if it projects nondegen-
erately onto Tx M under (πk )— , x = πk (θk ).
16 1. CLASSICAL SYMMETRIES

Remark 1.5. Let us consider the previous two examples from a bit dif-
ferent point of view. Namely, the horizontal distribution T (or the section
: J 0 (π) ’ J 1 (π), which is the same, as we saw above) may be understood
as a connection in the bundle π. By the latter we understand the following.
Let X be a vector ¬eld on the manifold M . Then, for any point x ∈ M ,
the vector Xx ∈ Tx M can be uniquely lifted up to a vector Xx ∈ Tθ E,
π(θ) = x, such that Xx ∈ T (θ). In such a way, we get the correspon-
dence D(M ) ’ D(E) which we shall denote by the same symbol . This
correspondence possesses the following properties:
(i) it is C ∞ (M )-linear, i.e., (f X + gY ) = f (X) + g (Y ), X, Y ∈
D(M ), f, g ∈ C ∞ (M );
(ii) for any X ∈ D(M ), the ¬eld (X) is projected onto M in a well-
de¬ned way and π— (X) = X.
Equation (1.20) is equivalent to ¬‚atness of the connection , which means
that
([X, Y ]) ’ [ (X), (Y )] = 0, X, Y ∈ M, (1.21)

i.e., that is a homomorphism of the Lie algebra D(M ) of vector ¬elds on
M to the Lie algebra D(E).
In Chapter 4 we shall deal with the concept of connection in a more
extensive and general manner. In particular, it will allow us to construct
equations (1.20) invariantly, without use of local coordinates.
Example 1.10. Let π : Rm —Rn+1 ’ Rn+1 be the trivial m-dimensional
bundle. Then the system of equations

uj n+1 = f j (x1 , . . . , xn+1 , . . . , u±1 ,...,σn ,0 , . . . ), (1.22)
σ
1

where j, ± = 1, . . . , m, is called evolutionary. In more conventional notations
this system is written down as
‚uj ‚ σ1 +···+σn u±
j
= f (x1 , . . . , xn , t, . . . , σ1 , . . . ),
‚x1 . . . ‚xσn
‚t n

where the independent variable t corresponds to xn+1 .
2.2. The Cartan distributions. Now we know what a di¬erential
equation is, but cannot speak about geometry of these equation. The rea-
son is that the notion of geometry implies the study of smooth manifolds
(spaces) enriched with some additional structures. In particular, transfor-
mation groups preserving these structures are of great interest as it was
stated in the Erlangen Program by Felix Klein [45].
Our nearest aim is to use this approach to PDE and the main question
to be answered is
What are the structures making di¬erential equations of smooth man-
ifolds?
2. NONLINEAR PDE 17

At ¬rst glance, the answer is clear: solutions are those entities for the sake of
which di¬erential equations are studied. But this viewpoint can hardly con-
sidered to be constructive: to implement it, one needs to know the solutions
of the equation at hand and this task, in general, is transcendental.
This means that we need to ¬nd a construction which, on one hand,
contains all essential information about solutions and, on the other hand,
can be e¬ciently studied by the tools of di¬erential geometry.
Definition 1.11. Let π : E ’ M be a vector bundle. Consider a point
θk ∈ J k (π) and the span Cθk ‚ Tθk (J k (π)) of all R-planes (see De¬nition
k

1.4) at the point θk .
(i) The correspondence C k = C k (π) : θk ’ Cθk is called the Cartan dis-
k

tribution on J k (π).
(ii) Let E ‚ J k (π) be a di¬erential equation of order k. The correspon-
dence C k (E) : E θk ’ Cθk © Tθk E ‚ Tθk E is called the Cartan dis-
k

tribution on E. We call elements of the Cartan distributions Cartan
planes.
k
(iii) A point θk ∈ E is called regular, if the Cartan plane Cθk (E) is of
maximal dimension. We say that E is a regular equation, if all its
points are regular.
In what follows, we deal with regular equations or in neighborhoods of
regular points6 .
We are now going to give an explicit description of Cartan distribu-
tions on J k (π) and to describe their integral manifolds. Let θk ∈ J k (π) be
represented in the form
θk = [•]k , • ∈ “(π), x = πk (θk ). (1.23)
x
k
Then, by de¬nition, the Cartan plain Cθk is spanned by the vectors
v ∈ Tx M,
jk (•)—,x (v), (1.24)
for all • ∈ “loc (π) satisfying (1.23).
Let x1 , . . . , xn , . . . , uj , . . . , j = 1, . . . , m, |σ| ¤ k, be a special coordinate
σ
def
system in a neighborhood of θk . Introduce the notation ‚xi = ‚/‚xi ,
def
‚uσ = ‚/‚uσ . Then the vectors of the form (1.24) can be expressed as
linear combinations of the vectors
m
‚ |σ|+1 •j j
‚xi + ‚uσ , (1.25)
‚xσ ‚xi

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