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)

.................................................