Ann(H) = {v ∈ Fθk | δw v = 0, ∀w ∈ H}.

Then, as it follows from Proposition 1.8, the following description of maximal

involutive subspaces takes place:

Corollary 1.9. Let θk = [•]k , • ∈ “loc (π). Then any maximal invo-

x

k

lutive subspace V ‚ Cθk (π) is of the form

V = jk (•)— (H) • Ann(H)

for some H ‚ Tx M .

If V is a maximal involutive subspace, then the corresponding space

H is obviously πk,— (V ). We call dimension of H the type of the maximal

involutive subspace V and denote it by tp(V ).

Proposition 1.10. Let V be a maximal involutive subspace. Then

n’r+k’1

dim V = m + r,

k

where n = dim M , m = dim π, r = tp(V ).

Proof. Choose local coordinates in M in such a way that the vectors

‚x1 , . . . , ‚xr form a basis in H. Then, in the corresponding special system in

J k (π), coordinates along Ann(H) will consist of those functions uj , |σ| = k,

σ

for which σ1 = · · · = σr = 0.

We can now describe maximal integral manifolds of the Cartan distri-

bution on J k (π).

Let N ‚ J k (π) be such a manifold θk ∈ N . Then the tangent plane to

N at the point θk is a maximal involutive plane. Assume that its type is

equal to r(θk ).

Definition 1.15. The number

def

tp(N ) = max r(θk ).

θk ∈N

is called the type of the maximal integral manifold N of the Cartan distri-

bution.

Obviously, the set

def

g(N ) = {θk ∈ N | r(θk ) = tp(N )}

is everywhere dense in N . We call the points θk ∈ g(N ) generic. Let θk be

such a point and U be its neighborhood in N consisting of generic points.

Then:

7

Using the linear structure, we identify the ¬ber Fθk of the bundle πk,k’1 with its

tangent space.

2. NONLINEAR PDE 21

(i) N = πk,k’1 (N ) is an integral manifold of the Cartan distribution on

J k’1 (π);

(ii) dim(N ) = tp(N );

(iii) πk’1 |N : N ’ M is an immersion.

Theorem 1.11. Let N ‚ J k’1 (π) be an integral manifold of the Cartan

distribution on J k (π) and U ‚ N be an open domain consisting of generic

points. Then

U = {θk ∈ J k (π) | Lθk ⊃ Tθk’1 U },

where θk’1 = πk,k’1 (θk ), U = πk,k’1 (U).

Proof. Let V = πk’1 (U ) ‚ M . Denote its dimension (which equals

the number tp(N )) by r and choose local coordinates in M in such a way

that the submanifold V is determined by the equations xr+1 = · · · = xn = 0

in these coordinates. Then, since U ‚ J k’1 (π) is an integral manifold

and πk’1 |U : U ’ V is a di¬eomorphism, in the corresponding special

coordinates the manifold U is given by the equations

± |σ| j

‚ •

, if σ = (σ1 , . . . , σr , 0, . . . , 0),

uj = ‚xσ

σ

0, otherwise,

for all j = 1, . . . , m, |σ| ¤ k’1 and some smooth function • = •(x1 , . . . , xr ).

Hence, the tangent plane H to U at θk’1 is spanned by the vectors of the

form (1.25) with i = 1, . . . , r. Consequently, a point θk , such that Lθk ⊃ H,

is determined by the coordinates

± |σ| j

‚ •

, if σ = (σ1 , . . . , σr , 0, . . . , 0),

j

uσ = ‚xσ

arbitrary real numbers, otherwise,

where j = 1, . . . , m, |σ| ¤ k. Hence, if θk , θk are two such points, then the

vector θk ’ θk lies in Ann(H), as it follows from the proof of Proposition

1.10. As it is easily seen, any integral manifold of the Cartan distribution

projecting onto U is contained in U, which ¬nishes the proof.

Remark 1.7. Note that maximal integral manifolds N of type dim M

are exactly graphs of jets jk (•), • ∈ “loc (π). On the other hand, if tp(N ) =

0, then N coincides with a ¬ber of the projection πk,k’1 : J k (π) ’ J k’1 (π).

2.3. Symmetries. The last remark shows that the Cartan distribution

on J k (π) is in a sense su¬cient to restore the structures speci¬c to the jet

manifolds. This motivates the following de¬nition:

Definition 1.16. Let U, U ‚ J k (π) be open domains.

(i) A di¬eomorphism F : U ’ U is called a Lie transformation, if it

preserves the Cartan distribution, i.e.,

k k

F— (Cθk ) = CF (θk )

for any point θk ∈ U.

22 1. CLASSICAL SYMMETRIES

Let E, E ‚ J k (π) be di¬erential equations.

(ii) A Lie transformation F : U ’ U is called a (local ) equivalence, if

F (U © E) = U © E .

(iii) A (local) equivalence is called a (local ) symmetry, if E = E and

U = U . Such symmetries are also called classical 8 .

Below we shall not distinguish between local and global versions of the

concepts introduced.

Remark 1.8. There is an alternative approach to the concept of a sym-

metry. Namely, we can introduce the Cartan distribution on E by setting

def

Cθ (E) = Cθ © Tθ E, θ ∈ E,

and de¬ne interior symmetries of E as a di¬eomorphism F : E ’ E preserv-

ing C(E). In general, the group of these symmetries does not coincide with

the above introduced. A detailed discussion of this matter can be found in

[60].

Example 1.11. Consider the case J 0 (π) = E. Then, since any n-di-

mensional horizontal plane in Tθ E is tangent to some section of the bundle

0

π, the Cartan plane Cθ coincides with the whole space Tθ E. Thus the Car-

tan distribution is trivial in this case and any di¬eomorphism of E is a Lie

transformation.

Example 1.12. Since the Cartan distribution on J k (π) is locally deter-

mined by the Cartan forms (1.27), the condition of F to be a Lie transfor-

mation cam be reformulated as

m

F — ωσ =

j

»j,± ω„ ,

±

|σ| < k,

j = 1, . . . , m, (1.28)

σ,„

±=1 |„ |<k

where »j,± are smooth functions on J k (π). Equations (1.28) are the base

σ,„

for computations in local coordinates.

In particular, if dim π = 1 and k = 1, equations (1.28) reduce to the only

condition F — ω = »ω, where ω = du ’ n u1i dxi . Hence, Lie transforma-

i=1

tions in this case are just contact transformations of the natural contact

structure in J 1 (π).

Example 1.13. Let F : J 0 (π) ’ J 0 (π) be a di¬eomorphism (which can

be considered as a general change of dependent and independent coordi-

nates). Let us construct a Lie transformation F (1) of J 1 (π) such that the

8

Contrary to higher, or generalized, symmetries which will be introduced in the next

chapter.

2. NONLINEAR PDE 23

diagram

F (1)

1

’ J 1 (π)

J (π)

π1,0 π1,0

“ “

F

J 0 (π) ’ J 0 (π)

is commutative, i.e., π1,0 —¦ F (1) = F —¦ π1,0 . To do this, introduce local

coordinates x1 , . . . , xn , u1 , . . . , um in J 0 (π) and consider the corresponding

special coordinates in J 1 (π) denoting the functions uj i by pj . Express the

1 i

transformation F in the form

xi ’ Xi (x1 , . . . , xn , u1 , . . . , um ), uj ’ U j (x1 , . . . , xn , u1 , . . . , um ),

i = 1, . . . , n, j = 1, . . . , m, in these coordinates. Then, due to (1.28), to ¬nd

F (1) : pj ’ Pij (x1 , . . . , xn , u1 , . . . , um , p1 , . . . , pm ),

1 n

i

one needs to solve the system

n m n

Pij

j j,± ±

p± dxi ),

dU ’ (du ’

dXi = » i

±=1

i=1 i=1

j = 1, . . . , m, with respect to the functions Pij for arbitrary smooth coe¬-

cients »j,± . Using matrix notation p = pj , P = Pij and » = »±β , we

i

see that

‚U ‚X

’P —¦

»=

‚u ‚u

and

’1

‚U ‚U ‚X ‚X

—¦p —¦ —¦p

P= + + , (1.29)

‚x ‚u ‚x ‚u

where

‚U ± ‚U ±

‚X ‚X± ‚X ‚X± ‚U ‚U

= , = , = , =

‚uβ ‚uβ

‚x ‚xβ ‚u ‚x ‚xβ ‚u

denote Jacobi matrices. Note that the transformation F (1) , as it follows

from (1.29), is unde¬ned at some points of J 1 (π), i.e., at the points where

the matrix ‚X/‚x + ‚X/‚u —¦ p is not invertible.

Example 1.14. Let π : Rn —Rn ’ Rn , i.e., dim π = dim M and consider

the transformation ui ’ xi , xi ’ ui , i = 1, . . . , n. This transformation is

called the hodograph transformation. From (1.29) it follows that the corre-

sponding transformation of the functions pj is de¬ned by P = p’1 .

i

24 1. CLASSICAL SYMMETRIES

Example 1.15. Let Ed be the equation determined by the de Rham

di¬erential (see Example 1.6), i.e., Ed = {dω = 0}, ω ∈ Λi (M ). Then for

any di¬eomorphism F : M ’ M one has F — (dω) = d(F — ω) which means

that F determines a symmetry of Ed . Symmetries of this type are called

gauge symmetries.

The construction of Example 1.13 can be naturally generalized. Let

F : J k (π) ’ J k (π) be a Lie transformation. Note that from the de¬-

nition it follows that for any maximal integral manifold N of the Cartan

distribution on J k (π), the manifold F (N ) possesses the same property. In

particular, graph of k-jets are taken to n-dimensional maximal integral man-

ifolds. Let now θk+1 be a point of J k+1 (π) and let us represent θk+1 as a pair

(θk , Lθk+1 ), or, which is the same, as a class of graphs of k-jets tangent to

each other at θk . Then, since di¬eomorphisms preserve tangency, the image

F— (Lθk+1 ) will almost always (cf. Example 1.13) be an R-plane at F (θk ).

Denote the corresponding point in J k+1 (π) by F (1) (θk+1 ).

Definition 1.17. Let F : J k (π) ’ J k (π) be a Lie transformation. The

above de¬ned mapping F (1) : J k+1 (π) ’ J k+1 (π) is called the 1-lifting of F .

The mapping F (1) is a Lie transformation at the domain of its de¬nition,

since almost everywhere it takes graphs of (k + 1)-jets to graphs of the same

def

kind. Hence, for any l ≥ 1 we can de¬ne F (l) = (F (l’1) )(1) and call this

map the l-lifting of F .

Theorem 1.12. Let π : E ’ M be an m-dimensional vector bundle over

an n-dimensional manifold M and F : J k (π) ’ J k (π) be a Lie transforma-

tion. Then:

(i) If m > 1 and k > 0, the mapping F is of the form F = G(k) for some

di¬eomorphism G : J 0 (π) ’ J 0 (π);

(ii) If m = 1 and k > 1, the mapping F is of the form F = G(k’1) for

some contact transformation G : J 1 (π) ’ J 1 (π).

Proof. Recall that ¬bers of the projection πk,k’1 : J k (π) ’ J k’1 (π)

for k ≥ 1 are the only maximal integral manifolds of the Cartan distribution

of type 0 (see Remark 1.7). Further, from Proposition 1.10 it follows that

in the cases m > 1, k > 0 and m = 1, k > 1 they are integral manifolds

of maximal dimension, provided n > 1. Therefore, the mapping F is πk,µ -

¬berwise, where µ = 0 for m > 1 and µ = 1 for m = 1.

Thus there exists a mapping G : J µ (π) ’ J µ (π) such that πk,µ —¦ F =

G —¦ πk,µ and G is a Lie transformation in an obvious way. Let us show

that F = G(k’µ) . To do this, note ¬rst that in fact, by the same reasons,

the transformation F generates a series of Lie transformations G l : J l (π) ’

J l (π), l = µ, . . . , k, satisfying πl,l’1 —¦Gl = Gl’1 —¦πl,l’1 and Gk = F , Gµ = G.

(1)

Let us compare the mappings F and Gk’1 .

From Proposition 1.6 and the de¬nition of Lie transformations we obtain

F— ((πk,k’1 )’1 (Lθk )) = F— (Cθk ) = CF (θk ) = (πk,k’1 )’1 (LF (θk ) )

k

— —

2. NONLINEAR PDE 25

for any θk ∈ J k (π). But F— ((πk,k’1 )’1 (Lθk )) = (πk,k’1 )’1 (Gk’1,— (Lθk )) and

— —

consequently Gk’1,— (Lθk ) = LF (θk ) . Hence, by the de¬nition of 1-lifting we

(1)

have F = Gk’1 . Using this fact as a base of elementary induction, we obtain

the result of the theorem for dim M > 1.

Consider the case n = 1, m = 1 now. Since all maximal integral man-

ifolds are one-dimensional in this case, it should treated in a special way.

Denote by V the distribution consisting of vector ¬elds tangent to the ¬bers

of the projection πk,k’1 . Then

F— V = V (1.30)

for any Lie transformation F , which is equivalent to F being πk,k’1 -¬berwise.

Let us prove (1.30). To do it, consider an arbitrary distribution P on a

manifold N and introduce the notation

PD = {X ∈ D(N ) | X lies in P} (1.31)

and

DP = {X ∈ D(N ) | [X, Y ] ∈ P, ∀Y ∈ PD}. (1.32)

Then one can show (using coordinate representation, for example) that

DV = DC k © D[DC k ,DC k ]

for k ≥ 2. But Lie transformations preserve the distributions at the right-

hand side of the last equality and consequently preserve DV.

We pass now to in¬nitesimal analogues of Lie transformations:

Definition 1.18. Let π : E ’ M be a vector bundle and E ‚ J k (π) be

a k-th order di¬erential equation.

(i) A vector ¬eld X on J k (π) is called a Lie ¬eld, if the corresponding

one-parameter group consists of Lie transformations.

(ii) A Lie ¬eld is called an in¬nitesimal classical symmetry of the equa-

tion E, if it is tangent to E.

It should be stressed that in¬nitesimal classical symmetries play an im-

portant role in applications of di¬erential geometry to particular equations.

Since in the sequel we shall deal with in¬nitesimal symmetries only, we

shall skip the adjective in¬nitesimal and call them just symmetries. By

de¬nition, one-parameter groups of transformations corresponding to sym-

metries preserve generalized solutions.

Remark 1.9. Similarly to the above considered situation, we may in-

troduce the concepts both of exterior and interior in¬nitesimal symmetries

(see Remark 1.8), but we do not treat the second ones below.

Let X be a Lie ¬eld on J k (π) and Ft : J k (π) ’ J k (π) be its one-param-

(l)

eter group. The we can construct l-liftings Ft : J k+l (π) ’ J k+l (π) and

the corresponding Lie ¬eld X (l) on J k+l (π). This ¬eld is called the l-lifting

of the ¬eld X. As we shall see a bit later, liftings of Lie ¬elds, as opposed

26 1. CLASSICAL SYMMETRIES

to those of Lie transformations, are de¬ned globally and can be described

explicitly.

An immediate consequence of the de¬nition and of Theorem 1.12 is the

following result:

Theorem 1.13. Let π : E ’ M be an m-dimensional vector bundle over

an n-dimensional manifold M and X be a Lie ¬eld on J k (π). Then:

(i) If m > 1 and k > 0, the ¬eld X is of the form X = Y (k) for some

vector ¬eld Y on J 0 (π);

(ii) If m = 1 and k > 1, the ¬eld X is of the form X = Y (k’1) for some

contact vector ¬eld Y on J 1 (π).

Coordinate expressions for Lie ¬elds can be obtained as follows. Let

x1 , . . . , xn , . . . , uj , . . . be a special coordinate system in J k (π) and ωσ be

j

σ

the corresponding Cartan forms. Then X is a Lie ¬eld if and only if the

following equations hold

m

j

»j,± ω„ ,

±

|σ| < k,

L X ωσ = j = 1, . . . , m, (1.33)

σ,„

±=1 |„ |<k

where »j,± are arbitrary smooth functions. Let the vector ¬eld X be repre-

σ,„

sented in the form

n m

‚ ‚

j

X= Xi + Xσ .

‚uj

‚xi σ

i=1 j=1 |σ|¤k

Then from (1.33) it follows that the coe¬cients of the ¬eld X are related by

the following recursion equalities

n

j

uj ± Di (X± ),

j

’

Xσ+1i = Di (Xσ ) (1.34)

σ+1

±=1

where

m

‚ ‚

uj i

Di = + (1.35)

σ+1

‚uj

‚xi σ

j=1 |σ|≥0

are the so-called total derivatives.

Recall now that a contact ¬eld X on J 1 (π), dim π = 1, is completely

def

determined by its generating function which is de¬ned as f = iX ω, where

ω = du’ i u1i dxi is the Cartan (contact) form on J 1 (π). The contact ¬eld

corresponding to a function f ∈ F1 (π) is denoted by Xf and is expressed as

n n

‚f ‚ ‚f ‚

Xf = ’ + f’ u1i

‚u1i ‚xi ‚u1i ‚u

i=1 i=1

n

‚f ‚f ‚

+ + u 1i (1.36)

‚xi ‚u ‚u1i

i=1

2. NONLINEAR PDE 27

in local coordinates.

Thus, starting with a ¬eld (1.36) in the case dim π = 1 or with an

arbitrary ¬eld on J 0 (π) for dim π > 1 and using (1.34), we can obtain

e¬cient expressions for Lie ¬elds.

Remark 1.10. Note that in the case dim π > 1 we can introduce

def

vector-valued generating functions by setting f j = iX ω j , where ω j =

duj ’ i uj i dxi are the Cartan forms on J 1 (π). Such a function may be

1

understood as an element of the module F1 (π, π). The local conditions that

a section f ∈ F1 (π, π) corresponds to a Lie ¬eld is as follows:

‚f ± ‚f β ‚f ±

= , = 0, ± = β.

‚u±i β

‚uβi

‚u1i

1 1

In Chapter 2 we shall generalize the theory and get rid of these conditions.

We call f the generating section (or generating function, depending on

the dimension of π) of the Lie ¬eld X, if X is a lifting of the ¬eld Xf .

Let us ¬nally write down the conditions of a Lie ¬eld to be a symmetry.

Assume that an equation E is given by the relations F 1 = 0, . . . , F r = 0,

where F j ∈ Fk (π). Then X is a symmetry of E if and only if

r

j

»j F ± ,