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

|σ|¤k j=1

where i = 1, . . . , n. Using this representation, we prove the following result:

Proposition 1.6. For any point θk ∈ J k (π), k ≥ 1, the Cartan plane

is of the form Cθk = (πk,k’1 )’1 (Lθk ), where Lθk is the R-plane at the

k k

Cθ k —

6

It is clear that for any regular point there exists a neighborhood of this point all

points of which are regular.

18 1. CLASSICAL SYMMETRIES

point πk,k’1 (θk ) ∈ J k’1 (π) determined by the point θk (see p. 5 for the

de¬nition of Lθk ).

k,•

Proof. Denote the vector (1.25) by vi . It is obvious that for any two

k,• k,•

sections •, • satisfying (1.23) the di¬erence vi ’ vi is a πk,k’1 -vertical

vector and any such a vector can be obtained in this way. On the other

k’1,•

hand, the vectors vi do not depend on • satisfying (1.23) and form a

basis in the space Lθk .

Remark 1.6. From the result proved it follows that the Cartan distri-

bution on J k (π) can be locally considered as generated by the vector ¬elds

m

[k]

uj i ‚uj , V„s = ‚us , |„ | = k, s = 1, . . . , m.

Di = ‚xi + σ „

σ+1

|σ|¤k’1 j=1

(1.26)

[k]

From here, by direct computations, it follows that [V„s , Di ] = V„s’1i , where

V(„1 ,...,„i ’1,...,„n ) , if „i > 0,

s

V(„1 ,...,„n )’1i =

0, otherwise.

j

But, as it follows from Proposition 1.6, vector ¬elds Vσ for |σ| ¤ k do not

lie in C k .

Let us consider the following 1-forms in special coordinates on J k+1 (π):

n

j def

uj i dxi ,

duj ’

ωσ = (1.27)

σ σ+1

i=1

where j = 1, . . . , m, |σ| < k. From the representation (1.26) we immediately

obtain the following important property of the forms introduced:

Proposition 1.7. The system of forms (1.27) annihilates the Cartan

j

distribution on J k (π), i.e., a vector ¬eld X lies in C k if and only if iX ωσ = 0

for all j = 1, . . . , m, |σ| < k.

Definition 1.12. The forms (1.27) are called the Cartan forms on

J k (π) associated to the special coordinate system xi , uj .

σ

Note that the Fk (π)-submodule generated in Λ1 (J k (π)) by the forms

(1.27) is independent of the choice of coordinates.

Definition 1.13. The Fk (π)-submodule generated in Λ1 (J k (π)) by the

Cartan forms is called the Cartan submodule. We denote this submodule by

CΛ1 (J k (π)).

Our last step is to describe maximal integral manifolds of the Cartan

distribution on J k (π). To do this, we start with the “in¬nitesimal estimate”.

Let N ‚ J k (π) be an integral manifold of the Cartan distribution. Then

from Proposition 1.7 it follows that the restriction of any Cartan form ω onto

2. NONLINEAR PDE 19

N vanishes. Similarly, the di¬erential dω vanishes on N . Therefore, if vector

¬elds X, Y are tangent to N , then dω |N (X, Y ) = 0.

Definition 1.14. Let Cθk be the Cartan plane at θ ∈ J k (π).

k

k

(i) We say that two vectors v, w ∈ Cθk are in involution, if the equality

dω |θk (v, w) = 0 holds for any ω ∈ CΛ1 (J k (π)).

k

(ii) A subspace W ‚ Cθk is said to be involutive, if any two vectors

v, w ∈ W are in involution.

(iii) An involutive subspace is called maximal , if it cannot be embedded

into other involutive subspace.

Consider a point θk = [•]k ∈ J k (π). Then from Proposition 1.7 it follows

x

that the direct sum decomposition

•

k v

Cθ k = T θ k • T θ k

v

is valid, where Tθk denotes the tangent plane to the ¬ber of the projection

•

πk,k’1 passing through the point θk , while Tθk is the tangent plane to the

graph of jk (•). Hence, the involutiveness is su¬cient to be checked for the

k

following pairs of vectors v, w ∈ Cθk :

v

(i) v, w ∈ Tθk ;

•

(ii) v, w ∈ Tθk ;

•

v

(iii) v ∈ Tθk , w ∈ Tθk .

v

Note now that the tangent space Tθk is identi¬ed with the tensor product

— —

S k (Tx ) — Ex , x = πk (θk ) ∈ M , where Tx is the ¬ber of the cotangent bundle

to M at the point x, Ex is the ¬ber of the bundle π at the same point while

S k denotes the k-th symmetric power. Then any tangent vector w ∈ Tx M

— —

determines the mapping δw : S k (Tx ) — Ex ’ S k’1 (Tx ) — Ex by

k

··· ρk ) — e = ··· ··· ρk — e,

δw (ρ1 ρ1 ρi , w

i=1

— —

where denotes multiplication in S k (Tx ), ρi ∈ Tx , e ∈ Ex , while ·, · is the

—

natural pairing between Tx and Tx .

k

Proposition 1.8. Let v, w ∈ Cθk . Then:

v

(i) All pairs v, w ∈ Tθk are in involution.

•

(ii) All pairs v, w ∈ Tθk are in involution too.

•

v

(iii) If v ∈ Tθk and w ∈ Tθk , then they are in involution if and only if

δπk,— (w) v = 0.

Proof. Note ¬rst that the involutiveness conditions are su¬cient to be

checked for the Cartan forms (1.27) only. All three results follow from the

representation (1.26) by straightforward computations.

20 1. CLASSICAL SYMMETRIES

Consider a point θk ∈ J k (π). Let Fθk be the ¬ber of the bundle πk,k’1

passing through the point θk and H ‚ Tx M be a subspace. De¬ne the