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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
|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

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