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space7
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 ± ,

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