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C(Xπ∞ (θ) ). Then, by construction, the ¬eld CX is projected by π∞,— to X
while the correspondence C : D(M ) ’ D(π) is C ∞ (M )-linear. In other
words, this correspondence is a connection in the bundle π∞ : J ∞ (π) ’ M .
Definition 2.3. The connection C : D(M ) ’ D(π) de¬ned above is
called the Cartan connection in J ∞ (π).
Let now E ‚ J k (π) be a formally integrable equation. Then, due to
the fact that Cθ (E ∞ ) = Cθ (π) at any point θ ∈ E ∞ , we see that the ¬elds
CX are tangent to E ∞ for all vector ¬elds X ∈ D(M ). Thus we obtain the
Cartan connection in the bundle π∞ : E ∞ ’ M which is denoted by the
same symbol C.
Let x1 , . . . , xn , . . . , uj , . . . be special coordinates in J ∞ (π) and X =
σ
X1 ‚/‚x1 + · · · + Xn ‚/‚xn be a vector ¬eld on M represented in this co-
ordinate system. Then the ¬eld CX is to be of the form CX = X + X v ,
where X v = j,σ Xσ ‚/‚uj is a π∞ -vertical ¬eld. The de¬ning conditions
j
σ
j j
iCX ωσ = 0, where ωσ are the Cartan forms on J ∞ (π), imply
« 
n n
‚ ‚
j
Xi 
CX = + uσ+1i j = Xi Di . (2.5)
‚xi ‚uσ
i=1 j,σ i=1

In particular, we see that C(‚/‚xi ) = Di , i.e., total derivatives are just
liftings to J ∞ (π) of the corresponding partial derivatives by the Cartan
connection.
To obtain a similar expression for the Cartan connection on E ∞ , it needs
only to obtain coordinate representation for total derivatives in internal
coordinates. For example, in the case of equations (1.22) (see p. 16) we have
n n
‚ ‚
C Xi +T = Xi Di + T D t ,
‚xi ‚t
i=1 i=1

where D1 , . . . , Dn , Dt are given by formulas (2.1) and Xi , T ∈ C ∞ (M ) are
the coe¬cients of the ¬eld X ∈ D(M ).
Consider the following construction now. Let V be a vector ¬eld on E ∞
and θ ∈ E ∞ be a point. Then the vector Vθ can be projected parallel to
the Cartan plane Cθ onto the ¬ber of the projection π∞ : E ∞ ’ M passing
through θ. Thus we get a vertical vector ¬eld V v . Hence, for any f ∈ F(E)
a di¬erential one-form UE (f ) ∈ Λ1 (E) is de¬ned by
def
iV (UE (f )) = V v (f ), V ∈ D(E). (2.6)
62 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

The correspondence f ’ UE (f ) is a derivation of the algebra F(E) with the
values in the F(E)-module Λ1 (E), i.e.,
UE (f g) = f UE (g) + gUE (f )
for all f, g ∈ F(E). This correspondence contains all essential data about
the equation E.
Definition 2.4. The derivation UE : F(E) ’ Λ1 (E) is called the struc-
tural element of the equation E.
For the “empty” equation, i.e., in the case E ∞ = J ∞ (π), the structural
element Uπ is locally represented in the form

j
ωσ — j ,
Uπ = (2.7)
‚uσ
j,σ
j
where ωσ are the Cartan forms on J ∞ (π). To obtain the expression in the
general case, one needs to rewrite (2.7) in local coordinates. For example,
in the case of evolution equations we get the same expression with σ =
j
(σ1 , . . . , σn , 0) and the forms ωσ given by (2.2). Contrary to the Cartan
forms, the structural element is independent of local coordinates.
We shall now give a “more algebraic” version of the Cartan connection
de¬nition.
Proposition 2.4. For any vector ¬eld X ∈ D(M ), the equality
j∞ (•)— (CX(f )) = X(j∞ (•)— (f )) (2.8)
takes place, where f ∈ F(π) and • ∈ “loc (π). Equality (2.8) uniquely deter-
mines the Cartan connection in J ∞ (π).
Proof. Both statements follow from the fact that in special coordinates
the right-hand side of (2.8) is of the form
‚ |σ| •j
‚f
X .
j ‚xσ
j,σ ‚uσ j∞ (•)(M )



Corollary 2.5. The Cartan connection in E ∞ is ¬‚at, i.e.,
C[X, Y ] = [CX, CY ]
for any X, Y ∈ D(M ).
Proof. Consider the case E ∞ = J ∞ (π). Then from Proposition 2.4 we
have
j∞ (•)— (C[X, Y ](f )) = [X, Y ](j∞ (•)— (f ))
= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))
for any • ∈ “loc (π), f ∈ F(π). On the other hand,
1. BASIC STRUCTURES 63

j∞ (•)— ([CX, CY ](f )) = j∞ (•)— (CX(CY (f )) ’ CY (CX(f )))
= X(j∞ (•)— (Y (f ))) ’ Y (j∞ (•)— (CX(f )))
= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))
To prove the statement for an arbitrary formally integrable equation E, it
su¬ces to note that the Cartan connection in E ∞ is obtained by restricting
the ¬elds CX onto in¬nite prolongation of E.
The construction of Proposition 2.4 can be generalized.
1.4. C-di¬erential operators. Let π : E ’ M be a vector bundle and
ξ1 : E1 ’ M , ξ2 : E2 ’ M be another two vector bundles.
Definition 2.5. Let ∆ : “(ξ1 ) ’ “(ξ2 ) be a linear di¬erential operator.
The lifting C∆ : F(π, ξ1 ) ’ F(π, ξ2 ) of the operator ∆ is de¬ned by
j∞ (•)— (C∆(f )) = ∆(j∞ (•)— (f )), (2.9)
where • ∈ “loc (π) and f ∈ F(π, ξ1 ) are arbitrary sections.
Immediately from the de¬nition, we obtain the following properties of
operators C∆:
Proposition 2.6. Let π : E ’ M , ξi : Ei ’ M , i = 1, 2, 3, be vector
bundles. Then
(i) For any C ∞ (M )-linear di¬erential operator ∆ : “(ξ1 ) ’ “(ξ2 ), the
operator C∆ is an F(π)-linear di¬erential operator of the same order.
(ii) For any ∆, : “(ξ1 ) ’ “(ξ2 ) and f, g ∈ F(π), one has
C(f ∆ + g ) = f C∆ + gC .
(iii) For ∆1 : “(ξ1 ) ’ “(ξ2 ), ∆2 : “(ξ2 ) ’ “(ξ3 ), one has
C(∆2 —¦ ∆1 ) = C∆2 —¦ C∆1 .
From this proposition and from Proposition 2.4 it follows that if ∆ is a
scalar di¬erential operator C ∞ (M ) ’ C ∞ (M ) locally represented as ∆ =
|σ| ∞
σ aσ ‚ /‚xσ , aσ ∈ C (M ), then

C∆ = a σ Dσ (2.10)
σ
in the corresponding special coordinates. If ∆ = ∆ij is a matrix operator,
then C∆ = C∆ij .
From Proposition 2.6 it follows that C∆ may be understood as a dif-
ferential operator acting from sections of the bundle π to linear di¬erential
operators from “(ξ1 ) to “(ξ2 ). This observation is generalized as follows.
Definition 2.6. An F(π)-linear di¬erential operator ∆ : F(π, ξ1 ) ’
F(π, ξ2 ) is called a C-di¬erential operator, if it admits restriction onto graphs
of in¬nite jets, i.e., if for any section • ∈ “(π) there exists an operator
∆• : “(ξ1 ) ’ “(ξ2 ) such that
j∞ (•)— (∆(f )) = ∆• (j∞ (•)— (f )) (2.11)
64 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

for all f ∈ F(π, ξ1 ).
Thus, C-di¬erential operators are nonlinear di¬erential operators taking
their values in C ∞ (M )-modules of linear di¬erential operators. The follow-
ing proposition gives a complete description of such operators.
Proposition 2.7. Let π, ξ1 , ξ2 be vector bundles over M . Then any C-
di¬erential operator ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) can be presented in the form
a± C∆± , a± ∈ F(π),
∆=
±
where ∆± are linear di¬erential operators acting from “(ξ1 ) to “(ξ2 ).
Proof. Recall ¬rst that we consider the ¬ltered theory; in particular,
there exists an integer l such that ∆(Fk (π, ξ1 )) ‚ Fk+l (π, ξ2 ) for all k.
Consequently, since “(ξ1 ) is embedded into F0 (π, ξ1 ), we have ∆(“(ξ1 )) ‚
¯
Fl (π, ξ2 ) and the restriction ∆ = ∆ “(ξ1 ) is a C ∞ (M )-di¬erential operator
taking its values in Fl (π, ξ2 ). Then one can easily see that the equality
¯
∆• = j∞ (•)— —¦ ∆ holds, where • ∈ “loc (π) and ∆• is the operator from
(2.11). It means that any C-di¬erential ∆ operator is completely determined
¯
by its restriction ∆.
¯ ¯
On the other hand, the operator ∆ is represented in the form ∆ =

± a± ∆± , a± ∈ Fl (π) and ∆± : “(ξ1 ) ’ “(ξ2 ) being C (M )-linear dif-
¯ def
ferential operators. Let us de¬ne C ∆ = ± a± C∆± . Then the di¬erence
¯
∆ ’ C ∆ is a C-di¬erential operator such that its restriction onto “(ξ1 ) van-
¯
ishes. Therefore ∆ = C ∆.
Remark 2.1. From the result obtained it follows that C-di¬erential op-
erators are operators “in total derivatives”. By this reason, they are called
total di¬erential operators sometimes.
Corollary 2.8. C-di¬erential operators admit restrictions onto in¬nite
prolongations: if ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) is a C-di¬erential operator and
E ‚ J k (π) is a k-th order equation, then there exists a linear di¬erential
operator ∆E : F(E, ξ1 ) ’ F(E, ξ2 ) such that
µ— —¦ ∆ = ∆ E —¦ µ — ,
where µ : E ∞ ’ J ∞ (π) is the natural embedding.
Proof. The result immediately follows from Example 2.1 and Proposi-
tion 2.7.
We shall now consider an example which will play a very important role
in the sequel.
Example 2.3. Let ξ1 = „i— , ξ2 = „i+1 , where „p : p T — M ’ M (see
— —

Example 1.2 on p. 6), and ∆ = d : Λi (M ) ’ Λi+1 (M ) be the de Rham
def
di¬erential. Then we obtain the ¬rst-order operator dh = Cd : Λi (π) ’
h
p
i+1 — ). Due Corollary 2.8, the
Λh (π), where Λh (π) denotes the module F(π, „p
operators d : Λi (E) ’ Λi+1 (E) are also de¬ned, where Λp (E) = F(E, „p ).

h h h
1. BASIC STRUCTURES 65

Definition 2.7. Let E ‚ J k (π) be an equation.
(i) Elements of the module Λi (E) are called horizontal i-forms on E ∞ .
h
i (E) ’ Λi+1 (E) is called the horizontal de Rham
(ii) The operator dh : Λh h
di¬erential on E ∞ .
(iii) The sequence
d d
0 ’ F(E) ’ Λ1 (E) ’ · · · ’ Λi (E) ’ Λi+1 (E) ’ · · ·
’h ’h
h
is called the horizontal de Rham sequence of the equation E.
From Proposition 2.6 (iii) it follows that d—¦d = 0 and hence the de Rham
sequence is a complex. It cohomologies are called the horizontal de Rham
— i
cohomologies of E and are denoted by Hh (E) = i≥0 Hh (E).
In local coordinates, horizontal forms of degree p on E ∞ are represented
as ω = i1 <···<ip ai1 ...ip dxi1 § · · · § dxip , where ai1 ...ip ∈ F(E), while the
horizontal de Rham di¬erential acts as
n
Di (ai1 ...ip ) dxi § dxi1 § · · · § dxip .
dh (ω) = (2.12)
i=1 i1 <···<ip

In particular, we see that both Λi (E) and Hh (E) vanish for i > dim M .
i
h

Remark 2.2. In fact, the above introduced cohomologies are horizontal
cohomologies with trivial coe¬cients. The case of more general coe¬cients
will be considered in Chapter 4 (see also [98, 52]). Below we make the ¬rst
step to deal with a nontrivial case.
Consider the algebra Λ— (E) of all di¬erential forms on E ∞ and let us note
that one has the embedding Λ— (E) ’ Λ— (E). Let us extend the horizontal
h
de Rham di¬erential onto this algebra as follows:
(i) dh (dω) = ’d(dh (ω)),
(ii) dh (ω § θ) = dh (ω) § θ + (’1)ω ω § dh (θ).
Obviously, conditions (i), (ii) de¬ne the di¬erential dh : Λi (E) ’ Λi+1 (E) and
its restriction onto Λ— (E) coincides with the horizontal de Rham di¬erential.
h
def
Let us also set dC = d ’ dh : Λ— (E) ’ Λ— (E). Then, by de¬nition,
h h
dh —¦ dh = dC —¦ dC = 0, dC —¦ dh + dh —¦ dC = 0.
d = d h + dC ,
In other words, the pair (dh , dC ) forms a bicomplex in Λ— (E) with the total
di¬erential d. Hence, the corresponding spectral sequence converges to the
de Rham cohomology of E ∞ .
Remark 2.3. We shall rede¬ne this bicomplex in a more general alge-
braic situation in Chapter 4. On the other hand, it should be noted that
the above mentioned spectral sequence (in the case, when dh is taken for the
¬rst di¬erential and dC for the second one) is a particular case of the Vino-
gradov C-spectral sequence (or the so-called variational bicomplex) which is
essential to the theory of conservation laws and Lagrangian formalism with
constraints; cf. Subsection 2.2 below.
66 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

To conclude this section, let us write down the coordinate representation
for the di¬erential dC and the extended dh . First note that by de¬nition and
due to (2.12), one has
n
uj i dxi ,
dC (uj ) d(uj ) dh (uj ) duj
’ ’
= =
σ σ σ σ σ+1
i=1

i.e., dC takes coordinate functions uj to the corresponding Cartan forms.
σ
Since obviously dC (xi ) = 0 for any coordinate function on the base, we
obtain
‚f j
f ∈ F(π).
dC (f ) = ωσ , (2.13)
j
j,σ ‚uσ

The same representation, written in internal coordinates, is valid on E ∞ .
Therefore, the image of dC spans the Cartan submodule CΛ1 (E) in Λ1 (E).
By this reason, we call dC the Cartan di¬erential on E ∞ . From the equality
d = dh + dC it follows that the direct sum decomposition
Λ1 (E) = Λ1 (E) • CΛ1 (E)
h

takes place which extends to the decomposition

Λq (E) — C p Λ(E).
Λi (E) = (2.14)
h
p+q=i

Here the notation
def
C p Λ(E) = CΛ1 (E) § · · · § CΛ1 (E)
p times

j
is used. Consequently, to ¬nish computations, it su¬ces to compute dh (ωσ ).
But we have
dh (ωσ ) = dh dC (uj ) = ’dC dh (uj )
j
σ σ

and thus
n
j
j
dh (ωσ ) = ’ ωσ+1i § dxi . (2.15)
i=1

Note that from the results obtained it follows, that
dh (Λq (E) — C p Λ(E)) ‚ Λq+1 (E) — C p Λ(E),
h h
dC (Λq (E) — C p Λ(E)) ‚ Λq (E) — C p+1 Λ(E).
h h

Remark 2.4. Note that the sequence dh : Λq (E) — C — (E) ’ Λq+1 (E) —
h h
— (E) can be considered as the horizontal de Rham complex with coe¬cients
C
in Cartan forms
2. HIGHER SYMMETRIES AND CONSERVATION LAWS 67

Remark 2.5. From (2.14) it follows that to any form ω ∈ Λ— (E) we
can put into correspondence its “purely horizontal” component ωh ∈ Λ— (E).
h
k (π), then, due to the equality duj =
Moreover, if the form ω “lives” on J σ
j
n — k+1 (π)). This correspon-
i=1 uσ+1i dxi + ωσ , the form ωh belongs to Λ (J
dence coincides with the one used in Example 1.7 on p. 14 to construct
Monge“Ampere equations.

2. Higher symmetries and conservation laws
In this section, we brie¬‚y expose the theory of higher (or Lie“B¨cklund)
a
symmetries and conservation laws for nonlinear partial di¬erential equations
(for more details and examples see [60, 12]).
2.1. Symmetries. Let π : E ’ M be a vector bundle and E ‚ J k (π)
be a di¬erential equation. We shall still assume E to be formally integrable,
though is it not restrictive in this context.
Consider a symmetry F of the equation E and let θk+1 be a point of E 1
such that πk+1,k (θk+1 ) = θk ∈ E. Then the R-plane Lθk+1 is taken to an
R-plane F— (Lθk+1 ) by F , since F is a Lie transformation, and F— (Lθk+1 ) ‚
TF (θk ) , since F is a symmetry. Consequently, the lifting F (1) : J k+1 (π) ’
J k+1 (π) is a symmetry of E 1 . By the same reasons, F (l) is a symmetry of the
l-th prolongation of E. From here it also follows that for any in¬nitesimal
symmetry X of the equation E, its l-th lifting is a symmetry of E l as well.
In fact, the following result is valid:
Proposition 2.9. Symmetries of a formally integrable equation E ‚
J k (π)
coincide with symmetries of any prolongation of this equation. The
same is valid for in¬nitesimal symmetries.
Proof. We have shown already that to any (in¬nitesimal) symmetry
of E there corresponds an (in¬nitesimal) symmetry of E l . Consider now an
(in¬nitesimal) symmetry of E l . Then, due to Theorems 1.12 and 1.13 (see
pp. 24 and 26), it is πk+l,k -¬berwise and therefore generates an (in¬nitesimal)
symmetry of E.
The result proved means that a symmetry of E generates a symmetry of
E∞ which preserves every prolongation up to ¬nite order. A natural step to
generalize the concept of symmetry is to consider “all symmetries” of E ∞ .
Let us clarify such a generalization.
First of all note that only in¬nitesimal point of view may be e¬cient
in the setting under consideration. Otherwise we would have to deal with
di¬eomorphisms of in¬nite-dimensional manifolds with all natural di¬cul-
ties arising as a consequence. Keeping this in mind, we proceed with the
following de¬nition. Recall the notation
def
CD(π) = {X ∈ D(π) | X lies in C(π)},
cf. (1.31) on p. 25.
68 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Definition 2.8. Let π be a vector bundle. A vector ¬eld X ∈ D(π) is
called a symmetry of the Cartan distribution C(π) on J ∞ (π), if [X, CD(π)] ‚
CD(π).
Thus, the set of symmetries coincides with DC (π) (see (1.32) on p. 25)
and forms a Lie algebra over R and a module over F(π). Note that since the
Cartan distribution on J ∞ (π) is integrable, one has CD(π) ‚ DC (π) and,
moreover, CD(π) is an ideal in the Lie algebra DC (π).
Note also that symmetries belonging to CD(π) are of a special type:
they are tangent to any integral manifold of the Cartan distribution. By
this reason, we call such symmetries trivial. Respectively, the elements of
the quotient Lie algebra
def
sym(π) = DC (π)/CD(π)
are called nontrivial symmetries of the Cartan distribution on J ∞ (π).
Let now E ∞ be the in¬nite prolongation of an equation E ‚ J k (π).
Then, since CD(π) is spanned by the ¬elds of the form CY , Y ∈ D(M ) (see
Example 2.1), any vector ¬eld from CD(π) is tangent to E ∞ . Consequently,
either all elements of the coset [X] = X mod CD(π), X ∈ D(π), are tangent
to E ∞ or neither of them is. In the ¬rst case we say that the coset [X] is
tangent to E ∞ .
Definition 2.9. An element [X] = X mod CD(π), X ∈ D(π), is called
a higher symmetry of E, if it is tangent to E ∞ .
The set of all higher symmetries forms a Lie algebra over R and is de-
noted by sym(E). We shall usually omit the adjective higher in the sequel.
Let us describe the algebra sym(E) in e¬cient terms. We start with
describing sym(π) as the ¬rst step. To do this, note the following. Consider
a vector ¬eld X ∈ D(π). Then, substituting X into the structural element
Uπ (see (2.7)), we obtain a ¬eld X v ∈ D(π). The correspondence Uπ : X ’
X v = X Uπ possesses the following properties:
(i) The ¬eld X v is vertical, i.e., X v (C ∞ (M )) = 0.
(ii) X v = X for any vertical ¬eld.
(iii) X v = 0 if and only if the ¬eld X lies in CD(π).
Therefore, we obtain the direct sum decomposition1
D(π) = D v (π) • CD(π),
where D v (π) denotes the Lie algebra of vertical ¬elds. A direct corollary of

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