Báo cáo toán học: "A Presentation of the Elements of the Quotient Sheaves Ωk /Θk in Variational Sequences" pdf

Introduction The notion of variational bicomplexes was introduced in studying the problem of characterizing the kernel and image of Euler-Lagrange mapping in the calcu-lus of variations.

R I

0 $ 7 + ( 0 $ 7 , & 6

‹ 9$67 

A Presentation of the Elements of the Quotient

Sheaves Ωk r /Θ k r in Variational Sequences

Nong Quoc Chinh

Thai Nguyen University, Thai Nguyen City, Vietnam

Received Ferbuary 05, 2004 Revised Ferbuary 17, 2005

AbstractIn this paper we give a concrete presentation of the elements of the quotient sheavesΩk /Θ k in the variational sequence

0→ R → Ω0

r

E0

−→ Ω1

r /Θ1r −→ Ω E1 2

r /Θ2r −→ E2 E −→ Ω P −2 P −1

r /Θ P −1 r →

E −→ Ω P −1 P

r /Θ P r −→ Ω E P P +1

r

E −→ → Ω P +1 N

r → 0.

1 Introduction

The notion of variational bicomplexes was introduced in studying the problem

of characterizing the kernel and image of Euler-Lagrange mapping in the calcu-lus of variations This problem has been considered by Anderson [1], Duchamp [2], Dedecker [4], Tulczyvew [7], Takens [6] and Krupka [5] The variational

bi-complex was mainly studied on the infinite jet prolongation J ∞ Y of the fibered

manifold Y with the base X and the projection π : Y → X, where dimX = n, dimY −n = m The variational bicomplexes contain Euler-Lagrange mapping as

one of its morphisms Then its developments in the theory of variational bicom-plexes have made an important role in many problems in caculus of variations

on manifolds, in diffrential geometry, in the theory of differential equations and

in mathematical physics

Krupka [5] studied the sheaves of differential forms on finite r-jet prolonga-tions J r Y Then he constructed the sequence of quotient sheaves Ω k /Θ k This

quotient sequence is called the variational sequence of order r over Y It is an acyclic resolution of the constant sheaf R over Y

0→ R → Ω0

r

E0

−→ Ω1

r /Θ1r −→ Ω E1 2

r /Θ2r −→ E2 E −→ Ω P −2 P −1

r /Θ P −1 r →

E −→ Ω P −1 P

r /Θ P r −→ Ω E P P +1

r

E −→ → Ω P +1 N

In the sequence (1), E n is the Euler-Lagrange mapping and E n+1is the Helmholtz-Sonin mapping In the study of variational sequence, it is very important to give

a concrete presentation of the elements [] ∈ Ω k /Θ k It also has been shown in

[5] that [] = p k r,0  for all positive integers k satisfying 1 ≤ k ≤ n,  ∈ Ω k, where

p k

r,0  is the horizontal component of k-form  Then Krupka [5] gave a concrete

presentation of [] for 1 ≤ k ≤ n.

The main purpose of this paper is to give a concrete presentation of elements

[] ∈ Ω k /Θ k for all positive integers k satisfying n + 1 ≤ k ≤ P For simplicity,

we will solve this problem in the case of r = 1 and r = 2 Then the other cases

follow by the same method

2 Notations and Premilinaries

Throughout this paper, the following notations will be used: (Y, π, X) is a fibered manifold with base X and the projection π : Y → X, where dimX = n, dimY −

n = m J r Y is the finite r-jet prolongation of the fibered manifold (Y, π, X) π r:

J r Y → X, π r,s : J r Y → J s Y are the canonical jet projections (V, Ψ) is a fiber

chart on Y , where Ψ = (x i , y σ ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m (V r , Ψ r) is the fiber chart

on r-jet prolongation J r Y associated with (V, Ψ), where V r = π −1 r,0 (V ), Ψ r =

(x i , y σ , y j σ1, , y j σ1j2 j r ), 1 ≤ i ≤ n, 1 ≤ σ ≤ m, 1 ≤ j1≤ ≤ j r ≤ n Ω k is the

sheave of k-forms over J r Y and π ∗ r+1,r is the pull-back of the mapping π r+1,r

We put

ω0= dx1∧ dx2∧ ∧ dx n ,

ω i= (−1) i−1 dx1∧ ∧ dx i−1 ∧ dx i+1 ∧ · · · ∧ dx n , 1 ≤ i ≤ n,

ω j σ1j2 j k = dy σ j1j2 j k − y σ

j1j2 j k i dx i , 1 ≤ k ≤ r − 1.

Note that the forms

dx i , ω σ j1j2 j k , dy σ j1j2 j r ,

for 0≤ k ≤ r − 1, define a basis of the space of linear forms on V r

Obviously we have

dω j σ1j2 j k ∧ ω i=−ω σ

j1j2 j k i ∧ ω0, for 0 ≤ k ≤ r − 2,

dω j σ1j2 j r−1 ∧ ω i=−dy σ

j1j2 j r−1 i ∧ ω0.

Let N = n + m



n + r n



, M = m



n + r − 1 n



, P = m



n + r − 1 n



+ 2n − 1.

It is clear that N = dimJ r Y and M is the number of linear independent

forms ω σ

j j j , where 0 ≤ k ≤ r − 1.

For any function f :V r → R we have h(df) =n

i=1 d i f.dx i , where

d i f = ∂f

∂x i +

m



σ=1

r



k=0

∂f

∂y σ

j1 j k

y σ j1 j k

For any k-form ρ ∈ Ω k , we denote by ρ0the horizontal component of ρ, and

ρ q , 1 ≤ q ≤ k, is the q-contact component of ρ Then for any ρ ∈ Ω kthere exists

a unique decomposition

π ∗ r+1,r ρ = ρ0+ ρ1+· · · + ρ k

We denote by p k

r,q : Ωk → Ω k

r+1 the morphism of sheaves defined by p k

r,q ρ = ρ q, for 0 ≤ q ≤ k Ω k

r(c) = ker p k

r,0 , 1≤ k ≤ n, is the sheave of contact k-forms.

Ωk

r(c) = ker p k

r,k−n , n + 1 ≤ k ≤ N , is the sheave of strongly contact k-forms.

Let Θk = dΩ k−1 r(c) + Ωk

r(c) In [3] and [5] the authours proved the softness of

the sheaves Θk, considered the quotient sheaves Ωk /Θ k, and they obtained the following short exact sequence

0→ Θ k r

i k r

→Ω k r

τ r k

→Ω k

r /Θ k r → 0,

where i k is the canonical injective, τ k is the canonical quotient mapping Especially, Krupka [5] constructed the following variational sequence of order

r over Y

0→ R → Ω0

r

E0

−→ Ω1

r /Θ1r −→ Ω E1 2

r /Θ2r −→ · · · E2 E −→ Ω P −2 P −1

r /Θ P −1 r →

E −→ Ω P −1 P

r /Θ P r −→ Ω E P P +1

r

E −→ · · · → Ω P +1 N

r → 0,

where the sheaf morphism E k : Ωk /Θ k → Ω k+1

r /Θ k+1

r is defined by the formula

E k ([]) = [d].

He proved that, the variational sequence of order r is an acyclic resolution

of the constant sheaf R over Y , and [] = p k

r,0  for all 1 ≤ k ≤ n and for all

 ∈ Ω k This is the horizontal component of k-form  Let 1 ≤ k ≤ n − 1 and

 ∈ Ω k, then

[] = 1

k! f i1···i k dx

i1∧ · · · ∧ dx i k , (2)

where f i1 i k are some functions on V r

Let k = n and  ∈ Ω n

r Then

where f is some function on V r In this case [] is called the Lagrange class of  Let n + 1 ≤ k ≤ P For every s > r, Krupka [5] proved that

Ωk /Θ k ≈ Im(τ k π ∗ s,r+1 p k r,k−n ), and this implies that for every  ∈ Ω k

Let k = n + 1 For each element  ∈ Ω n+1

r , [] is called the Euler-Lagrange

class of (n + 1)-forms .

Let k = n + 2 For each element  ∈ Ω n+2 r , [] is called the Helmiholtz-Sonin

class of (n + 2)-forms .

Below we give a concrete presentation of the elements in Ωk /Θ k for all

pos-itive integer k satisfying n + 1 ≤ k ≤ P in the case of r = 1 and r = 2.

3 The Case of r = 1

Theorem 1.

a) Let  ∈ Ω n+11 be a germ Suppose that in the fiber chart (V, ψ), ψ = (x i , y σ ),

p n+1 1,1  = f σ w σ ∧ w0+ f σ i w i σ ∧ w0, (5)

where 1 ≤ σ ≤ m, 1 ≤ i ≤ n , and f σ , f i

σ are functions defined on V2 ⊂ J2Y Then we have

b) Let  ∈ Ω n+21 be a germ Suppose that in the fiber chart (V, ψ), ψ = (x i , y σ ),

p n+2 1,2  = f σν w σ ∧ w ν ∧ w0+ f σν i w σ i ∧ w ν ∧ w0+ f σν ij w σ i ∧ w ν

j ∧ w0, (7)

where 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n, and f σν , f i

σν , f ij

σν are functions defined on

V2⊂ J2Y Then

[] = ((f σν w σ + (f σν i − d i f σν ij )w σ i)− f ij

σν w σ ij)∧ w ν ∧ w0. (8)

c) Let n + 3 ≤ k ≤ P and  ∈ Ω k

1 be a germ Suppose that in the fiber chart

(V, ψ), ψ = (x i , y σ ),

p k 1,k−n  = f σ1 σ k−n w σ1 ∧ w σ2∧ ∧ ∧ w σ k−n ∧ w0

+ f i1

σ1 σ k−n w σ1

i1 ∧ w σ2∧ ∧ ∧ w σ k−n ∧ w0

+

+ f i1 i k−n−1

σ1 σ k−n w σ1

i1 ∧ w σ2

i2 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w0

+ f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ w σ2

i2 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n

i k−n ∧ w0,

(9)

where 1 ≤ σ1, , σ k−n ≤ m, 1 ≤ i1, i k−n ≤ n and each function defined

on V2⊂ J2Y Then we have

[] =

k−n−2

h=0

f i1 i h

σ1 σ k−n w σ1

i1 ∧ ∧ w σ h

i h ∧ w σ h+1 ∧ ∧ w σ k−n−1

+ (f i1 i k−n−1

σ1 σ k−n − d i k−n f i1 i k−n

σ1 σ k−n )w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1

−

k−n−1

t=1

f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ t−1

i t−1 ∧ w σ t

i t i k−n ∧ w σ t+1

i t+1 ∧

∧ w σ i k−n−1 k−n−1



∧w σ k−n ∧ w0,

(10)

where each function is well defined on V3⊂ J3Y

Proof.

a) Considering all the factors in (5) which contains w σ

i, we get

f σ i w σ i ∧ w0=−f i

σ d(w σ ∧ w i) =−d(f i

σ w σ ∧ w i ) + df σ i ∧ w σ ∧ w i (11)

Since f σ i w σ ∧ w i ∈ Θ n

2 and

π 3,2 ∗ (df σ i ∧ w σ ∧ w i ) = h(df σ i)∧ w σ ∧ w i + p(df σ i)∧ w σ ∧ w i ,

we have

[] = τ3n+1 p n+1 2,1 π ∗ 2,1 () = f σ w σ ∧ w0+ h(df σ i)∧ w σ ∧ w i

= (f σ − d i f σ i )w σ ∧ w0.

b) Considering all the factors in (7) which contains w σ

i ∧ w ν

j , we get

f ij

σν w σ

i ∧ w ν

j ∧ w0=−f ij

σν w σ

i ∧ d(w ν ∧ w j) =

= f σν ij d(w i σ ∧ w ν ∧ w j)− f ij

σν dw i σ ∧ w ν ∧ w j

= d(f σν ij w σ i ∧ w ν ∧ w j)− df ij

σν ∧ w σ

i ∧ w ν ∧ w j+ +

n



l=1

f σν ij dy σ il ∧ dx l ∧ w ν ∧ w j

(12)

Since f ij

σν w σ

i ∧ w ν ∧ w j ∈ Θ n+1

2 , we have

[] =τ3n+2 p n+2 2,2 π 2,1 ∗ ()

= f σν w σ ∧ w ν ∧ w0+ f σν i w i σ ∧ w ν ∧ w0

− d j f σν ij w σ i ∧ w ν ∧ w0− f ij

σν w σ ij ∧ w ν ∧ w0.

(13)

Therefore we have

[] = ((f σν w σ + (f σν i − d i f σν ij )w σ i)− f ij

σν w σ ij)∧ w ν ∧ w0.

c) In the formula (9), we consider all the factors containing w i σ k−n k−n They are

f σ i11 i σ k−n k−n w σ1

i ∧ ∧ w σ k−n

i ∧ w0 We get

f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ k−n

i k−n ∧ w0

= − f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ d(w σ k−n ∧ w i k−n)

= (−1) k−n f i1 i k−n

σ1 σ k−n d(w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n) +

k−n−1

t=1

(−1) k−n+t f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ t−1

i t−1 ∧ dw σ t

i t ∧

∧ w σ t+1

i t+1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n

= (−1) k−n d(f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n)

− (−1) k−n df i1 i k−n

σ1 σ k−n ∧ w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n

+

k−n−1

t=1

n



l=1

(−1) k−n+t+1 f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ t−1

i t−1 ∧ dy σ t

i t l ∧ dx l ∧

∧ w σ t+1

i t+1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n

(14)

Since f σ i11 i σ k−n k−n w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w i k−n ∈ Θ k−1

2 , we have

τ3k p k 2,k−n (f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ k−n

i k−n ∧ w0

= − d i k−n f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w0

− k−n−1

t=1

f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ t−1

i t−1 ∧ w σ t

i t i k−n ∧ w σ t+1

i t+1 ∧ ∧ w σ k−n−1

i k−n−1 ∧ w σ k−n ∧ w0.

(15)

Then [] can be presented in the following form

[] =

k−n−2

h=0

f i1 i h

σ1 σ k−n w σ1

i1 ∧ ∧ w σ h

i h ∧ w σ k+1 ∧ ∧ w σ k−n−1

+ (f i1 i k−n−1

σ1 σ k−n − d i k−n f i1 i k−n

σ1 σ k−n )w σ1

i1 ∧ ∧ w σ k−n−1

i k−n−1

−

k−n−1

t=1

f i1 i k−n

σ1 σ k−n w σ1

i1 ∧ ∧ w σ t−1

i t−1 ∧ w σ t

i t i t−1 ∧ w σ t+1

i t+1 ∧

∧ w σ i k−n−1 k−n−1



∧w σ k−n ∧ w0,

4 The Case of r = 2

Theorem 2.

a) Let  ∈ Ω n+12 be a germ Suppose that in the fiber chart (V, ψ), ψ = (x i , y σ ),

p n+1 2,1  = f σ w σ ∧ w0+ f σ i w σ i ∧ w0+ f σ ij w σ ij ∧ w0, (16)

where 1 ≤ σ ≤ m, 1 ≤ i, j ≤ m, f σ , f i

σ , f ij

σ are functions well defined on V3 ⊂

J3Y Then we have

[] = (f σ − d i f σ i + d i d j f σ ij )w σ ∧ w0, (17)

where each function is well defined on V5⊂ J5Y

b) Let  ∈ Ω n+22 be a germ Suppose that in the fiber chart (V, ψ), ψ = (x i , y σ ),

p n+2 2,2  = (f σν J w σ J ∧ w ν ∧ w0+ f σν Ji w J σ ∧ w ν

i ∧ w0+ f σν Jij w σ J ∧ w ν

ij ∧ w0, (18)

where 0 ≤ |J| ≤ 2, 1 ≤ σ, ν ≤ m, 1 ≤ i, j ≤ n, and functions f J

σν , f Ji

σν , f Jij

σν are well defined on V3⊂ J3Y Then we have

[] = ((f σν J − d i f σν Ji + d i d j f σν Jij )w J σ

+(−f Ji

σν + 2d j f σν Jij )w σ Ji + f σν Jij w σ Jij)∧ w ν ∧ w0, (19)

where each function is well defined on V5⊂ J5Y

c) Let n + 3 ≤ k ≤ P and  ∈ Ω k2 be a germ Suppose that in the fiber chart

(V, ψ), ψ = (x i , y σ ) we have

p k 2,k−n  =

2



q=0

f J1 J k−n−1 j1 j q

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n

j1 j q ∧ w0, (20)

where 0 ≤ |J1|, , |J k−n−1 | ≤ 2, 1 ≤ σ1, , σ k−n ≤ m, 1 ≤ j1, , j q ≤ n, and every function f σ J11 σ J k−n k−n−1 j1 j q is well defined on V3⊂ J3Y Then we have

[] =



(f J1 J k−n−1

σ1 σ k−n − d j1f J1 J k−n−1 j1

σ1 σ k−n + d j1d j2f J1 J k−n−1 j1j2

σ1 σ k−n )w σ1

J1 ∧ ∧ w σ J k−n−1 k−n−1+

k−n−1

t=1

(−f J1 J k−n−1 j1

σ1 σ k−n + 2d j2f J1 J k−n−1 j1j2

σ1 σ k−n )w σ1

J1 ∧ ∧ w σ J t−1 t−1 ∧ w σ t

J t j1∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

+

k−n−1

h=1

h=t

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ h−1

J h−1 ∧ w σ h

J h j1∧ w σ h+1

J h+1 ∧ ∧ w σ J t−1 t−1 ∧ w σ t

J t j2∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1

+

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ t−1

J t−1 ∧ w σ t

J t j1j2∧ w σ t+1

J t+1 ∧

∧ w σ J k−n−1 k−n−1



∧w σ k−n ∧ w0,

where each function is well defined on V5⊂ J5Y

a) In the formula (16) we consider the factors containing w σ

i They aref i

σ w σ

i ∧w0.

We get

f σ i w i σ ∧ w0=−f i

σ d(w σ ∧ w i) =−d(f i

σ w σ ∧ w i ) + df σ i ∧ w σ ∧ w i

(22)

Since f σ i w σ ∧ w i ∈ Ω n

3 = Θn3 and

π 4,3 ∗ (df σ i ∧ w σ ∧ w i ) = h(df σ i)∧ w σ ∧ w i + p(df σ i)∧ w σ ∧ w i ,

it implies that

τ4n+1 p n+1 3,1 (f σ i w σ i ∧ w0) =−d i f σ i ∧ w σ ∧ w0. (23)

Now we consider all the factors containing w ij σ in the formula (16) Then we have

f σ ij w σ ij ∧ w0=−d(f ij

σ w σ i ∧ w j ) + df σ ij ∧ w σ

This implies that

τ4n+1 p n+1 3,1 (f σ ij w σ ij ∧ w0) =−d j f σ ij w σ i ∧ w0

= d(d j f σ ij w σ ∧ w i)− d(d j f σ ij)∧ w σ ∧ w i (25) Therefore we have

τ5n+1 p n+1 4,1 π 4,3 ∗ (f σ ij w ij σ ∧ w0) = d i d j f σ ij w σ ∧ w0. (26) Since (16), (23) and (26) we get

[] = τ5n+1 p n+1 4,1 π 4,2 ∗ () = (f σ − d i f i

σ + d i d j f ij

σ )w σ ∧ w0,

where each function is defined on V5⊂ J5Y

b) In the formula (18) we consider all the factors containing w ν

i Then we get

f σν Ji w σ J ∧ w ν

i ∧ w0=−f Ji

σν w σ J ∧ d(w ν ∧ w i)

= f σν Ji d(w σ J ∧ w ν ∧ w i)− f Ji

σν dw J σ ∧ w ν ∧ w i

= d(f σν Ji w σ ∧ w ν ∧ w i)− df Ji

σν ∧ w σ ∧ w ν ∧ w i

+

n



l=1

df σν Ji dy σ Jl ∧ dx l ∧ w ν ∧ w i

Therefore

τ4n+2 p n+2 3,2 (f σν Ji w σ J ∧ w ν

i ∧ w0) =− d i f σν Ji w σ J ∧ w ν ∧ w0

− f Ji

σν w Ji σ ∧ w ν ∧ w0. (27)

Now we consider all the factors containing w ν

ij in the formula (16) Then we have

f σν Jij w σ J ∧ w ν

ij ∧ w0=−f Jij

σν w σ J ∧ d(w ν

i ∧ w j)

= d(f σν Jij w σ J ∧ w ν

i ∧ w j)− df Jij

σν ∧ w σ

J ∧ w ν

i ∧ w j

+

n



l=1

f σν Jij dy Jl σ dx l ∧ w ν

i ∧ w j

τ4n+2 .p n+2 3,2

−→ −d j f σν Jij w σ J ∧ w ν

i ∧ w0− f Jij

σν w σ Jj ∧ w ν

i ∧ w0

τ5n+2 .p n+2 4,2

−→ d i d j f σν Jij w σ J ∧ w ν ∧ w0+ d j f σν Jij w Ji σ ∧ w ν ∧ w0

+ d i f σν Jij w σ Jj ∧ w ν ∧ w0+ f σν Jij w σ Jij ∧ w ν ∧ w0,

(28)

where τ4n+2 p n+2 3,2 ( resp τ5n+2 p n+2 4,2 ) are morphisms of sheaves Since the formu-las (18), (27), (28) and the symmetry of indexes i, j we have

[] = τ5n+2 p n+2 4,2 π ∗ 4,2 () = ((f σν J − d i f σν Ji + d i d j f σν Jij )w σ J

+ (−f Ji

σν + 2d j f σν Jij )w Ji σ + f σν Jij w σ Jij)∧ w ν ∧ w0,

where each function is defined on V5⊂ J5Y

c) We consider the factors containing w σ J1k−n in the formula (20) We have

f J1 J k−n−1 j1

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n

j1 ∧ w0

= − f J1 J k−n−1 j1

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ d(w σ k−n ∧ w j1)

= (−1) k−n f J1 J k−n−1 j1

σ1 σ k−n d(w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w j1) +

k−n−1

t=1

(−1) k−n+t f J1 J k−n−1 j1

σ1 σ k−n w σ1

J1 ∧ ∧ w σ t−1

J t−1 ∧

∧ dw σ t

J t ∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w j1

= (−1) k−n d(f J1 J k−n−1 j1

σ1 σ k−n w σ1

J1∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w j1)

− (−1) k−n df J1 J k−n−1 j1

σ1 σ k−n ∧ w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w j1

−

k−n−1

t=1

n



l=1

(−1) k−n+t f J1 J k−n−1 j1

σ1 σ k−n w σ1

J1∧ ∧ w σ t−1

J t−1 ∧

∧ dy σ t

J t l ∧ dx l ∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w j1

τ4k .p k 3,k−n

−→ −d j1f J1 J k−n−1

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w0

− k−n−1

t=1

f J1 J k−n−1

σ1 σ k−n w σ1

J1∧ ∧ w σ t−1

J t−1 ∧ w σ t

J t j1∧

∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w0.

(29)

We consider all the factors containing w σ j k−n j in the formula (20) We have

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n

j1j2 ∧ w0τ

k

4.p k 3,k−n

−→

− d j2f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n

j1 ∧ w0

−

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ t−1

J t−1 ∧ w σ t

J t j2∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n

j1 ∧ w0

τ5k .p −→ k 4,k−n d j1 d j2f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w0

+

k−n−1

h=1

d j2f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1∧ ∧ w σ h−1

J h−1 ∧ w σ h

J h j1∧ w σ h+1

J h+1 ∧ ∧ w J σ k−n−1 k−n−1 ∧ w σ k−n ∧ w0

+

k−n−1

t=1

d j1f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1∧ ∧ w σ t−1

J t−1 ∧ w σ t

J t j2∧ w σ t+1

J t+1 ∧ ∧ w J σ k−n−1 k−n−1 ∧ w σ k−n ∧ w0

+

k−n−1

h=1

h=t

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1∧ ∧ w σ h−1

J h−1 ∧ w σ h

J h j1∧ w σ h+1

J h+1 ∧ ∧ w J σ t−1 t−1 ∧ w σ t

J t j2∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1 ∧ w σ k−n ∧ w0

+

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1 ∧ ∧ w σ t−1

J t−1 ∧ w σ t

J t j1j2∧ w σ t+1

J t+1 ∧ ∧ w J σ k−n−1 k−n−1 ∧ w σ k−n ∧ w0.

(30)

Since the formulas (20), (29), (30) and the symmetry of indexes j1, j2 we have

[] = τ5k p k 4,k−n π 4,2 ∗ ()

=



(f J1 J k−n−1

σ1 σ k−n − d j1f J1 J k−n−1 j1

σ1 σ k−n + d j1d j2f J1 J k−n−1 j1j2

σ1 σ k−n )w σ1

J1∧ ∧ w J σ k−n−1 k−n−1+

k−n−1

t=1

(−f J1 J k−n−1 j1

σ1 σ k−n + 2d j2f J1 J k−n−1 j1j2

σ1 σ k−n )w σ1

J1∧ ∧ w J σ t−1 t−1 ∧ w σ t

J t j1∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1

+

k−n−1

h=1

h=t

k−n−1

t=1

f J1 J k−n−1 j1j2

σ1 σ k−n w σ1

J1∧ ∧ w σ h−1

J h−1 ∧ w σ h

J h j1∧ w σ h+1

J h+1 ∧ ∧ w J σ t−1 t−1 ∧ w σ t

J t j2∧ w σ t+1

J t+1 ∧ ∧ w σ k−n−1

J k−n−1