Abstract. Consider a (1, 1) tensor field J , defined on a real or complex m-dimensional manifold M , whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions f

(1)

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1996

input nmsymb

CLASSIFICATION OF (1,1) TENSOR FIELDS AND BIHAMILTONIAN STRUCTURES

F R A N C I S C O J A V I E R T U R I E L Secci´ on de Matem´ aticas, Facultad de Ciencias

A.P. 59, 29080 M´ alaga, Spain E-mail: turiel@ccuma.sci.uma.es

Abstract. Consider a (1, 1) tensor field J , defined on a real or complex m-dimensional manifold M , whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions f

1

, . . . , f

m

, defined around p, such that (df

1

∧. . .∧df

m

)(p) 6= 0 and d(df

_j

(J ( )))(p) = 0, j = 1, . . . , m. Then there exists a dense open set such that we can find coordinates, around each of its points, on which J is written with affine coefficients. This result is obtained by associating to J a bihamiltonian structure on T

^∗

M .

Introduction. Consider a (1, 1) tensor field J , defined on a real or complex m- dimensional manifold M , whose Nijenhuis torsion vanishes. Suppose that for each point p ∈ M there exist functions f

1

, . . . , f

m

, defined around p, such that (df

1

∧. . .∧df

m

)(p) 6= 0 and d(df

j

◦ J )(p) = 0 , j = 1, . . . , m [here df ◦ J means df (J ( ))]. In this paper we give a complete local classification of J on a dense open set that we call the regular open set.

Moreover, near each regular point, i.e. each element of the regular open set, J is written with affine coefficients on a suitable coordinate system.

To express the condition about functions f

1

, . . . , f

m

, stated above, in a simple compu- tational way we introduce the invariant P

_J

(see section 1). This invariant only depends on the 1-jet of J at each point, and P

J

(p) = 0 iff functions f

1

, . . . , f

m

as before exist. When J defines a G-structure, the first-order structure function being zero implies P

J

= 0 and N

_J

= 0 (this last property is well known). Besides all points of M are regular; therefore this work generalizes the main result of [5]. On the other hand N

J

and P

J

both together can be considered as a generalization of the first-order structure function.

This kind of tensor fields appear in a natural way in Differential Geometry. For ex- ample, on the base space of a bilagrangian fibration (see [1]) there exists a tensor field J ,

1991 Mathematics Subject Classification: Primary 53C15; Secondary 58H05, 35N99.

Key words and phrases: (1,1) tensor field, bihamiltonian structure.

Supported by DGICYT under grant PB91-0412.

The paper is in final form and no version of it will be published elsewhere.

[449]

(2)

with N

J

= 0, such that if (x

1

, . . . , x

m

) are action coordinates then each dx

j

◦ J is closed;

so P

J

= 0. From a wider viewpoint, when N

J

= 0, we can study the equation:

(1) d(df ◦ J ) = 0;

i.e. the existence of conservation laws for J . Our classification shows that the existence, close to p, of m functionally independent solutions to equation (1) is equivalent to P

_J

= 0 near p.

Partial answers to the foregoing question may be found in [2], [6] and [7]. In [4], by using eigenvalues and eigenspaces, J. Grifone and M. Mehdi give an elegant necessary and sufficient condition for the existence of enough local solutions to equations (1) when J is real analytic. With the Grifone-Mehdi condition all points are regular and a calculation shows that it implies P

J

= 0. Therefore the Grifone-Mehdi result follows from ours.

Finally, let us sketch the way for classifying J . As N

_J

= 0 we can construct a bi- hamiltonian structure on T

^∗

M and from it a (1, 1) tensor field J

^∗

, prolongation of J to T

^∗

M (see [8]). The main result of [9] gives us the local model of J

^∗

on a dense open set and now a J

^∗

-invariant cross section of T

^∗

M allows us to find a model of J . This cross section exists because P

J

= 0 implies that the behaviour of J

^∗

does not change along each fiber of T

^∗

M .

In a forthcoming paper we will study some cases where P

J

6= 0.

1. The first step. Consider a (1, 1) tensor field J on a real or complex manifold M of dimension m. We recall that the Nijenhuis torsion of J is the (1, 2) tensor field given by the formula

N

J

(X, Y ) = [J X, J Y ] + J

²

[X, Y ] − J [X, J Y ] − J [J X, Y ].

If τ is a 1-form τ ◦ J will mean the 1-form defined by (τ ◦ J )(X) = τ (J X).

For each p ∈ M let F (2, J )(p) be the vector subspace of all the 2-forms β

σ

defined by β

σ

(v, w) = σ(J v, w) − σ(v, J w) where v, w ∈ T

p

M and σ is a symmetric bilinear form on T

_p

M . Observe that F (2, J

^k

)(p) ⊂ F (2, J )(p) for each k ∈ N. Set

F

J

(p) = Λ

²

T

_p^∗

M F (2, J )(p) .

Given α ∈ T

_p^∗

M and a function f defined around p such that df (p) = α, the class of d(df ◦J )(p) on F

_J

(p) only depends on α. That defines a linear map P

_J

(p) : T

_p^∗

M → F

_J

(p) or, from a global viewpoint, P

J

: T

^∗

M → F

J

where F

J

is the disjoint union of all F

J

(p).

Note that P

J

(p) = 0 if and only if there exist functions f

1

, . . . , f

m

, defined around p, such that (df

₁

∧ . . . ∧ df

m

)(p) 6= 0 and d(df

_j

◦ J )(p) = 0, j = 1, . . . m. When the characteristic polynomial of J (p) equals its minimal polynomial, i.e. when T

p

M is cyclic, then F (2, J )(p) = Λ

²

T

_p^∗

M and automatically P

_J

(p) = 0. If J

²

= − Id a straightforward calculation shows that N

J

= 0 implies P

J

= 0. However J can be semisimple, N

J

= 0 and P

_J

6= 0; e.g. on R

^m

, m ≥ 2, J = e

^x¹

Id.

Let K

^N

[t] be the polynomial algebra in one variable over the ring of differentiable

functions on a manifold N . Here differentiable means C

^∞

if N is a real manifold (K =

R) and holomorphic in the complex case (K = C). A polynomial ϕ ∈ K

N

[t] is called

irreducible if it is irreducible at each point of N . We shall say that ϕ, ρ ∈ K

^N

[t] are

(3)

relatively prime if they are at each point. Consider an endomorphism field H of a vector bundle π : V → N , i.e. a cross section of V ⊗ V

^∗

. We will say that H has constant algebraic type if there exist relatively prime irreducible polynomials ϕ

₁

, . . . , ϕ

_`

∈ K

N

[t]

and natural numbers a

ij

, i = 1, . . . , r

j

, j = 1, . . . , `, such that for each p ∈ N the family {ϕ

^a_j^ij

(p)}, i = 1, . . . , r

_j

, j = 1, . . . , `, is the family of elementary divisors of H(p).

Suppose that J defines a G-structure, i.e. J has constant algebraic type on M and ϕ

₁

, . . . , ϕ

_`

∈ K[t]. If its first-order structure function vanishes then P

J

= 0. Indeed, around each point p ∈ M there exists a linear connection ∇, whose torsion at p vanishes, such that ∇J = 0. Let f

₁

, . . . , f

_m

be normal coordinates with origin p; then d(df

_j

◦ J )(p) = 0 and P

J

(p) = 0. Conversely N

J

= 0 and P

J

= 0 imply that the first-order structure function equals zero. In a word, the invariants N

J

and P

J

can be seen as a generalization of the first-order structure function to the case where J does not define a G-structure.

Henceforth we shall suppose N

_J

= 0. Set g

_k

= trace(J

^k

) and E = T

m

j=1

Ker dg

_j

. It is well known that (k + 1)dg

k

◦ J = kdg

k+1

and J E ⊂ E (see [9]).

We say that a point p ∈ M is regular if there exists an open neighbourhood A of p such that:

(1) J has constant algebraic type on A,

(2) E, restricted to A, is a vector subbundle of T A.

(3) The restriction of J to E has constant algebraic type on A.

The set of all regular points is a dense open set of M which we shall call the regular open set. Our local classification of J only refers to the regular open set.

Now suppose that on an open neighbourhood of a regular point p the characteristic polynomial ϕ of J is the product ϕ

₁

· ϕ

₂

of two monic relatively prime polynomials ϕ

₁

and ϕ

2

. Then around p the structure (M, J ) decomposes into a product of two similar structures (M

1

, J

1

) × (M

2

, J

2

), where ϕ

1

is the characteristic polynomial of J

1

(more exactly ϕ

1

is the pull-back of the characteristic polynomial of J

1

) and ϕ

2

that of J

2

(see [3] and [9]). Moreover N

J₁

= 0, N

J₂

= 0, and p

1

and p

2

are regular points where p = (p

₁

, p

₂

). On the other hand P

_J₁

= 0 and P

_J₂

= 0 if P

_J

= 0.

This splitting property reduces the classification to the case where the characteristic polynomial ϕ of J is a power of an irreducible one. Therefore we have only two possibil- ities: ϕ = (t + f )

^m

, or ϕ = (t

²

+ f t + g)

ⁿ

where m = 2n and M is a real manifold.

2. The case ϕ = (t + f )

^m

. In this section, by associating to J a bihamiltonian structure on T

^∗

M , we prove the following result:

Theorem 1. Consider a (1, 1) tensor field J such that N

^J

= 0 and P

J

= 0. Suppose that its characteristic polynomial is (t+f )

^m

. Then around each regular point p there exist coordinates ((x

^j_i

), y) with origin p, i.e. p ≡ 0, such that :

(a) i = 1, . . . , r

_j

and r

₁

≥ r

2

≥ . . . ≥ r

`

. Moreover we also consider the case with no

coordinates (x

^j_i

), i.e. ` = 0, and the case with coordinates (x

^j_i

) only.

(4)

(b) J = (y + a) Id +H + Y ⊗ dy where H =

`

X

j=1

^r^j⁻¹

X

i=1

∂

∂x

^j_i+1

⊗ dx

^j_i

and Y = ∂

∂x

¹₁

+

`

X

j=1

^r^j

X

i=2

(1 − i)x

^j_i

∂

∂x

^j_i

. R e m a r k. In the first special case m = 1 and J = (y + a) Id; in the second one m = r

1

+ . . . + r

`

and J = a Id + P

`

j=1

( P

rj−1

i=1

∂/∂x

^j_i+1

⊗ dx

^j_i

). The elementary divisors of J determine its model completely. If there is no coordinate y, i.e. if J defines a G-structure, they are: {(t − a)

^r^j

}, j = 1, . . . , `. Otherwise they are: (t−(y + a))

^r¹⁺¹

; {(t − (y + a))

^r^j

}, j = 2, . . . , `.

Let c

J

: T

^∗

M → T

^∗

M be the morphism of T

^∗

M defined by c

J

(τ ) = τ ◦ J and let ω be the Liouville symplectic form of T

^∗

M . Set ω

1

= (c

J

)

^∗

ω where c

J

is regarded as a differentiable map. Consider the (1, 1) tensor field J

^∗

, on T

^∗

M , defined by ω

₁

(X, Y ) = ω(J

^∗

X, Y ). Then N

J^∗

= 0, because N

J

= 0, and {ω, ω

1

} is a bihamiltonian structure (see [8]). If (x

₁

, . . . , x

_m

) are coordinates on M , (x

₁

, . . . , x

_m

, z

₁

, . . . , z

_m

) the associated coordinates on T

^∗

M , and J = P

m

i,j=1

f

ij

∂/∂x

i

⊗ dx

j

then J

^∗

=

m

X

i,j=1

f

ij

∂

∂x

_i

⊗ dx

j

+ ∂

∂z

_j

⊗ dz

i

+

m

X

i,j,k=1

z

i

∂f

ij

∂x

_k

− ∂f

ik

∂x

_j

∂

∂z

_j

⊗ dx

k

. Hence π

∗

◦ J

^∗

= J ◦ π

∗

.

Throughout the rest of this section J is as in theorem 1. By the local expression of J

^∗

given above, its characteristic polynomial is ϕ

^∗

= (t + f ◦ π)

^2m

. Since P

_J

= 0, around each regular point p ∈ M there exist coordinates (x

1

, . . . , x

m

) such that d(dx

i

◦J )(p) = 0, i = 1, . . . , m. Even more if df (p) 6= 0 [regularity implies df (p) = 0 iff f is constant near p]

we can suppose f = x

₁

because g

₁

= −mf and dg

₁

◦ J =

^dg₂²

. But dx

_i

◦ J = P

m

j=1

f

_ij

dx

_j

, then

^∂f_∂x^ij

k

(p) =

^∂f_∂x^ik

j

(p) and J

^∗

(p, z) =

m

X

i,j=1

f

_ij

(p)

∂

∂x

_i

⊗ dx

_j

+ ∂

∂z

_j

⊗ dz

_i

(p, z).

Therefore the elementary divisors of J (p) and (J

_|E

)(p) determine those of J

^∗

(p, z) and (J

^∗_|E^∗

)(p, z) completely, and the pull-back of the regular open set of J is included in the regular open set of J

^∗

. This is the role of the assumption P

_J

= 0 while N

_J

= 0 assures us that {ω, ω

1

} is bihamiltonian.

The zero cross section allows us to consider M as a submanifold of T

^∗

M . Take a regular point p ∈ M such that df (p) = 0, i.e. f constant near p. By theorem 3 of [9] there exist coordinates (y

1

, . . . , y

2m

) on an open neighbourhood A of p, with origin this point, on which ω and ω

₁

are written with constant coefficients and J

^∗

as well. By rearranging coordinates (y

1

, . . . , y

2m

) if necessary, we can suppose that {

_∂y^∂

1

(p), . . . ,

_∂y^∂

m

(p)} spans T

_p

M and {

_∂y^∂

m+1

(p), . . . ,

_∂y^∂

2m

(p)} spans the vertical subspace Ker π

_∗

(p) at p. Both sub- spaces are J

^∗

-invariant as the local expression of J

^∗

shows. Set A

0

= {y ∈ A : y

m+1

= . . . = y

_2m

= 0}. As rank((π

_|A₀

)(p)) = m we can choose an open neighbourhood B of p on A

0

such that π(B) is open and π : B → π(B) a diffeomorphism.

By construction J

^∗

(T A

0

) ⊂ T A

0

. Let J

⁰

be the restriction of J

^∗

to A

0

. The tensor

(5)

field J

⁰

is written with constant coefficients on A

0

. Moreover (π

_|A₀

)

∗

◦ J

⁰

= J ◦ (π

_|A₀

)

∗

since π

_∗

◦ J

^∗

= J ◦ π

_∗

. Then J is written with constant coefficients on π(B), which proves theorem 1 when df (p) = 0.

The proof of the other case is basically the same but we have to rearrange coordinates in a more sophisticated way. Let V be a real or complex vector space of dimension 2n. Consider α, α

₁

∈ Λ

²

V

^∗

such that α

ⁿ

6= 0. Let ˜ J be the endomorphism of V given by α

1

(v, w) = α( ˜ J v, w). Suppose ˜ J nilpotent (see proposition 1 of [9] for the model of {α, α

₁

}). An n-dimensional vector subspace W of V is called bilagrangian if α(v, w) = α

1

(v, w) = 0 for all v, w ∈ W ; in other words W is lagrangian for α and J W ⊂ W . When W is bilagrangian and there exists another bilagrangian subspace W

⁰

such that V = W ⊕ W

⁰

we shall say that W is superlagrangian. A bilagrangian subspace W is superlagrangian if and only if the elementary divisors of J

_|W

are half those of J ; i.e. if {t

^r^j

}, j = 1, . . . , `, are the elementary divisors of J

_|W

then {t

^r^j

, t

^r^j

}, j = 1, . . . , `, are those of J .

Lemma 1. Consider a basis {e

^ji

}, i = 1, . . . , 2r

_j

, j = 1, . . . , `, of V such that

α =

`

X

j=1

rj

X

k=1

e

^∗j_2k−1

∧ e

^∗j_2k

and α

1

=

`

X

j=1

rj−1

X

k=1

e

^∗j_2k−1

∧ e

^∗j_2k+2

.

Let W be the vector subspace spanned by {e

^j_2k−1

}, k = 1, . . . , r

j

, j = 1, . . . , `. Then for each superlagrangian subspace W

⁰

of V there exists T ∈ GL(V ) such that T

^∗

α = α, T

^∗

α

1

= α

1

and W ∩ T W

⁰

= {0}. Moreover if e

¹_2r₁₋₁

6∈ W

⁰

we can choose T in such a way that T e

¹₁

= e

¹₁

.

Now take a regular point p ∈ M . Suppose df (p) 6= 0. By theorem 3 of [9] there exist coordinates (x, y) = ((x

^j_i

), y

1

, y

2

), i = 1, . . . , 2r

j

and r

1

≥ r

2

≥ . . . ≥ r

`

, with origin p, such that

ω =

`

X

j=1

r_j

X

k=1

dx

^j_2k−1

∧ dx

^j_2k

+ dy

1

∧ dy

2

and ω

1

= (y

2

+ a)ω + τ + α ∧ dy

2

where

τ =

`

X

j=1

rj−1

X

k=1

dx

^j_2k−1

∧ dx

^j_2k+2

and

α = dx

¹₂

+

`

X

j=1

r_j

X

k=1

[(k + 1/2)x

^j_2k

dx

^j_2k−1

+ (k − 1/2)x

^j_2k−1

dx

^j_2k

] .

Hence J

^∗

= (y

₂

+ a) Id +H

^∗

+

_∂y^∂

1

⊗ α − Z ⊗ dy

₂

where H

^∗

=

`

X

j=1

^r^j⁻¹

X

k=1

∂

∂x

^j_2k+1

⊗ dx

^j_2k−1

+

rj

X

k=2

∂

∂x

^j_2k−2

⊗ dx

^j_2k

(6)

and

Z = ∂

∂x

¹₁

+

`

X

j=1

^r^j

X

k=1

(k − 1/2)x

^j_2k−1

∂

∂x

^j_2k−1

− (k + 1/2)x

^j_2k

∂

∂x

^j_2k

.

Lemma 2. The vector

_∂x1^∂ 2r1−1

(p) does not belong to the vertical subspace Ker π

∗

(p).

P r o o f. By the local expression of J

^∗

in the coordinates (x

₁

, . . . , x

_m

, z

₁

, . . . , z

_m

) given at the beginning of this section, Ker π

∗

(p) and T

p

M are J

^∗

(p)-invariant, and J

_{|Ker π}_∗(p)

and J

_|T_p_M

have the same elementary divisors. As p ≡ 0 in coordinates (x, y), the ele- mentary divisors of J

^∗

(p) are (t − a)

^r¹⁺¹

; (t − a)

^r¹⁺¹

; {(t − a)

^r^j

, (t − a)

^r^j

}, j = 2, . . . , `.

Therefore there exists v ∈ T

p

M spanning a cyclic subspace U of dimension r

1

+ 1 such that U ∩ Ker π

_∗

(p) = {0}.

Moreover v = a

_∂y^∂

2

(p) + b

_∂x^∂1 2r1

(p) + v

1

where (J

^∗

(p) − a Id)

^r¹

v

1

= 0.

By construction

(J

^∗

(p) − a Id)

^r¹

v = a ∂

∂x

¹_2r

1−1

(p) + b ∂

∂y

₁

(p)

does not belong to Ker π

_∗

(p). As ω(∂/∂y

₁

, ) = dy

₂

= −d(f ◦ π) and ω = P

m

j=1

dz

_j

∧ dx

j

in coordinates (x

1

, . . . , x

m

, z

1

, . . . , z

m

) of T

^∗

M , the vector

_∂y^∂

1

(p) belongs to Ker π

∗

(p).

So

_∂x₁^∂

2r1−1

(p) 6∈ Ker π

_∗

(p).

Set ω

⁰

= P

` j=1

( P

rj

k=1

dx

^j_2k−1

∧ dx

^j_2k

).

Lemma 3. The vector subspace (Ker π

∗

∩ Ker dy

₁

∩ Ker dy

₂

)(p), regarded as a subspace of T

0

K

^2m−2

, is superlagrangian with respect to {ω

⁰

(0), τ (0)}.

P r o o f. As f ◦π = −(y

₂

+a), Ker π

_∗

(p) ⊂ Ker dy

₂

(p) = Ker d(f ◦π)(p). Now note that ((J

^∗

− a Id)

^r¹

Ker π

∗

)(p) is a 1-dimensional subspace of Ker π

∗

(p) ∩ K{

∂x¹^∂

2r1−1

(p),

_∂y^∂

1

(p)}

(here K{v

¹

, . . . , v

s

} is the space spanned by {v

1

, . . . , v

s

}). So ((J

^∗

− a Id)

^r¹

Ker π

_∗

)(p) = K{

_∂y^∂₁

(p)} since

_∂x₁^∂

2r1−1

(p) 6∈ Ker π

_∗

(p).

On the other hand T

0

K

^2m−2

can be seen as the quotient space Ker dy

2

(p)/K{

∂y^∂₁

(p)}, which identifies (Ker π

∗

∩ Ker dy

1

∩ Ker dy

2

)(p) with Ker π

∗

(p)/K{

∂y^∂1

(p)}, and (H

^∗

+ a Id)(0) as the endomorphism induced by J

^∗_{|Ker dy}₂_(p)

. Therefore the elementary divisors of H

^∗_{|(Ker π}_∗_{∩Ker dy}₁_{∩Ker dy}₂_)(p)

are {t

^r^j

}, j = 1, . . . , `.

Lemma 4. Let {e

^ji

}, i = 1, . . . , 2r

j

, j = 1, . . . , `, be the canonical basis of K

^2m−2

= K

^2r¹

×. . .×K

^2r^`

. Set α = P

`

j=1

( P

rj

k=1

e

^∗j_2k−1

∧e

^∗j_2k

) and α

₁

= P

`

j=1

( P

rj−1

k=1

e

^∗j_2k−1

∧e

^∗j_2k+2

).

Given T ∈ GL(K

^2m−2

) if T e

¹₁

= e

¹₁

; T

^∗

α = α and T

^∗

α

1

= α

1

, there exists a germ of diffeomorphism ˜ G : (K

^2m

, 0) → (K

^2m

, 0) such that ˜ G(x, y) = (G(x), y); ˜ G

^∗

ω = ω;

G ˜

^∗

ω

1

= ω

1

and G

_∗

(0) = T .

P r o o f. We will adapt to our case the proof of proposition 3 of [9]. Consider the map G

T

: K

^2m

→ K

^2m

given by G

T

(x, y) = (T x, y). Then G

^∗_T

ω = ω and G

^∗_T

ω

1

= ω

1

+dg ∧dy

2

where g is a quadratic function such that d(dg ◦ H

^∗

) = 0. Indeed G

_T

preserves dx

¹₂

(0) = ω(

_∂x^∂1

1

, )(0) and H

^∗

, and d(α ◦ H

^∗

) = −2τ .

(7)

Let D and L be the exterior derivative and the Lie derivative with respect to the variables x only. We begin searching for a vector field X

t

= P

`

j=1

( P

^2rj

i=1

ϕ

^j_i

(x, t)

^∂

∂x^j_i

), defined on an open neighbourhood of the compact {0} × [0, 1] ⊂ K

^2m−2

× K, such that:

(1) L

^Xt

ω

⁰

= L

^Xt

τ = 0.

(2) L

Xt

(α + tDg) = Dg (remark that dg = Dg).

(3) For each i = 1, . . . , 2r

j

and j = 1, . . . , `, ϕ

^j_i

and Dϕ

^j_i

vanish on {0} × [0, 1].

Consider the vector field Z

_t

given by ω

⁰

(Z

_t

, ) = α + tDg. Take a function f (x, t), defined around {0} × [0, 1], such that:

(I) Z

t

f = −f − g.

(II) D(Df ◦ H

^∗

) = 0.

(III) For all i = 1, . . . , 2r

j

, j = 1, . . . , `, k = 1, . . . , 2r

s

and s = 1, . . . , `, the partial derivatives ∂f /∂x

^j_i

and ∂

²

f /∂x

^s_k

∂x

^j_i

vanish on {0} × [0, 1].

Let X

t

the vector field defined by ω

⁰

(X

t

, ) = Df . Then X

t

satisfies conditions (1), (2) and (3). By proposition 1.A (see the appendix) this kind of functions exists because g is quadratic, D(Dg ◦ H

^∗

) = 0, Z

t

(0) = ∂/∂x

¹₁

, and L

^Zt

H

^∗

= −H

^∗

since L

^Zt

ω

⁰

= D(α + tDg) = −ω

⁰

and L

^Zt

τ = D(α ◦ H

^∗

+ tDg ◦ H

^∗

) = −2τ .

By integrating the vector field −X

t

we obtain a germ of diffeomorphism F : (K

^2m−2

, 0)

→ (K

^2m−2

, 0) such that F

^∗

ω

⁰

= ω

⁰

; F

^∗

τ = τ ; F

^∗

(α + Dg) = α and F

_∗

(0) = Id. Now set G = ˜ ˜ F ◦ G

_T

where ˜ F (x, y) = (F (x), y).

Let W be the subspace of T

p

T

^∗

M spanned by {

^∂

∂x^j_2k−1

(p)}, k = 1, . . . , r

j

, j = 1, . . . , `. By lemmas 1, 2, 3 and 4 we can suppose, without loss of generality, W ∩ (Ker π

∗

∩ dy

1

∩ dy

2

)(p) = {0}, which implies (W ⊕ K{

∂y^∂2

(p)}) ∩ Kerπ

∗

(p) = {0}. Indeed dim(Ker π

_∗

∩ dy

1

∩ dy

2

)(p) = m − 1 (lemma 3) and

_∂y^∂

1

(p) ∈ Ker π

_∗

(p) (lemma 2, proof);

then Ker π

_∗

(p) = K{

∂y^∂₁

(p)} ⊕ (Ker π

_∗

∩ dy

₁

∩ dy

₂

)(p).

Set A

0

= {(x, y) ∈ A : x

^j_2k

= y

1

= 0, k = 1, . . . , r

j

, j = 1, . . . , `} where A is the domain of coordinates (x, y). Then J

^∗

(T A

₀

) ⊂ T A

₀

and T

_p

A

₀

⊕ Ker π

∗

(p) = T

_p

T

^∗

M . Finally, by reasoning as in the case df (p) = 0 we can state:

Proposition 1. Under the assumptions of theorem 1, if df (p) 6= 0 then there exist coordinates ((x

^j_i

), y) as in this theorem such that J = (y + a) Id +H + Y ⊗ dy where

H =

`

X

j=1

^r^j⁻¹

X

i=1

∂

∂x

^j_i+1

⊗ dx

^j_i

and Y = ∂

∂x

¹₁

+

`

X

j=1

^r^j

X

i=1

(1/2 − i)x

^j_i

∂

∂x

^j_i

.

When df (p) 6= 0, proposition 1 shows that the local model of J only depends on its elementary divisors.

Lemma 5. Consider on K

^m

= K

^r¹

× . . . × K

^r^`

× K, with r

1

≥ . . . ≥ r

`

if ` > 0, coordinates ((x

^j_i

), y). Let L be the Lie derivative with respect to variables (x

^ji

) only. Set J = (y + a) Id +H + Y ⊗ dy where Y is a vector field defined around the origin such that dy(Y ) = 0 and H = P

`

j=1

P

rj−1 i=1

∂

∂x^j_i+1

⊗ dx

^j_i

. If L

Y

H = H and H

^r¹⁻¹

Y (0) 6= 0, then

N

J

= 0 and close to the origin P

J

= 0 and J has constant algebraic type.

(8)

The elementary divisors of J , near the origin, are the same both for proposition 1 and lemma 5: (t − (y + a))

^r¹⁺¹

; {(t − (y + a))

^r^j

}, j = 2, . . . , `. So their models are equivalent.

We finish the proof of theorem 1 by taking Y = ∂

∂x

¹₁

+

`

X

j=1

^r^j

X

i=2

(1 − i)x

^j_i

∂

∂x

^j_i

.

The model announced by the author in a lecture at the Banach Center is obtained by setting

Y = ∂

∂x

¹₁

−

`

X

j=1

^r^j

X

i=1

ix

^j_i

∂

∂x

^j_i

. Another interesting model is given by taking

Y = ∂

∂x

¹₁

+

`

X

j=1

^r^j

X

i=1

(r

_j

+ 1 − i)x

^j_i

∂

∂x

^j_i

.

For this model the forms dy ◦ J = (y + a)dy and dx

^j_r_j

◦ J = (y + a)dx

^j_r_j

+ x

^j_r_j

dy + dx

^j_r

j−1

are closed. As N

_J

= 0 all the forms dx

^j_r

j

◦J

^k

are closed too. Therefore if the characteristic polynomial of J is (t + f )

^m

, for each regular point p and for all λ

0

∈ T

_p^∗

M there exists a closed 1-form λ, defined near p, such that λ(p) = λ

0

and d(λ ◦ J ) = 0; usually λ is called a conservation law. In other words, the equation d(df ◦ J ) = 0 has enough local solutions on the regular open set.

3. The case ϕ = (t

²

+f t+g)

ⁿ

. Since our problem is local we can suppose M connected and all of its points regular. Set J

₀

= 2(4g − f

²

)

⁻¹²

J + f (4g − f

²

)

⁻¹²

Id which makes sense because f

²

− 4g < 0. By construction J

0

defines a G-structure and (J

₀²

+ Id)

ⁿ

= 0. Let H be the semisimple part of J

₀

. Then H is a complex structure, J a holomorphic tensor field and (t + h)

ⁿ

its complex characteristic polynomial, where h =

¹₂

(f − i(4g − f

²

)

¹²

) is holomorphic.

Indeed, consider {ω, ω

1

} and J

^∗

on T

^∗

M as in section 2. Now the characteristic polynomial of J

^∗

is ϕ

^∗

= (t

²

+ (f ◦ π)t + (g ◦ π))

²ⁿ

. Let A be the regular open set of J

^∗

. Set J

₀^∗

= 2((4g − f

²

)

⁻¹²

◦ π)J

^∗

+ ((f (4g − f

²

)

⁻¹²

) ◦ π) Id. On each connected component of A the tensor field J

₀^∗

defines a G-structure; moreover ((J

₀^∗

)

²

+ Id)

²ⁿ

= 0.

Let H

^∗

be the semisimple part of J . In section 6 of [9] we showed that H

^∗

is a complex structure, J

^∗

holomorphic and (t + h

^∗

)

²ⁿ

its complex characteristic polynomial, where h

^∗

=

¹₂

(f ◦π−i(4g−f

²

)

¹²

◦π) is a holomorphic function. On the other hand π

_∗

◦J

₀^∗

= J

0

◦π

_∗

and π

_∗

◦ H

^∗

= H ◦ π

_∗

because π

_∗

◦ J

^∗

= J ◦ π

_∗

. So holomorphy holds on π(A), and on M as well since A is dense on T

^∗

M and π(A) on M .

The complex regular set of J is M (see section 6 of [9] again).

Suppose P

J

= 0. Let f = f

1

+ if

2

a holomorphic function. Then d(df ◦ J ) = d(df

1

◦

J ) + i(d(df

₂

◦ J )) is a holomorphic 2-form, so d(df

₁

◦ J )(HX, Y ) = d(df

₁

◦ J )(X, HY )

and d(df

2

◦ J )(X, Y ) = −d(df

1

◦ J )(HX, Y ). As P

J

(p) = 0 from the real viewpoint,

there exists a real symmetric bilinear form σ on T

p

M such that d(df

1

◦ J )(p)(v, w) =

σ(J (p)v, w) − σ(v, J (p)w). Set σ

₁

(v, w) =

¹₂

(σ(v, w) − σ(H(p)v, H(p)w)) and ˜ σ(v, w) =

σ

1

(v, w) − iσ

1

(H(p)v, w). As J and H commute ˜ σ is a complex symmetric bilinear

(9)

form and d(df ◦ J )(p)(v, w) = ˜ σ(J (p)v, w) − ˜ σ(v, J (p)w). In other words P

J

= 0 from the complex viewpoint. So to find a model of J , regard M as a complex manifold of dimension n and apply theorem 1. Then forget the complex structure and regard J as a real tensor field.

Theorem 2. Suppose N

J

= 0 and P

_J

= 0. Then the local model of J around each regular point is a finite product of models chosen among:

(a) For a complex manifold , those of theorem 1.

(b) For a real manifold , those of theorem 1 and those obtained considering the complex models of that theorem from the real viewpoint.

The local model of J is completely determined by its elementary divisors.

R e m a r k. Suppose N

J

= 0. Let p be a regular point. By theorem 2 there exist enough solutions to the equation d(df ◦J ) = 0, i.e. conservation laws, near p iff P

J

vanishes around this point. Nevertheless the existence of this kind of functions does not imply N

J

= 0;

e.g. on K

²

consider J = e

^x²

Id +∂/∂x

2

⊗ dx

1

; f

1

= x

1

− e

^x²

and f

2

= x

2

.

Appendix. Consider an open set A of K

ⁿ

endowed with a nilpotent constant coef- ficient (1, 1) tensor field H. Let B be a differentiable manifold (the parameter space).

Elements of A × B will be denoted by (x, y) while by D, D

⁽²⁾

and L we mean the exterior derivative, the second-order differential and the Lie derivative, all of them with respect to the variables (x

1

, . . . , x

n

) only. Let Z be a vector field on A depending on the parameter y ∈ B. We say that Z is generic at a point (x, y) if the dimension of the cyclic subspace spanned by Z(x, y) equals the degree of the minimal polynomial of H.

Proposition 1.A. Suppose given p ∈ A, a compact set K ⊂ B, a scalar a ∈ K and a function g : A × B → K, such that: (1) L

^Z

H = cH where c ∈ K; (2) Z is generic on {p} × K; (3) D(Dg ◦ H) = 0, g({p} × B) = 0 and Dg({p} × B) = 0.

Then there exist an open neighbourhood U of p, an open set V ⊃ K and a function f : U × V → K such that: (I) Zf = af + g; (II) D(Df ◦ H) = 0; (III) Df ({p} × V ) = 0 and D

⁽²⁾

f ({p} × V ) = 0. Moreover if Dg(Ker H

^r

) = 0 we can choose f in such a way that Df (Ker H

^r

) = 0.

The proof of this result is essentially that of proposition 1.A of [9]. Before lemma 2.A no change is needed at all. This last result should be replaced with:

Lemma 2

⁰

.A. Consider a function h

1

: A × B → K. Suppose Dh

¹

(KerH) = 0 and D(Dh

₁

◦ H) = 0. Then there exist an open neighbourhood U of p and a function h : U × B → K such that: (1) Dh ◦ H = Dh

¹

; (2) h({p} × B) = 0; (3) Dh(p, y) = 0 for all y ∈ B such that Dh

₁

(p, y) = 0; D

⁽²⁾

h(p, y) = 0 for each y ∈ B such that Dh

₁

(p, y) = 0 and D

⁽²⁾

h

1

(p, y) = 0.

P r o o f. There exist a vector subbundle E of T A and a morphism ρ : T A → T A such that T A = E ⊕ Ker H and (ρ ◦ H)

_|E

= Id. Set α = Dh

1

◦ ρ. Obviously α ◦ H = Dh

1

. Let C be the set of all y ∈ B such that Dh

1

(p, y) = 0 and D

⁽²⁾

h

1

(p, y) = 0. Suppose α = P

n

j=1

g

_j

dx

_j

. Then g

_j

({p} × C) = 0 and Dg

_j

({p} × C) = 0, j = 1, . . . , n.

(10)

By rearranging coordinates (x

1

, . . . , x

n

) we can suppose the foliation Ker H given by dx

1

= . . . = dx

k

= 0. From lemma 1.A, Dα(Im H, Im H) = 0 so Dα = P

k

j=1

( P

n

i=1

f

ij

dx

i

)

∧ dx

j

where each f

_ij

equals zero on {p} × C.

Let U = Q

n

i=1

U

i

be an open neighbourhood of p, product of intervals (K = R) or disks (K = C). As Dα is closed, there exist functions ˜ f

_j

: U × B → K such that ∂ ˜ f

_j

/∂x

_i

= f

_ij

and ˜ f

j

(U

1

× . . . × U

k

× {(p

k+1

, . . . , p

n

)} × B) = 0, i = k + 1, . . . , n, j = 1, . . . , k, where p = (p

1

, . . . , p

n

). Therefore ˜ f

j

({p} × B) = 0 and D ˜ f

j

({p} × C) = 0.

Set β = Dα − D( P

k

j=1

f ˜

j

dx

j

) = P

k

i,`=1

e

i`

dx

i

∧ dx

`

. As Dβ = 0, the functions e

i`

do not depend on (x

k+1

, . . . , x

n

). By construction e

i`

({p} × C) = 0.

Now we can find functions e

₂

, . . . , e

_k

: U × B → K, which do not depend on (x

k+1

, . . . . . . , x

n

), such that ∂e

j

/∂x

1

= e

1j

and e

j

({p

1

} × U

2

× . . . × U

n

× B) = 0, j = 2, . . . , k.

So e

_j

({p} × B) = 0 and De

_j

({p} × C) = 0. Set β

⁰

= P

k

j=2

e

_j

dx

_j

. Then β

₁

= β − Dβ

⁰

is closed and β

1

({p} × C) = 0. Moreover β

1

only involves the variables (x

2

, . . . , x

k

) and differentials dx

2

, . . . , dx

k

. By induction we construct ˜ β = P

k

j=1

e ˜

j

dx

j

such that D ˜ β = β,

˜

e

j

({p} × B) = 0 and D˜ e

j

({p} × C) = 0, j = 1, . . . , k.

Set α

1

= P

k

j=1

f

j

dx

j

where f

j

= ˜ f

j

+ ˜ e

j

. Again f

j

({p}×B) = 0 and Df

j

({p}×C) = 0, j = 1, . . . , k. By construction α

₁

◦ H = 0 and D(α − α

1

) = 0. Therefore there exists a function h : U × B → K such that h({p} × B) = 0 and Dh = α − α

¹

. Now Dh ◦ H = α ◦ H = Dh

₁

and Dh(p, y) = α(p, y) = (Dh

₁

◦ ρ)(p, y), which proves (1), (2) and (3).

Finally, note that Dh = P

k

j=1

(g

j

− f

j

)dx

j

+ P

n

j=k+1

g

j

dx

j

so D

⁽²⁾

h({p} × C) = 0.

Beyond this point both propositions have the same proof (lemma 2

⁰

.A assures us that Dg

0

({p} × B) = 0).

References

[1] R. B r o u z e t, P. M o l i n o et F. J. T u r i e l, G´ eom´ etrie des syst` emes bihamiltoniens, Indag.

Math. 4 (3) (1993), 269–296.

[2] P. C a b a u, J. G r i f o n e et M. M e h d i, Existence de lois de conservation dans le cas cyclique, Ann. Inst. H. Poincar´ e Phys. Th´ eor. 55 (1991), 789–803.

[3] A. F r ¨ o l i c h e r and A. N i j e n h u i s, Theory of vector-valued differential forms, Part I, Indag. Math. 18 (1956), 338–359.

[4] J. G r i f o n e and M. M e h d i, Existence of conservation laws and characterization of recur- sion operators for completely integrable systems, preprint, Univ. Toulouse II, 1993.

[5] J. L e h m a n n - L e j e u n e, Int´ egrabilit´ e des G-structures d´ efinies par une 1-forme 0-d´ efor- mable ` a valeurs dans le fibr´ e tangent, Ann. Inst. Fourier (Grenoble) 16 (1966), 329–387.