A Remarkable Transformation Group on Cotangent Bundle

Monica Purcaru

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

The present paper deals with the almost symplectic structure on T^{∗}M. The set of all almost symplectic d-linear connections are determined for the case when the nonlinear connection is arbitrary and its structure is discussed. The important invariants are determined for the transformation group of almost symplectic d-linear connections corresponding to the same nonlinear connection N. The problem of integrability of the almost symplectic structure on T^{∗}M is solved by means of these invariants, obtaining two single integrability typs I,II and one combined type εI + II, where ε ≠ 0 is real number.

Mathematics Subject Classification: 53C05 Key words: cotangent bundle, d-linear connection, almost symplectic structure, invariants, integrability.

1 Introduction

The geometrical structures on T^{∗}M have imposed themselves naturally, once the cotangent bundle geometry was approachead. Moreover, the necessity on studying them has been put forward by R.Miron, who pointed out the connection between the metrical structures on T^{∗}M and the geometry of [4, 5, 6] Hamilton spaces.

The study of the cotangent bundle geometry is important also because it provides a natural geometrical structure for the Gauge theories of theoretical physics.

The the cotangent bundle geometry has been studied by R.Miron, S.Watana- be and S.Ikeda in [9], by K.Yano and S.Ishihara in [12], by Gh.Atanasiu and F.Klepp in [2], C.Udrişte and O.Şandru [11], by R.Miron, D.Hrimiuc, H.Shimada and S.Sabău in [8], and others.

Concerning the terminology and notations, we use those from [7].

Let M be a real C^{∞}-differentiable manifold with dimension n, and let (T^{∗}M, π^{∗}, M) be its cotangent bundle.

If (x^{i}) is a local coordinates system on a domain U of a chart on M, the induced system of coordonates on π^{∗^{−1}}(U) is (x^{i}, p_{i}), (i = 1, ..., n).

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 49-56.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

Let N be a nonlinear connection on T^{∗}M, with the coefficients N_{ij}(x, p).

We consider on T^{∗}M an almost symplectic structure A:

A(x, p) = \frac{1}{2}a_{ij}(x, p)dx^{i} ∧ dx^{j} + a^{ij}(x, p)δp_{i} ∧ δp_{j,} (1.1)

where \{dx^{i}, δp_{i}\}, (i = 1, ..., n) is the dual basis of \{\frac{δ}{δx^{i}} , \frac{∂}{∂p_{i}} \}, and (a_{ij}(x, p), a^{ij}(x, p)) is a pair of given d-tensor fields on T^{∗}M, of the type (0,2), and (2,0) respectively, each of them alternate and nondegenerate.

We asociate to the lift A the Obata’s operators:

\{ \begin{matrix}Φ_{sj}^{ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} − a_{sj}a^{ir}), Φ_{sj}^{∗ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} + a_{sj}a^{ir}), \\ Φ̃_{sj}^{ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} − ã_{sj}a^{ir}), Φ̃_{sj}^{∗ir} = \frac{1}{2}(δ_{s}^{i}δ_{j}^{r} + ã_{sj}a^{ir}). \end{matrix} (1.2)

Obata’s operators have the same properties as the ones associated with a Finsler space [7].

The results obtained in the particular case of the normal d-linear connections support the findings of G.Atanasiu in his paper [1].

2 Almost symplectic d-linear connections on T^{∗}M

Definition 2.1 A d-linear connection, D^{∗}, on T^{∗}M, with local coefficients D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃ _{j}^{i k}, C_{i}^{jk}), for which:

\{ \begin{matrix}a_{ij|k} = 0, a_{ij}|^{k} = 0, \\ a_{|k}^{ij} = 0, a^{ij}|^{k} = 0, \end{matrix} (2.1)

where and denote the h-and v-covariant derivatives with respect to D^{∗},is called almost symplectic d-linear connections on T^{∗}M, or compatible whith the almost symplectic structure A, (1.1) and is denoted by: D^{∗}Γ(N).

We shall determine the set of all almost symplectic d-linear connections on T^{∗}M.

Let N^{0} be an other nonlinear connection on T^{∗}M, with the coefficients N_{ij}^{0} (x, p).

Let D^{∗}^{0} Γ(N^{0}) = (H_{jk}^{i}^{0}, H̃_{i k}^{j}^{0}, C̃_{j}^{i k}^{0}, C_{i}^{jk}^{0}) be the local coefficients of a fixed d-linear

connection, D^{∗},^{0} on T^{∗}M. Then any d-linear connection, D^{∗}, on T^{∗}M , can be expresed in the form:

\{ \begin{matrix}N_{ij} =N_{ij}^{0} −A_{ij}, \\ H_{jk}^{i} =H_{jk}^{i}^{0} +A_{lk} C̃_{j}^{i l}^{0} −B_{jk}^{i}, \\ H̃_{i k}^{j} =H̃_{i k}^{j}^{0} +A_{lk} C_{i}^{jl}^{0} −B̃_{i k}^{j}, \\ C̃_{j}^{i k} =C̃_{j}^{i k}^{0} −D̃_{j}^{i k}, \\ C_{i}^{jk} =C_{i}^{jk}^{0} −D_{i}^{jk}, \\ A_{ij |^{0}k} = 0, \end{matrix} (2.2)

where (A_{ij}, B_{jk}^{i}, B̃_{i k}^{j}, D̃_{j}^{i k}, D_{i}^{jk}) are components of the difference tensor fields of

D^{∗}Γ(N) from D^{∗}^{0} Γ(N^{0}), [3].

Theorem 2.1 Let D^{∗}^{0} be a given d-linear connection on T^{∗}M, with local coefficients

D^{∗}^{0} Γ(N^{0}) = (H_{jk}^{i}^{0}, H̃_{i k}^{j}^{0}, C̃_{j}^{i k}^{0}, C_{i}^{jk}^{0}). The set of all almost symplectic d-linear connections on T^{∗}M, with local coefficients D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) is given by:

\{ \begin{matrix}N_{ij} =N_{ij}^{0} −X_{ij}, \\ H_{jk}^{i} =H_{jk}^{i}^{0} +X_{lk} C̃_{j}^{i l}^{0} +\frac{1}{2}a^{ir}(a_{rj |^{0}k} + a_{rj} |^{0}^{l} X_{lk}) + Φ_{mj}^{ir}X_{rk}^{m}, \\ H̃_{i k}^{j} =H̃_{i k}^{j}^{0} +X_{lk} C_{i}^{jl}^{0} −\frac{1}{2} ã_{ir}(a_{|^{0}k}^{rj} + a^{rj} |^{0}^{l} X_{lk}) + Φ̃_{ir}^{mj}X̃_{m k}^{r}, \\ C̃_{j}^{i k} =C̃_{j}^{i k}^{0} +\frac{1}{2}a^{ir}a_{rj} |^{0}^{k} +Φ_{mj}^{ir}Ỹ _{r}^{m k}, \\ C_{i}^{jk} =C_{i}^{jk}^{0} −\frac{1}{2} ã_{ir}a^{rj} |^{0}^{k} +Φ̃_{ir}^{mj}Y_{m}^{rk}, \\ X_{ij |^{0}k} = 0, \end{matrix} (2.3)

where ^{0} and ^{0} denote the h-and v-covariant derivatives with respect to D^{∗}^{0}, and X_{ij}, X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk} are arbitrary tensor fields on T^{∗}M.

Particular cases:

1. If X_{ij} = X_{jk}^{i} = X̃_{i k}^{j} = Ỹ _{j}^{i k} = Y_{i}^{jk} = 0, in Theorem 2.1 we have:

Theorem 2.2 Let D^{∗}^{0} be a given d-linear connection on T^{∗}M. Then the following d-linear connection K^{∗}, with local coefficients K^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) given by (2.4) is an almost symplectic d-linear connection on T^{∗}M.

\{ \begin{matrix}H_{jk}^{i} =H_{jk}^{i}^{0} +\frac{1}{2}a^{ir}a_{rj |^{0}k}, \\ H̃_{i k}^{j} =H̃_{i k}^{j}^{0} −\frac{1}{2} ã_{ir}a_{|^{0}k}^{rj}, \\ C̃_{j}^{i k} =C̃_{j}^{i k}^{0} +\frac{1}{2}a^{ir}a_{rj} |^{0}^{k}, \\ C_{i}^{jk} =C_{i}^{jk}^{0} −\frac{1}{2} ã_{ir}a^{rj} |^{0}^{k}, \end{matrix} (2.4)

where ^{0} and ^{0} denote the h-and v-covariant derivatives with respect to the given d-linear connection, D^{∗}^{0}, on T^{∗}M.

2. If we take an almost symplectic d-linear connection on T^{∗}M (e.g. K^{∗} ) as D^{∗}^{0}, in Theorem 2.1., we have:

Theorem 2.3 Let D^{∗}^{0} be a fixed almost symplectic d-linear connection on T^{∗}M, with

local coefficients: D^{∗}^{0} Γ(N^{0}) = (H_{jk}^{i}^{0}, H̃_{i k}^{j}^{0}, C̃_{j}^{i k}^{0}, C_{i}^{jk}^{0}). The set of all almost symplectic d-linear connections on T^{∗}M, with local coefficients: D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) is given by:

\{ \begin{matrix}N_{ij} =N_{ij}^{0} −X_{ij}, \\ H_{jk}^{i} =H_{jk}^{i}^{0} +X_{lk} C̃_{j}^{i l}^{0} +Φ_{mj}^{ir}X_{rk}^{m}, \\ H̃_{i k}^{j} =H̃_{i k}^{j}^{0} +X_{lk} C_{i}^{jl}^{0} +Φ̃_{ir}^{mj}X̃_{m k}^{r}, \\ C̃_{j}^{i k} =C̃_{j}^{i k}^{0} +Φ_{mj}^{ir}Ỹ _{r}^{m k}, \\ C_{i}^{jk} =C_{i}^{jk}^{0} +Φ̃_{ir}^{mj}Y_{m}^{rk}, \\ X_{ij |^{0}k} = 0, \end{matrix} (2.5)

where ^{0} and ^{0} denote the h-and v-covariant derivatives with respect to D^{∗}^{0}, and X_{ij}, X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk} are arbitrary tensor fields on T^{∗}M.

3. If we take X_{ij} = 0, in Theorem 2.3 we obtain:

Theorem 2.4 Let D^{∗}^{0} be a fixed almost symplectic d-linear connection on T^{∗}M, with

local coefficients: D^{∗}^{0} Γ(N^{0}) = (H_{jk}^{i}^{0}, H̃_{i k}^{j}^{0}, C̃_{j}^{i k}^{0}, C_{i}^{jk}^{0}). The set of all almost symplectic d-linear connections on T^{∗}M, corresponding to the same nonlinear connection N, with local coefficients:D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) is given by:

\{ \begin{matrix}H_{jk}^{i} =H_{jk}^{i}^{0} +Φ_{mj}^{ir}X_{rk}^{m}, \\ H̃_{i k}^{j} =H̃_{i k}^{j}^{0} +Φ̃_{ir}^{mj}X̃_{m k}^{r}, \\ C̃_{j}^{i k} =C̃_{j}^{i k}^{0} +Φ_{mj}^{ir}Ỹ _{r}^{m k}, \\ C_{i}^{jk} =C_{i}^{jk}^{0} +Φ̃_{ir}^{mj}Y_{m}^{rk}, \end{matrix} (2.6)

where X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk} are arbitrary tensor fields on T^{∗}M.

3 The group of transformations of almost symplectic d-linear connections on T^{∗}M

We study the transformations D^{∗}Γ(N) → D̄^{∗}Γ(N) of the almost symplectic dlinear connections on T^{∗}M, corresponding to the same nonlinear connection N.

If we replace D^{∗}^{0} Γ(N^{0}) and D^{∗}Γ(N) in Theorem 2.3 by D^{∗}Γ(N) and D̄^{∗}Γ(N), respectively, two almost symplectic d-linear connections, we obtain:

Theorem 3.1 Two almost symplectic d-linear connections D^{∗}, D̄^{∗} with local coefficients D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) and

D̄^{∗}Γ(N̄) = (H̄_{jk}^{i}, \overline{H̃}_{i k}^{j}, \overline{C̃}_{j}^{i k}, C̄_{i}^{jk}) respectively, are related as follows:

\{ \begin{matrix}H̄_{jk}^{i} = H_{jk}^{i} + Φ_{sj}^{ir}X_{rk}^{s}, \\ \overline{H̃}_{i k}^{j} = H̃_{i k}^{j} + Φ̃_{ir}^{sj}X̃_{s k}^{r}, \\ \overline{C̃}_{j}^{i k} = C̃_{j}^{i k} + Φ_{sj}^{ir}Ỹ _{r}^{s k}, \\ C̄_{i}^{jk} = C_{i}^{jk} + Φ̃_{ir}^{sj}Y_{s}^{rk}, \end{matrix} (3.1)

where X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk}are arbitrary tensor fields on T^{∗}M.

Conversely, given the tensor fields X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk} the above (3.1) is thought to be a transformation of an almost symplectic d- linear connection D^{∗} to an almost symplectic d- linear connection D̄^{∗}.

We shall denote this transformation by t(X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k} , Y_{i}^{jk}). Thus we have:

Theorem 3.2 The set G_{as} of all transformations t(X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk}) given by (3.1) is a transformation group of the set of all almost symplectic d-linear connections corresponding to the same nonlinear connection N, on T^{∗}M together with the mapping product: t(X_{jk}^{′ i}, X̃_{i k}^{′ j}, Ỹ _{j}^{′ i k}, Y _{i}^{′ jk})∘ t(X_{jk}^{i}, X̃_{i k}^{j}, Ỹ _{j}^{i k}, Y_{i}^{jk}) = (X_{jk}^{i} + X_{jk}^{′ i}, X̃_{i k}^{j} + X̃_{i k}^{′ j}, Ỹ _{j}^{i k} + Ỹ _{j}^{′ i k}, Y_{i}^{jk} + Y _{i}^{′ jk}).

This group, G_{as}, is an abelian group wich is isomorphic to the additive group of quadruples of tensor fields (Φ_{sj}^{ir}X_{rk}^{s}, Φ̃_{ir}^{sj}X̃_{s k}^{r}, Φ_{sj}^{ir}Ỹ _{r}^{s k}, Φ̃_{ir}^{sj}Y_{s}^{rk}).

We determine the invariants of the group G_{as}. We denote with:

t_{ij}^{k} = A_{jk}\{\frac{∂N_{ij}}{∂p_{k}} \}, (3.2)

where A_{jk}{...} denotes the alternate summation. Since R_{ijk} and the tensor field t_{ij}^{k} depend on N only, they are invariants of the group G_{as}.

We make some notations:

\{ \begin{matrix}T _{ijk}^{∗} = S_{ijk}\{a_{im}T _{jk}^{m}\}, \\ S^{∗ijk} = S_{ijk}\{a^{im}S_{m}^{jk}\}, \\ R_{ij}^{∗k} = S_{ijk}\{a^{km}R_{mij}\}, \\ χ_{ij}^{k} = A_{ij}\{a_{im}C̃_{j}^{m k}\}, \\ ν_{i}^{j k} = A_{jk}\{a^{mj}P_{mi}^{k}\}, \end{matrix} (3.3)

where S_{ijk}{. . . ...} denotes the cyclic summation. By direct calculations we have:

Theorem 3.3 The tensor fields t_{ij}^{k}, R_{ij}^{∗k}, R _{ijk}, T _{ijk}^{∗}, S^{∗ijk}, χ_{ij}^{k}, ν_{i}^{j k} are invariants of the group G_{as}.

Theorem 3.4 Let N be a nonlinear connection on T^{∗}M. The invariant T_{ijk}^{∗} (resp. S^{∗ijk}) vanishes if and only if there exists an almost symplectic d-linear connection, D^{∗}, having the local coefficients D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}), with T _{jk}^{i} = 0 (resp. S_{i}^{jk} = 0).

Proof. Let X_{jk}^{i} =αT _{jk}^{i}, α ∈ R in (3.1), then we have: T̄ _{jk}^{i} = T _{jk}^{i} + \frac{α}{2}A_{jk}\{Φ_{mj}^{ir}T _{rk}^{m}\} = (1 + \frac{3}{2}α)T _{jk}^{i} − \frac{α}{2}a^{ir}T_{rjk}^{∗}. Taking α = −\frac{2}{3}, T_{ijk}^{∗} = 0 implies T̄ _{jk}^{i} = 0. The converse is evident. The statement about S^{∗ijk} is proved in the same way.

Theorem 3.5 a_{ij} does not depend on p, if and only if χ_{ij}^{k} = 0.

Proof. From a_{ij}|^{k} = 0 we have \frac{∂a_{ij}}{∂p_{k}} − χ_{ij}^{k} = 0.

4 The integrability of an almost symplectic structure in the cotangent bundle

Let Λ^{k}(T^{∗}M) be the F-module of all k-forms on the cotangent bundle (T^{∗}(M), π^{∗}, M) where F is the ring of all differentiable functions on T^{∗}(M). If N is a given nonlinear connection, then \{dx^{i}, δp_{i}\} is a local basis of Λ^{1}(T^{∗}(M)), which is dual to \{\frac{δ}{δx^{i}} , \frac{∂}{∂p_{i}} \}.

If f ∈ F, then the 1-form df is written as:

df = \frac{δf}{δx^{i}} dx^{i} + \frac{∂f}{∂p_{i}} δp_{i}, (4.1)

and the exterior differential of δp_{i} is given by:

d(δp_{i}) = \frac{1}{2}R_{ijk}dx^{k} ∧ dx^{j} − \frac{∂N_{ij}}{∂p_{k}} δp_{k} ∧ dx^{j}. (4.2)

In general, ω ∈ Λ^{2}(T^{∗}(M)) is written in the form:

ω = \frac{1}{2}ã_{ij}dx^{i} ∧ dx^{j} + b̃_{i}^{j}dx^{i} ∧ δp_{j} + \frac{1}{2}c̃^{ij}δp_{i} ∧ δp_{j}, (4.3)

where ã_{ij} = −ã_{ji}, c̃^{ij} = −c̃^{ji}.

The exterior differential dω is given by:

\begin{matrix}dω = \frac{1}{6} ω_{ijk}^{1} dx^{i} ∧ dx^{j} ∧ dx^{k} + \frac{1}{2} ω^{2}_{ij}^{k} dx^{i} ∧ dx^{j} ∧ δp_{k}+ \\ +\frac{1}{2} ω^{3}_{i}^{jk} dx^{i} ∧ δp_{j} ∧ δp_{k} + \frac{1}{6} ω^{4}^{ijk} δp_{i} ∧ δp_{j} ∧ δp_{k}, \end{matrix} (4.4)

where:

\{ \begin{matrix}ω_{ijk}^{1}= S_{ijk}\{\frac{δã_{ij}}{δx^{k}} + b̃_{i}^{m}R_{mjk}\}, \\ ω^{2}_{ij}^{k}= \frac{∂ã_{ij}}{∂p_{k}} + c̃^{km}R_{mij} + A_{ij}\{\frac{δb̃_{j}^{k}}{δx^{i}} − b̃_{i}^{m} \frac{∂N_{mj}}{∂p_{k}} \}, \\ ω^{3}_{i}^{jk}= \frac{δc̃^{jk}}{δx^{i}} + A_{jk}\{\frac{∂b̃_{i}^{j}}{δp_{k}} − c̃^{km} \frac{∂N_{mi}}{∂p_{j}} \}, \\ ω^{4}^{ijk}= S_{ijk}\{\frac{∂c̃^{ij}}{∂p_{k}} \}. \end{matrix} (4.5)

Proposition 4.1 If a d-linear connection D^{∗} is given on T^{∗}M, with local coefficients:

D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}), then the coefficients ω_{ijk}^{1}, ω^{2}_{ij}^{k}, ω^{3}_{i}^{jk}, ω^{4}^{ijk} have the

following expressions:

\{ \begin{matrix}ω_{ijk}^{1}= S_{ijk}\{ã_{ij|k} + ã_{im}T _{jk}^{m} + b̃_{i}^{m}R_{mjk}\}, \\ ω^{2}_{ij}^{k}= ã_{ij}|^{k} + b̃_{m}^{k}T _{ji}^{m} + c̃^{km}R_{mij} + A_{ij}\{b̃_{j |i}^{k} + ã_{im}C̃_{j}^{m k} + b̃_{i}^{m}P̃_{mj}^{k}\}, \\ ω^{3}_{i}^{jk}= c̃_{|i}^{jk} − b̃_{i}^{m}S_{m}^{jk} + A_{jk}\{b̃_{i}^{j}|^{k} + b̃_{m}^{j}C̃_{i}^{m k} + c̃^{mj}P̃_{mi}^{k}\}, \\ ω^{4}^{ijk}= S_{ijk}\{c̃^{ij}|^{k} + c̃^{im}S_{m}^{jk}\}. \end{matrix} (4.6)

Definition 4.1 A 2-form ω ∈ Λ^{2}(T^{∗}M), for which the matrix A=( \begin{matrix}ã_{ij} & b̃_{i}^{j} \\ −b̃_{i}^{j} & c̃^{ij} \end{matrix} ) is

nondegenerate, is called integrable if: dω=0.

Theorem 4.1 A 2-form ω ∈ Λ^{2}(T^{∗}M), for which the matrix A is nondegenerate, is

integrable, if and only if the tensor fields ω_{ijk}^{1}= 0, ω^{2}_{ij}^{k}= 0, ω^{3}_{i}^{jk}= 0, and ω^{4}^{ijk}= 0

Assume that a nonlinear connection N on T^{∗}M is given, then an almost symplectic structure on the base manifold M is lifted to a 2-form ω on T^{∗}M thus: we consider the following 2-forms ω of two single types I, II and one combined type εI+II, ε ∈ R^{∗} :

I: ω = \frac{1}{2}a_{ij}dx^{i} ∧ dx^{j}; II: ω =\frac{1}{2}ã^{ij}δp_{i}∧δp_{j},

εI+II: ω = εa_{ij}dx^{i} ∧ dx^{j} + \frac{1}{2}ã^{ij}δp_{i} ∧ δp_{j},

Proposition 4.2 Each 2-form ω of the type εI + II is nondegenerate and defines an almost symplectic structure on T^{∗}M.

Proposition 4.3 The coefficients ω_{ijk}^{1}, ω^{2}_{ij}^{k}, ω^{3}_{i}^{jk}, ω^{4}^{ijk} of the exterior differential of the 2-form given in Proposition 4.2 are invariants of the group G_{as} and are given in the following form

εI+II: ω_{ijk}^{1}= εT_{ijk}^{∗},ω^{2}_{ij}^{k}= a^{km}R_{mij} + εχ_{ij}^{k},

ω^{3}_{i}^{jk}= ν_{i}^{j k},  ω^{4 ijk} = S^{∗ijk}.

Proof. Using the relations (4.6) and (3.3) we calculate the coefficients of the exterior differentials of the 2-forms of the type I and II:

I: ω^{1}_{ijk}= T_{ijk}^{∗}, ω^{2}_{ij}^{k}= χ_{ij}^{k}, ω^{3}_{i}^{jk}=  ω^{4 ijk} = 0,
II: ω^{1}_{ijk}= 0, ω^{2}_{ij}^{k}= a^{km}R_{mij}, ω^{3}_{i}^{jk}= ν_{i}^{j k},  ω^{4 ijk} = S^{∗ijk}.

Definition 4.2 An almost symplectic structure on a differentiable manifold M is called integrable of the type εI + II, if there exists an almost symplectic d-linear connection D^{∗} on T^{∗}M such that the corresponding lifted 2-form on T^{∗}M is integrable.

Then from Theorems (3.4), (4.1) and from Definition 4.2 we have:

Theorem 4.2 An almost symplectic structure on a differentiable manifold M is integrable of the type ϵI + II if and only if there exists an almost symplectic d-linear connection D^{∗} on T^{∗}M with local coefficients D^{∗}Γ(N) = (H_{jk}^{i}, H̃_{i k}^{j}, C̃_{j}^{i k}, C_{i}^{jk}) satisfying the following conditions:

εI+II: T _{jk}^{i} = S_{i}^{jk} = 0, a^{km}R_{mij} + εχ_{ij}^{k} = 0, ν_{i}^{j k} = 0.

Theorem 4.3 An almost symplectic structure on a differentiable manifold M, integrable of the type εI + II, ε ∈ R^{∗}, does not depend on p if and only if R_{ijk} = 0.

The proof follows from a^{km}R_{mij} + εχ_{ij}^{k} = 0 and Theorem 3.5.

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”Transilvania” University of Braşov Department of Algebra and Geometry Iuliu Maniu 50 2200 Braşov, ROMANIA Fax: 0040-068-150274 email: mpurcaru@unitbv.ro