The homogeneous structure in a Cartan space

The homogeneus almost product structure on the Finsler space have Lieviu Popescu studied. In this paper we study the integrability conditions for the homogeneus product structure in Cartan space with Miron connection.

Let (T * M, π, M ) be the contangent bundle, where M is a differentiable, real n-dimensional manifold. If (U, ϕ) is a local chart on M , then the coordinates of a point u = (x, y) ∈ π −1 (U ) ∈ T * M will be denotes (x i , y i ), i, j, . . . = 1, 2, . . . , n. The natural basis of the module X(T * M ) is given by ( ∂ ∂x i , ∂ ∂yi ) = (∂ i , ∂ i ). Given H(x, y) -the metrical function T * M → R, the 2-homogeneus with respect to y i , g ij = 1 2 ∂ i ∂ j H -the metrical tensor, then pair (T * M, g) is Cartan space. The linear connection are given by respectively, such that δ i = ∂ i − L ik ∂ k , and (δ i , ∂ i ) is a local basis of X(T * 0 M ) = X(T * M )\{0} which is called the adapted basis to L ij . The vector fields δ i and ∂ i are 1 and 0 -homogeneous with respect to y i . The tensor of curvature of linear connection L are given by then differential geometric object (L ij , Γ k ij , 0) will be called the Miron connection of Cartan spaces [1].
Let L : (T * M, g) → R a differentiable function which is 1-homogeneous with respect to y i , r > 0 is a constant. We define linear mapping P : Proposition 1.
(a) P is an almost product structure, P 2 = I, (b) P preserves the property of homogeneity of vector fields from X(T * 0 M ). The proof is evident.

Proposition 2. Obtain such the identity
The almost product structure are integrable, if and only if is Nijenhuis tensor equal zero.
there ∇ -the operator of covariant derivative respect the Miron connection.
Proof. Let N be the Nijenhuis tensor of the homogeneous almost product structure P In the adapted basis we have where Analogous where Also where Let N = 0, from (8) follow (4) and (5). The calculation give, as from (4) and (5) follow Eqs. (10) and (12).
Then the following class of Riemanian metrics may be considered on X(T * 0 M ) where Dy i = dy i − L ik dx k . We have We define the two form Φ by the class of almost product structure P [2]: for all vector fields X, Y on X(T * 0 M ).

Theorem 2. The class of almost product structures P is Kähler, if and only if, L is constant.
Proof. From the (16) we have local expression the 2-form Φ: As From the Ricci identity follows R ikh dx k ∧ dx h ∧ dx i = 0, from Eqs. (1) and (2) follows Γ h ik = Γ h ki , and Γ h ik dx k ∧ Dy h ∧ dx i = 0. Then As r > 0, then 1 r + r L 2 = 0. There for DΦ = 0, if and only if, δ k L = 0, ∂ k L = 0.

Corollary 1.
If the almost product structure P is Kähler, it is integrable.