COHOMOLOGY OF FOLIATED FINSLER MANIFOLDS. Adelina MANEA 1

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1 Bulletin of the Transilvania University of Braşov Vol 4(53), No Series III: Mathematics, Informatics, Physics, COHOMOLOGY OF FOLIATED FINSLER MANIFOLDS Adelina MANEA 1 Communicated to: Finsler Extensions of Relativity Theory, August 29 - September 4, 2011, Braşov, Romania Abstract We consider a foliation F of a Finsler manifold M. Its tangent manifold T M is Riemannian with respect to the Sasaki-Finsler metric and admits a natural foliation as a fibered manifold, called vertical foliation. Foliation on M determines a foliation F T on T M. The bundle T T M has two decompositions with respect to vertical foliation and to F T, respectively. We define the vertical-tangent, the horizontal-tangent, the vertical-transversal and the horizontal-transversal vector fields on T M. We also define a new type of differential forms on T M. The exterior derivative on T M admits a decomposition into some operators, one of them satisfies a Poincare type lemma, and we established a de Rham theorem for this particular one Mathematics Subject Classification: 53C12, 53C60. Key words: Finsler manifold, foliation, cohomology. 1 Preliminaries Finding new topological invariants of differentiable manifolds is still an open problem for geometries. The cohomology groups are such invariants. The Finsler manifolds are interesting models for some physical phenomena, so their properties are also useful to investigate, [1], [2], [3], [5]. The cohomology groups of manifolds, related sometimes to some foliations on the manifolds, have been studied in the last decades, [8], [9]. A study of foliations on a Lagrangian manifolds could be found in [7]. Our present work intends to develop the study of the Finsler manifolds and the foliated structures on the tangent bundle of such a manifold. For the beginning, we present the verical foliation on the tangent manifold T M of a n-dimensional Finsler manifold (M, F ), following [1]. In this paper the indices take the values i, j, i 1, j 1,... = 1, n. Let (M, F ) be a n-dimensional Finsler manifold and G be the Sasaki-Finsler metric on its tangent manifold T M. The vertical bundle V T M of T M is the tangent (structural) bundle to vertical foliation F V determined by the fibers of π : T M M. If (x i, y i ) i=1,n are local coordinates on T M, then V T M is locally spanned 1 Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania, amanea28@yahoo.com

2 24 Adelina Manea by { y i } i. A canonical transversal (also called horizontal) distribution is constructed in [1] as follows. We denote by (g ij (x, y)) i,j the inverse matrix of g = (g ij (x, y)) i,j, where g ij (x, y) = 1 2 F 2 2 y i (x, y), (1.1) yj and F is the fundamental function of the Finsler manifold. Obviously, we have the equalities g ij = g y k ik = g jk. y j y i We locally define the functions G i = 1 ( 2 F 2 4 gik y k x h yh F 2 ) x k, G j i = Gj y i. There exists on T M a n distribution HT M locally spanned by the vector fields x i = x i Gj i The Riemannian metric G on T M is satisfying, ( )i = 1, n. (1.2) yj G( x i, x j ) = G( y i, y j ) = g ij, G( x i, ) = 0, ( )i, j. (1.3) yj The local basis { decomposition x i, y i } i is called adapted to vertical foliation F V and we have the 2 Vertical cohomology of T M T T M = HT M V T M. (1.4) Corresponding to decomposition (1.4) we have the decomposition of module of vector fields on T M into horizontal vector fiels X h (T M) locally spanned by { } and vertical vector x i fiels X v (T M) locally spanned by { }. We also have two distributions, the vertical one y i being integrable. The Poisson braket of two horizontal vector fields depends on [ ] x i, x j = Rij k y k, Rk ij = Gi j x k Gi k x j. (2.1) The horizontal distribution is integrable iff Rij k = 0 for all indices. We apply theory of differential forms on foliated manifolds, [9], to the Riemannian foliated manifold (T M, G, F V ) and we consider the (p, q)-forms on T M being (p + q)- forms on T M which are non-zero only for p-arguments in X h (T M) and q-arguments in X v (T M). The module of (p, q)- forms is denoted by Ω p,q (T M) and we have the following decomposition of module of r-forms on T M: Ω r (T M) = p+q=r Ω p,q (T M).

3 Cohomology of foliated Finsler manifolds 25 Local cobasis adapted to vertical foliation is {dx i, y i = dy i + G i j dxj }. Locally, ω Ω p,q (T M) has the following expression: ω = α i1 i 2...i pj 1...j q dx i 1... dx ip y j 1... y jq. We compute dy i = k<j Ri jk dxk dx j + G i j k,j y k dx j, hence the exterior derivative on T M admits a decomposition into three y k operators: locally given by: d 10 : Ω p,q (T M) Ω p+1,q (T M); d = d 10 + d 2, 1 + d 01, d 2, 1 : Ω p,q (T M) Ω p+2,q 1 (T M); d 01 : Ω p,q (T M) Ω p,q+1 (T M), d 10 ω = α i 1 i 2...i pj 1...j q x i dx i dx i 1... dx ip y j 1... y jq + G j k +α i i1 i 2...i pj 1...j q y l y l dx i dx i 1... dx ip y j 1... where the symbol ˆ y j k means that this element is missing. y ˆj k... y j q, d 2, 1 ω = α i1 i 2...i pj 1...j q R j k il dx l dx i dx i 1... dx ip y j 1... y ˆj k... y j q, d 01 ω = α i 1 i 2...i pj 1...j q y i y i dx i 1... dx ip y j 1... y jq. From d 2 = 0 it follows d 2 01 = 0, this operator being exactly the foliated derivative of (T M, F V ) from [9]. It is known that the foliated derivative satisfies a Poincare type lemma. Let us denote by Φ v the sheaf of germs of basic functions on T M, that means f Ω 0 (T M) such that d 01 f = 0. The de Rham cohomology groups of foliated manifold (T M, F V ) are the quotient groups of the following semiexact sequence Φ v (T M) Ω 0 (T M) d 01 Ω 0,1 (T M) d 01 Ω 0,2 (T M) d 01..., H q v(t M) = Kerd 01 d 01 (Ω 0,q 1 (T M)), and these groups are so called v-cohomology of T M. Here Kerd 01 = {ω Ω 0,q (T M),d 01 ω = 0} is the set of d 01 -closed (0, q)-forms which is usually denoted by Z 0,q (T M). The set d 01 (Ω 0,q 1 (T M)) is usually denoted by B 0,q (T M) and it is called the set of d 01 -exact (0, q)-forms. The sheaves sequence 0 Φ v Ω 0 d 01 Ω 0,1 d 01 Ω 0,2 d 01..., is a fine resolution of Φ v. A de Rham theorem is also true, H q v(t M) is isomorphic with the q-dimensional Cech cohomology group of T M with coefficients in Φ v. All these results come from the theory of foliated manifolds applied to (T M, F V ).

4 26 Adelina Manea 3 Foliated Finsler manifold In this section we consider the Finsler manifold (M, F ) from the previous sections endowed with a m-codimensional foliation F. It follows that there is a partition of M into n m -dimensional submanifolds, called leaves. In the following, the indices take the values u, v, u 1, v 1,... = m + 1, n and a, b, a 1, b 1,...= 1, m. There is an atlas on M adapted to this foliation with local adapted charts (U, (x a, x u )) such that the leaves are locally defined by x a = constant, for all a = 1, m. The local coordinates on the tangent manifold T M are (x a, x u, y a, y u ). Generally, for two local charts (U, (x i )) and Ũ, ( xi 1 )), whose domains overlap, on T M, in U Ũ we have ỹ i 1 = xi 1 x i yi. Now, the above relations give the following changing coordinates rules on T M: x a 1 = x a 1 (x a ); x u 1 = x u 1 (x a, x u ), ỹ a 1 = xa 1 x a ya ; ỹ u 1 = xu1 x a ya + xu1 x u yu. Foliation F on M determine a 2m-codimensional foliation F T locally defined by x a = constant, y a = constant. on T M, whose leaves are Taking into account decomposition (1.4) of T T M, the local base { x a, x u, y a, y u } adapted to vertical foliation satisfies the following relations in U Ũ: x a 1 = xa x a 1 x a + xu x a 1 x u, ỹ a 1 = xa x a 1 x u 1 ỹ u 1 y a + xu x a 1 = xu x u 1 x u, = xu x u 1 y u, y u, Returning now to the foliation F T, the tangent bundle T F T to leaves, also called the structural bundle of this foliation, is locally spanned by { x, u y } and it is a subbundle u of T T M. Definition 3.1. We say that the foliation F of M is compatible with the Finsler structure F on M if, in every local chart around a point (x, y) T M, the matrix (g uv ) u,v, g uv (x, y) = 1 2 F 2 2 y u (x, y), yv

5 Cohomology of foliated Finsler manifolds 27 is nondegenerate and the functions G a are satisfying the relation for all a = 1, m, u = m + 1, n. G a y u = 0, Remark 3.1. The condition that a foliation is compatible to F has geometrical meaning. Indeed, in U Ũ we have and the matix ( xu x u 1 ) u,u 1 g u1,v 1 = g uv x u x u 1 x v x v 1, is obviously nondegenerate in every local chart. Proposition 3.1. If the foliation F of M is compatible with the Finsler structure F on M, then the vector fields on T M locally given by ξ a = x a tu a x u, ζ a = y a tu a y u, (3.1) are orthogonal to { x }, { u y }, with respect to Sasaki-Finsler metric G from (1.3), where u {t u a} are solutions of the system g av t u ag uv = 0. Proof. The compatibility between foliation on M and the Finsler structure of this manifold assures the solvability of the system g av t u ag uv = 0. Moreover, computing G(ξ a, x u ) = g au t a vg vu, G(ζ a, y u ) = g au t a vg vu, it results that these vector fields are orthogonal. As a consequence of the above Proposition, for a vector X T M we have the followig decomposition: X = X i x i + Y i y i = The basis = X a x a + Xu x u + Y a y a + Y u y u = = X a ξ a + (X u + t u ax a ) x u + Y a ζ a + (Y u + t u ay a ) y u. {ξ a, x u, ζ a, y u } (3.2) is adapted to F T and to vertical foliation, too. We denote by T F T the orthogonal bundle to T F T with respect to Sasaki-Finsler metric on T M.

6 28 Adelina Manea Definition 3.2. The vertical component of a section of T F T is called a verticaltangent vector field on T M. Their collection is denoted by V F T. The horizontal component of a section of T F T is called a horizontal-tangent vector field on T M. Their collection is denoted by HF T. The vertical component of a section of T F T is called a vertical-transversal vector field on T M. Their collection is denoted by V F T. The horizontal component of a section of T F T is called a horizontal-transversal vector field on T M. Their collection is denoted by H F T. From the above definition we have the decomposition of module of vector fields on T M into H F T HF T V F T V F T. Locally, H F T, HF T, V F T, V F T are spanned by {ξ a }, { x }, {ζ u a }, { y }, respectively. u The cobasis dual to basis (3.2) is {dx a, θ u, y a, η u }, where θ u = dx u + t u adx a, η u = y u + t u ay a. Definition 3.3. A (0, q)-form ω on T M is called a (0, s, t)-form if s + t = q and if it is nonzero only for s-arguments in V F T and t-arguments in V F T. We denote the space of (0, s, t)-forms by Ω 0,s,t (T M) and we have Ω 0,q (T M) = s+t=q Ω 0,s,t (T M). The foliated derivative d 01 acts on a (0, s, t)-form locally given by as it follows: ω = α a1,a 2,...,a s,u 1,..,u t y a 1... y as η u 1... η ut d 01 ω = ζ a (α a1,a 2,...,a s,u 1,..,u t )y a y a 1... y as η u 1... η ut + +( 1) s α a 1,a 2,...,a s,u 1,..,u t y u y a 1... y as η u η u 1... η ut. Hence there are two operators d 0,1,0 : Ω 0,s,t (T M) Ω 0,s+1,t (T M), d 0,0,1 : Ω 0,s,t (T M) Ω 0,s,t+1 (T M), with and d 01 = d 0,1,0 + d 0,0,1, d 0,0,1 ω = ( 1) s α a 1,a 2,...,a s,u 1,..,u t y u y a 1... y as η u η u 1... η ut. The equality d 2 01 = 0 implies d2 0,0,1 = 0. A function f Ω0 (T M) is called vertical basic if d 0,0,1 f = 0. We denote by Φ 0,0 the sheaf of germs of vertical basic functions. The following sequence

7 Cohomology of foliated Finsler manifolds 29 Φ 0,0 (T M) Ω 0 (T M) d 0,0,1 Ω 0,0,1 (T M) d 0,0,1 Ω 0,0,2 (T M) d 0,0,1..., is semiexact and we call the quotient group H q v,t (T M) = Z0,0,q (T M) B 0,0,q (T M), the v,t-cohomology of T M. Here Z 0,0,q (T M) = {ω Ω 0,0,q (T M), d 0,0,1 ω = 0} is the set of d 0,0,1 -closed (0, 0, q)-forms. The set B 0,q (T M) = d 0,0,1 (Ω 0,0,q 1 (T M)) is called the set of d 0,0,1 -exact (0, 0, q)-forms. Remark 3.2. We have the following strict inclusions: Z 0,q (T M) Ω 0,0,q (T M) Z 0,0,q (T M), B 0,q (T M) Ω 0,0,q (T M) B 0,0,q (T M). Another property of the operator d 0,0,1 is a Poincare type lemma, namely Theorem 3.1. Let ω be a d 0,0,1 -closed (0, 0, t)-form. For every open domain U T M there is γ Ω 0,0,t 1 (U) such that in U we have the equality ω = d 0,0,1 γ. Proof. Since ω is a (0, 0, t)-form, its local expression is ω = α u1,u 2,...,u t η u 1... η ut. From d 0,0,1 ω = 0 it follows d 01 ω = ζ a (α u1,u 2,...,u t )y a η u 1... η ut, so d 01 ω = 0 (modulo y 1, y 2,..., y m n ) But the operator d 01 satisfies a Poincare type lemma, hence in space y a = 0 we have that for every open domain U of T M there is a (0, t 1)-form ϕ such that in U ω = d 0,0,1 ϕ (modulo y 1, y 2,..., y m n ). Let γ be the (0, 0, t 1)-component of ϕ. Then we have the following equality in U: ω = d 0,0,1 γ + µ a y a, with µ a (0, t 1)-forms on U. By equalizing the terms of the same type in the above equality we obtain ω = d 0,0,1 γ, so every d 0,0,1 -closed form is locally d 0,0,1 -exact. Let Ω 0,0,q the sheaf of germs of (0, 0, q)-forms and i : Φ 0,0 Ω 0 the natural inclusion. The shaves Ω 0,0,q are fins, so the shaves sequence 0 Φ 0,0 Ω 0 d 0,0,1 Ω 0,0,1 d 0,0,1 Ω 0,0,2 d 0,0,1..., is a fine resolution of the sheaf Φ 0,0. Applying now a well-known theorem of algebraic topology it results a de Rham type theorem for the v, t-cohomology of T M: Theorem 3.2. H q v,t (T M) is isomorphic with the q-dimensional Cech cohomology group of T M with coefficients in Φ 0,0.

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