Cohomology of foliated Finsler manifolds

Abstract

We consider a foliation F of a Finsler manifold M. Its tangent manifold TM 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 FT on TM. The bundle TTM has two decompositions with respect to vertical foliation and to FT, respectively. We define the vertical-tangent, the horizontal-tangent, the vertical-transversal, and the horizontal-transversal vector fields on TM. We also define a new type of differential form on TM. The exterior derivative on TM 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.