The Vaisman connection on the vertical bundle of a Finsler manifold

Abstract

The slit tangent manifold of a Finsler manifold is endowed with two foliations: the vertical foliation and the Liouville foliation, the last being a subfoliation of the first one, [1]. We give an adapted basis on the vertical bundle of such a manifold. In this paper we give the Vaisman connection on the vertical bundle with respect to the Liouville foliation and we compute its coefficients with respect to that adapted basis. We prove that the leaves of the vertical foliation are Reinhart spaces.