Adapted basic connections to a certain subfoliation on the tangent manifold of a Finsler space

TL;DR

This paper studies a specific subfoliation on the tangent bundle of a Finsler space and introduces a set of basic connections adapted to this structure, enhancing understanding of the geometric properties involved.

Contribution

It constructs a triplet of basic connections adapted to a particular subfoliation on the tangent manifold of a Finsler space, extending previous foliation studies.

Findings

Defined a subfoliation on the tangent bundle of a Finsler space.

Constructed basic connections adapted to the subfoliation.

Provided geometric insights into the structure of the tangent bundle.

Abstract

On the slit tangent manifold $TM^0$ of a Finsler space $(M,F)$ there are given some natural foliations as vertical foliation and some other fundamental foliations produced by the vertical and horizontal Liouville vector fields, see [A. Bejancu, H. R. Farran, Finsler Geometry and Natural Foliations on the Tangent Bundle, Rep. Math. Physics 58, No. 1 (2006), 131-146]. In this paper we consider a $(n,2n-1)$-codimensional subfoliation $(\mathcal{F}_V,\mathcal{F}_{\Gamma})$ on $TM^0$ given by vertical foliation $\mathcal{F}_V$ and the line foliation spanned by vertical Liouville vector field $\Gamma$ and we give a triplet of basic connections adapted to this subfoliation.