Two new families of Finsler connections on even-dimensional manifolds

TL;DR

This paper introduces two novel families of Finsler connections on even-dimensional pseudo-Finsler manifolds by constructing an almost hypercomplex structure and analyzing its properties.

Contribution

The authors develop new Finsler connections using an almost hypercomplex structure on even-dimensional manifolds, expanding the geometric tools available in Finsler geometry.

Findings

Construction of an almost hypercomplex structure on the manifold.

Definition of two new families of Finsler connections.

Existence of a linear connection parallelizing the hypercomplex structure.

Abstract

Let $F^{2n}=(M,M',F^{\ast})$ be an even-dimensional pseudo-Finsler manifold. We construct an almost hypercomplex structure on any chart domain of a certain atlas of $M'$ by using a considered non-linear connection. Then by using the almost hypercomplex structure we define two new families of Finsler connections. Also we show that for any Finsler connection $\nabla$ there exists a linear connection $D$ such that the local almost hypercomplex structure is parallel with respect to it.