Convex Neighbourhoods and Complete Finsler Spaces

TL;DR

This paper demonstrates that various connections on a sub-bundle of a Finsler manifold can be used to analyze convex neighborhoods, establishing that complete Finsler spaces are geodesically connected.

Contribution

It introduces a framework using multiple connections to study convex neighborhoods in Finsler geometry, proving geodesic connectivity of complete Finsler spaces.

Findings

Connections on sub-bundles effectively characterize convex neighborhoods.

Complete Finsler spaces are proven to be geodesically connected.

Framework simplifies analysis of Finsler geometric properties.

Abstract

In this paper, it is shown that a large set of connections on a suitable sub-bundle of the tangent bundle of a Finsler Manifold can be used to study all the properties of convex neighbourhoods with respect to the Finsler Metric, which are needed to see that any Complete Finsler Space is Geodesically Connected.