Convex domains of Finsler and Riemannian manifolds

TL;DR

This paper explores convexity in Finsler and Riemannian manifolds, establishing equivalence of local and infinitesimal convexity notions, and extends classical results to more general settings with fewer differentiability requirements.

Contribution

It introduces a new approach to convexity in Finsler manifolds that generalizes Bishop's Riemannian results and reduces differentiability constraints.

Findings

Infinitesimal and local convexity notions are equivalent in Finsler manifolds.

The new approach extends classical convexity results to Finsler settings.

Results imply multiple geodesics can connect points in convex domains.

Abstract

A detailed study of the notions of convexity for a hypersurface in a Finsler manifold is carried out. In particular, the infinitesimal and local notions of convexity are shown to be equivalent. Our approach differs from Bishop's one in his classical result (Bishop, Indiana Univ Math J 24:169-172, 1974) for the Riemannian case. Ours not only can be extended to the Finsler setting but it also reduces the typical requirements of differentiability for the metric and it yields consequences on the multiplicity of connecting geodesics in the convex domain defined by the hypersurface.