Geodesic Convexity Types in Riemannian Manifolds

TL;DR

This paper explores different types of geodesic convexity in Riemannian manifolds, characterizing manifolds where these convexity types coincide, thus extending classical Euclidean convexity concepts to more general geometric contexts.

Contribution

It introduces a characterization of complete Riemannian manifolds where various convexity types are equivalent, advancing understanding of convexity in non-Euclidean geometries.

Findings

Characterization of manifolds with coinciding convexity types

Extension of Euclidean convexity notions to Riemannian settings

Identification of conditions for convexity type equivalence

Abstract

The usual notion of set-convexity, valid in the classical Euclidean context, metamorphoses into several distinct convexity types in the more general Riemannian setting. By studying this phenomenon in reverse, we characterize complete manifolds for which certain convexity types are assumed a priori to coincide.