What do `convexities' imply on Hadamard manifolds?

TL;DR

This paper demonstrates that certain convexity conditions on Hadamard manifolds are equivalent and only hold in Euclidean space, revealing that many results in the literature are essentially Euclidean in nature.

Contribution

It proves the equivalence of various convexity assumptions on Hadamard manifolds and characterizes when these conditions imply the manifold is Euclidean.

Findings

Convexity conditions are mutually equivalent.

These conditions hold only if the manifold is Euclidean.

Many results on Hadamard manifolds are Euclidean in disguise.

Abstract

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper we prove that these conditions are mutually equivalent and they hold if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well known Euclidean counterparts.