On Some Basic Results Related to Affine Functions on Riemmanian Manifolds

TL;DR

This paper investigates the properties of a specific affine function on Hadamard manifolds, providing characterizations, counterexamples, and analyzing convexity of sub-level sets on manifolds with constant curvature.

Contribution

It characterizes when the function is affine, refutes previous claims about its properties, and explores convexity of sub-level sets on curved manifolds.

Findings

The function is affine if and only if certain conditions are met.

Counterexample on Poincaré plane shows previous assertions are false.

Sub-level sets' convexity depends on the manifold's curvature.

Abstract

We study some basic properties of the function $f_0:M\rightarrow\IR$ on Hadamard manifolds defined by $$ f_0(x):=\langle u_0,\exp_{x_0}^{-1}x\rangle\quad\mbox{for any $x\in M$}. $$ A characterization for the function to be linear affine is given and a counterexample on Poincar\'e plane is provided, which in particular, shows that assertions (i) and (ii) claimed in \cite[Proposition 3.4]{Papa2009} are not true, and that the function $f_0$ is indeed not quasi-convex. Furthermore, we discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures.