On Some Basic Results Related to Affine Functions on Riemmanian Manifolds

We study some basic properties of the function $f_0:M\to \text{IR}$ on Hadamard manifolds defined by f_0(x):=\langle u_0,\exp_{x_0}^{-1}x\rangle\quadfor 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.