On Some Basic Results Related to Affine Functions on Riemmanian Manifolds

2015/09/24 by Xiangmei Wang, Chong Li, Wang, Xiangmei +3
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Analysis and Curvature Flows #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1509.07264

openalex publication_date 2015/09/24 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We study some basic properties of the function f₀:M→\IR on Hadamard manifolds defined by f₀(x):=⟨ u₀,exp_x0^-1x⟩\quadfor any x∈ M. A characterization for the function to be linear affine is given and a counterexample on Poincaré 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 f₀ 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.

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