Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study some basic properties related to affine functions on Riemannian manifolds. A characterization for a function to be linear affine is given and a counterexample on Poincare plane is provided, which, in particular, shows that assertions (i) and (ii) claimed by Papa Quiroz in (J Convex Anal 16(1):49---69, 2009, Proposition 3.4) are not true, and that the function involved in assertion (ii) is indeed not quasi-convex. Furthermore,  discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures.
['Xiangmei Wang', 'Chong Li', 'Jen-Chih Yao']
I am writing a research paper on a idea On Some Basic Results Related to Affine Functions on Riemannian Manifolds
[{"from": "human", "value": "I am writing a research paper on a idea On Some Basic Results Related to Affine Functions on Riemannian Manifolds"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study some basic properties related to affine functions on Riemannian manifolds. A characterization for a function to be linear affine is given and a counterexample on Poincare plane is provided, which, in particular, shows that assertions (i) and (ii) claimed by Papa Quiroz in (J Convex Anal 16(1):49---69, 2009, Proposition 3.4) are not true, and that the function involved in assertion (ii) is indeed not quasi-convex. Furthermore,  discuss the convexity properties of the sub-level sets of the function on Riemannian manifolds with constant sectional curvatures."}]