Convexity and Some Geometric Properties

João Xavier da Cruz Neto (), Ítalo Dowell Lira Melo () and Paulo Alexandre Araújo Sousa ()
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João Xavier da Cruz Neto: Universidade Federal do Piauí
Ítalo Dowell Lira Melo: Universidade Federal do Piauí
Paulo Alexandre Araújo Sousa: Universidade Federal do Piauí

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 2, No 5, 459-470

Abstract: Abstract The main goal of this paper is to present results of existence and nonexistence of convex functions on Riemannian manifolds, and in the case of the existence, we associate such functions to the geometry of the manifold. Precisely, we prove that the conservativity of the geodesic flow on a Riemannian manifold with infinite volume is an obstruction to the existence of convex functions. Next, we present a geometric condition that ensures the existence of (strictly) convex functions on a particular class of complete manifolds, and we use this fact to construct a manifold whose sectional curvature assumes any real value greater than a negative constant and admits a strictly convex function. In the last result, we relate the geometry of a Riemannian manifold of positive sectional curvature with the set of minimum points of a convex function defined on the manifold.

Keywords: Convex function; Geodesic flow; Conformal fields; Soul of a manifold; 26B25; 53C20 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-017-1087-2

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