What Do ‘Convexities’ Imply on Hadamard Manifolds?

Kristály, Alexandru and Li, Chong and Lopez, Genaro and Nicolae, Adriana (2016) What Do ‘Convexities’ Imply on Hadamard Manifolds? Journal of Optimization Theory and Applications, 170 (3). pp. 1068-1074. ISSN 0022-3239

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Abstract

Various results based on some convexity assumptions (involving the exponential map along with affine maps, geodesics and convex hulls) have been recently established on Hadamard manifolds. In this paper, we prove that these conditions are mutually equivalent and they hold, if and only if the Hadamard manifold is isometric to the Euclidean space. In this way, we show that some results in the literature obtained on Hadamard manifolds are actually nothing but their well-known Euclidean counterparts.

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