Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds |
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Authors: | Changyu Xia |
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Affiliation: | (1) Departamento de Matemática – IE, Fundação Universidade de Brasília, Campus Universitário, 70910-900 Brasília, DF, Brasi |
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Abstract: | In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature KM 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with KM 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere. |
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Keywords: | Riemannian manifolds critical points distance function topology rigidity |
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