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Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

Published online by Cambridge University Press:  04 December 2007

Changyu Xia
Changyu Xia
Affiliation:
Departamento de Matemática – IE, Fundação Universidade de Brasília, Campus Universitário, 70910-900, Brasília, DF, Brasi. E-mail: xia@ipe.mat.unb.br
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Abstract

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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 [ges ] 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than π/2 and contains a geodesic loop of length 2π. We also prove that if M is an n([ges ]3)-dimensional complete connected Riemannian manifold with KM [ges ] 1 and radius bigger than π/2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.

Keywords

Riemannian manifoldscritical pointsdistance functiontopologyrigidity
Type
Research Article
Information
Compositio Mathematica , Volume 132 , Issue 1 , May 2002 , pp. 49 - 55
Copyright
© 2002 Kluwer Academic Publishers