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 K M ≥ 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(≥3)-dimensional complete connected Riemannian manifold with K M ≥ 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.
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Xia, C. Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds. Compositio Mathematica 132, 49–55 (2002). https://doi.org/10.1023/A:1016053731232
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DOI: https://doi.org/10.1023/A:1016053731232