Geometric and topological rigidity for compact submanifolds of odd dimension |
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Authors: | HongWei Xu Yan Leng JuanRu Gu |
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Affiliation: | 1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China
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Abstract: | We investigate rigidity problems for odd-dimensional compact submanifolds. We show that if M n (n ? 5) is an odd-dimensional compact submanifold with parallel mean curvature in S n+p , and if Ric M > (n?2? (tfrac{1} {n}) )(1+H 2) and H < δ n , where δ n is an explicit positive constant depending only on n, then M is a totally umbilical sphere. Here H is the mean curvature of M. Moreover, we prove that if M n (n ? 5) is an odd-dimensional compact submanifold in the space form F n+p (c) with c ? 0, and if Ric M > (n?2?? n )(c+H 2), where ? n is an explicit positive constant depending only on n, then M is homeomorphic to a sphere. |
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Keywords: | submanifolds geometric and topological rigidity Ricci curvature stable currents homology group |
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