Abstract
Let M n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an (n + p)-dimensional locally symmetric Riemannian manifold N n+p. We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu [15].
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Supported by the National Natural Science Foundation of China (11531012, 11371315, 11301476), the Trans- Century Training Programme Foundation for Talents by the Ministry of Education of China, and the Postdoctoral Science Foundation of Zhejiang Province (Bsh1202060).
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Gu, Jr., Leng, Y. & Xu, Hw. Rigidity of closed submanifolds in a locally symmetric Riemannian manifold. Appl. Math. J. Chin. Univ. 31, 237–252 (2016). https://doi.org/10.1007/s11766-016-3227-0
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DOI: https://doi.org/10.1007/s11766-016-3227-0