摘要
Let Mn（n ≥ 4） be an oriented closed submanifold with parallel mean curvature in an （n ＋ p）-dimensional locally symmetric Riemannian manifold Nn＋p. We prove that if the sectional curvature of N is positively pinched in [5, 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].
Let Mn（n ≥ 4） be an oriented closed submanifold with parallel mean curvature in an （n ＋ p）-dimensional locally symmetric Riemannian manifold Nn＋p. We prove that if the sectional curvature of N is positively pinched in [5, 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].
基金
Supported by the National Natural Science Foundation of China(11531012,11371315,11301476)
the TransCentury Training Programme Foundation for Talents by the Ministry of Education of China
the Postdoctoral Science Foundation of Zhejiang Province(Bsh1202060)
