Differentiable sphere theorems for submanifolds of positive k-th ricci curvature

作者：Xu, Hong-Wei; Ye, Fei^*

来源：Manuscripta Mathematica, 2012, 138(3-4): 529-543.

DOI：10.1007/s00229-011-0508-z

• HOMOLOGY VANISHING THEOREMS
• MEAN-CURVATURE
• COMPLETE MANIFOLDS
• MINIMAL SUBMANIFOLDS
• SPACE-FORMS
• HYPERSURFACES
• DIMENSIONS
• TOPOLOGY
• CURRENTS

We investigate the differentiable pinching problem for compact immersed submanifolds of positive -th Ricci curvature, and prove that if is simply connected and the -th Ricci curvature of is bounded below by a quantity involving the mean curvature of and the curvature of the ambient manifold, then is diffeomorphic to the standard sphere . For the case where the ambient manifold is a space form with nonnegative constant curvature, we prove a differentiable sphere theorem without the assumption that the submanifold is simply connected. Motivated by a geometric rigidity theorem due to S. T. Yau and U. Simon, we prove a topological rigidity theorem for submanifolds in a space form.

• 出版日期2012-7
• 单位浙江大学