Communications in Analysis and Geometry

Volume 17 (2009)

Number 3

Topological and differentiable sphere theorems for complete submanifolds

Pages: 565 – 585

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n3.a6

Authors

Hong-Wei Xu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

En-Tao Zhao (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

We investigate topological and differentiable structures ofsubmanifolds under extrinsic restrictions. We first obtaina topological sphere theorem for compact submanifolds in aRiemannian manifold. Secondly, we prove an optimaldifferentiable sphere theorem for 4-dimensional completesubmanifolds in a space form, which provides a partialsolution of the smooth Poincaré conjecture. Finally, weprove some new differentiable sphere theorems for$n$-dimensional submanifolds in a Riemannian manifold.

Full Text (PDF format)