An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions

作者：Shiohama, Katsuhiro; Xu, Hong-Wei^*

来源：Journal of Geometric Analysis, 2011, 21(2): 241-251.

DOI：10.1007/s12220-010-9116-5

• Isometric immersion
• Morse function
• Critical points
• Betti numbers
• Sphere theorems

We have established (see Shiohama and Xu in J. Geom. Anal. 7:377-386, 1997; Lemma) an integral formula on the absolute Lipschitz-Killing curvature and critical points of height functions of an isometrically immersed compact Riemannian n-manifold into Rn+q . Making use of this formula, we prove a topological sphere theorem and a differentiable sphere theorem for hypersurfaces with bounded L-n/2 Ricci curvature norm in Rn+1. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.

• 出版日期2011-4
• 单位浙江大学