Abstract
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 Ln/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.
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