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

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 R ^n+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 R ⁿ⁺¹. We show that the theorems of Gauss-Bonnet-Chern, Chern-Lashof and the Willmore inequality are all its consequences.