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 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allendoerfer, C.B.: The Euler number of a Riemannian manifold. Am. J. Math. 62, 243–248 (1940)
Allendoerfer, C.B., Weil, A.: The Gauss-Bonnet theorem for Riemannian polyhedra. Trans. Am. Math. Soc. 53, 101–129 (1943)
Anderson, M.T.: Degeneration of metrics with bounded curvature and applications to critical metrics of Riemannian functionals. In: Green, R., Yau, S.-T. (eds.) Differential Geometry. Proc. Symp. Pure Math., vol. 54, pp. 53–79. Am. Math. Soc., Providence (1993)
Chen, B.Y.: On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof. Math. Ann. 194, 19–26 (1971)
Chern, S.S.: A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. Math. 45, 747–752 (1944)
Chern, S.S., Lashof, R.K.: On the total curvature of immersed manifolds II. Mich. Math. J. 5, 5–12 (1958)
Demailly, J.P.: Champs magnétiques et inégalités de Morse pour la d″-cohomologie. Ann. Inst. Fourier (Grenoble) 35(4), 189–229 (1985) (in French)
Fenchel, W.: On total curvatures of Riemannian manifolds. J. Lond. Math. Soc. 15, 15–22 (1940)
Grove, K., Shiohama, K.: A generalized sphere theorem. Ann. Math. 106, 201–211 (1977)
Milnor, J.W.: Morse Theory. Princeton University Press, Princeton (1963)
Patodi, V.K.: Curvature and the eigenforms of the Laplace operator. J. Differ. Geom. 5, 233–249 (1971)
Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology I. International Press, Cambridge (1994)
Shiohama, K., Xu, H.W.: Lower bound for L n/2 curvature norm and its application. J. Geom. Anal. 7, 377–386 (1997)
Shiohama, K., Xu, H.W.: The topological sphere theorem for complete submanifolds. Compos. Math. 107, 221–232 (1997)
Shiohama, K., Xu, H.W.: Rigidity and sphere theorems for submanifolds I. Kyushu J. Math. 48, 291–306 (1994)
Shiohama, K., Xu, H.W.: Rigidity and sphere theorems for submanifolds II. Kyushu J. Math. 54, 103–109 (2000)
Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)
Xu, H.W., Zhao, E.T.: Topological and differentiable sphere theorems for complete submanifolds. Commun. Anal. Geom. 17, 565–585 (2009)
Yu, Y.L.: Combinatorial Gauss-Bonnet-Chern formula. Topology 22, 153–163 (1983)
Zhang, W.P.: Lectures on Chern-Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, vol. 4. World Scientific, Singapore (2001)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Robert Greene.
Research supported by the NSFC, Grant No. 10771187; the Trans-Century Training Program Foundation for Talents by the Ministry of Education of China; and the Natural Science Foundation of Zhejiang Province, Grant No. 101037.
Rights and permissions
About this article
Cite this article
Shiohama, K., Xu, HW. An Integral Formula for Lipschitz-Killing Curvature and the Critical Points of Height Functions. J Geom Anal 21, 241–251 (2011). https://doi.org/10.1007/s12220-010-9116-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-010-9116-5