On the sectional curvature of compact hypersurfaces. (English) Zbl 0703.53046
It is well known that the maximum sectional curvature of compact hypersurfaces of a Riemannian manifold can be bounded from below by the curvature of the ambient space and the outer radius. In the paper under review the authors prove the following theorem: Let \(\bar M\) be an \((d+1)\)-dimensional Riemannian manifold (d\(\geq 3)\) whose sectional curvature \(\bar K\) is bounded above by C and let M be a compact immersed hypersurface of \(\bar M.\) Suppose that M is contained in a closed normal ball \(\bar B_{\rho}\) in \(\bar M\) of radius \(\rho\). Then there exists a positive number \(\mu_ c(\rho)\) such that if the sectional curvature K of M satisfies the condition \(K\leq K+(\mu_ c(\rho)/\rho)^ 2,\) then \(\bar B_{\rho}\) is isometric to a closed normal ball of radius \(\rho\) in a space form of curvature c and \(M=\partial \bar B_{\rho}\). Similar rigidity phenomena were previously observed for the estimate of the mean curvature by D. Koutroufiotis [Arch. Math. 24, 97-99 (1973; Zbl 0252.53050)] and S. Markvorsen [Math. Z. 183, 407-411 (1983; Zbl 0498.53024)].
Reviewer: T.Hasanis
MSC:
| 53C40 | Global submanifolds |
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
References:
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