zbMATH — the first resource for mathematics

An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds. (English) Zbl 1200.53040
The authors prove the following theorem: Let \(M\) be a Riemannian manifold with sectional curvature \(\geq \kappa\). Then any convex hypersurface \(F\) in \(M\) equipped with the induced intrinsic metric is an Alexandrov space with curvature \(\geq \kappa\). This is a global version of an earlier theorem by S. V. Buyalo [J. Sov. Math. 12, 73–85 (1979; Zbl 0405.53041)] showing that such \(F\) is locally an Alexandrov space.

53C20 Global Riemannian geometry, including pinching
53B25 Local submanifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
Full Text: Euclid arXiv
[1] S. Buyalo, Shortest paths on convex hypersurface of a Riemannian manifold (Russian), Studies in Topology, Zap. Nauchn. Sem. LOMI 66 (1976), 114–132; translated in J. of Soviet Math. 12 (1979), 73–85.
[2] R. E. Greene and H.-H. Wu, On the subharmonicity and plurisubharmonicity of geodesically convex functions , Indiana Univ. Math. J. 22 (1972/1973), 641–653. · Zbl 0235.53039
[3] A. D. Milka, Shortest arcs on convex surfaces (Russian), Dokl. Akad. Nauk SSSR 248 (1979), 34–36; translated in Soviet Math. Dokl. 20 (1979), 949–952. · Zbl 0441.53047
[4] P. Petersen, Riemannian geometry , Springer, New York, 1998. · Zbl 0914.53001
[5] A. Petrunin, Applications of quasigeodesics and gradient curves , Comparison geometry (Berkeley, CA, 1993–94), Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203–219. · Zbl 0892.53026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.