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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.

##### MSC:
 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)
##### Keywords:
Alexandrov space; convex hypersurface
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##### References:
 [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
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