Geodesic
Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 435-445.

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In this paper we study the topological and metric rigidity of hypersurfaces in ${\mathbb H}^{n+1}$, the $(n+1)$-dimensional hyperbolic space of sectional curvature $-1$. We find conditions to ensure a complete connected oriented hypersurface in ${\mathbb H}^{n+1}$ to be diffeomorphic to a Euclidean sphere. We also give sufficient conditions for a complete connected oriented closed hypersurface with constant norm of the second fundamental form to be totally umbilic.
Classification : 53C20, 53C24, 53C40, 53C42
Keywords: rigidity; hypersurfaces; topology; hyperbolic space 
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     title = {Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space},
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Wang, Qiaoling; Xia, Changyu. Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 435-445. http://geodesic.mathdoc.fr/item/CMJ_2007__57_1_a32/