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Topological and metric rigidity teorems for hypersurfaces in a hyperbolic space
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 1, pp. 435-445 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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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/

[1] A. D. Alexandrov: A characteristic property of spheres. Ann. Mat. Pura Appl. 58 (1962), 303–315. | DOI | MR

[2] H. Alencar, M. do  Carmo, and W. Santos: A gap theorem for hypersurfaces of the sphere with constant scalar curvature one. Comment. Math. Helv. 77 (2002), 549–562. | DOI | MR

[3] S. Alexander: Locally convex hypersurfaces of negatively curved spaces. Proc. Amer. Math. Soc. 64 (1977), 321–325. | DOI | MR | Zbl

[4] S. S. Chern, M. do  Carmo, and S. Kobayashi: Minimal submanifolds of a sphere with second fundamental form of constant length. In: Functional Analysis and Related Fields, F.  Browder (ed.), Springer-Verlag, Berlin, 1970, pp. 59–75. | MR

[5] M. P. do  Carmo: Riemannian Geometry. Bikhäuser-Verlag, Boston, 1993.

[6] M. P. do  Carmo, F. W. Warner: Rigidity and convexity of hypersurfaces in spheres. J.  Diff. Geom. 4 (1970), 133–144. | DOI | MR

[7] S. Y. Cheng, S. T. Yau: Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), 195–204. | DOI | MR

[8] P. Hartman, L. Nirenberg: On spherical image maps whose Jacobians do not change sign. Amer. J.  Math. 81 (1959), 901–920. | DOI | MR

[9] H. Z. Li: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305 (1996), 665–672. | DOI | MR | Zbl

[10] W. S.  Massey: Algebraic Topology: An Introduction. 4th corr. print. Graduate Texts in Mathematics Vol. 56. Springer-Verlag, New York-Heidelberg-Berlin, 1967. | MR

[11] S. Montiel, A. Ros: Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures. Differential geometry. Pitman Monographs. Surveys Pure Appl. Math.,  52, Longman Sci. Tech., Harlow, 1991, pp. 279–296. | MR

[12] M. Okumura: Hypersurfaces and a pinching problem on the second fundamental tensor. Amer. J.  Math. 96 (1974), 207–214. | DOI | MR | Zbl

[13] C. K. Peng, C. L. Terng: Minimal hypersurfaces of spheres with constant scalar curvature. Ann. Math. Stud. 103 (1983), 177–198. | MR

[14] C. K. Peng, C. L. Terng: The scalar curvature of minimal hypersurfaces in spheres. Math. Ann. 266 (1983), 105–113. | DOI | MR

[15] A. Ros: Compact hypersurfaces with constant scalar curvature and a congruence theorem. J.  Differential Geom. 27 (1988), 215–223. | DOI | MR | Zbl

[16] A. Ros: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3 (1987), 447–453. | MR | Zbl

[17] R. Sacksteder: On hypersurfaces with no negative sectional curvatures. Amer. J.  Math. 82 (1960), 609–630. | DOI | MR | Zbl

[18] J. Simons: Minimal varieties in Riemannian manifolds. Ann. Math. 88 (1968), 62–105. | DOI | MR | Zbl