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  • Cited by 2
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Brasil, Aldir Colares, A. Gervasio and Palmas, Oscar 2010. Complete hypersurfaces with constant scalar curvature in spheres. Monatshefte für Mathematik, Vol. 161, Issue. 4, p. 369.


    Deshmukh, Sharief 2010. Minimal hypersurfaces in a nearly Kaehler 6-sphere. Journal of Geometry and Physics, Vol. 60, Issue. 4, p. 623.


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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 135, Issue 6
  • December 2005, pp. 1129-1137

Compact hypersurfaces in a unit sphere

  • Qing-Ming Cheng (a1), Shichang Shu (a2) and Young Jin Suh (a3)
  • DOI: http://dx.doi.org/10.1017/S0308210500004303
  • Published online: 12 July 2007
Abstract

We study curvature structures of compact hypersurfaces in the unit sphere Sn+1(1) with two distinct principal curvatures. First of all, we prove that the Riemannian product is the only compact hypersurface in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies where n(n − 1)r is the scalar curvature of hypersurfaces and c2 = (n − 2)/nr. This generalized the result of Cheng, where the scalar curvature is constant is assumed. Secondly, we prove that the Riemannian product is the only compact hypersurface with non-zero mean curvature in Sn+1(1) with two distinct principal curvatures, one of which is simple and satisfies This gives a partial answer for the problem proposed by Cheng.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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