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Monatshefte für Mathematik

, 164:225 | Cite as

Hypersurfaces with two distinct principal curvatures in a real space form

  • Shichang ShuEmail author
  • Sanyang Liu
Article
  • 70 Downloads

Abstract

In this paper, we study hypersurfaces with two distinct principal curvatures in a real space form M n+1(c). Denote by \({\phi_{ij}}\) the trace free part of the second fundamental form of M n , and let ρ 2 be the square of the length of \({\phi_{ij}}\). If ρ 2 is constant, we obtain two rigidity results and give some characterization of the Riemannian products in \({M^{n+1}(c): S^k(a) \times S^{n-k}(\sqrt{1-a^2})}\) for c = 1, R k × S n-k (a) for c = 0 and \({H^k(\tanh^2 \varrho-1) \times S^{n-k}(\coth^2 \varrho-1)}\) for c = −1, where 1 ≤ k ≤ n − 1.

Keywords

Hypersurface Trace free tensor Mean curvature Principal curvature 

Mathematics Subject Classification (2000)

53C42 53A10 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsXianyang Normal UniversityXianyangPeople’s Republic of China
  2. 2.Department of Applied MathematicsXidian UniversityXi’anPeople’s Republic of China

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