Monatshefte für Mathematik

, 164:225 | Cite as

Hypersurfaces with two distinct principal curvatures in a real space form

  • Shichang ShuEmail author
  • Sanyang Liu


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.


Hypersurface Trace free tensor Mean curvature Principal curvature 

Mathematics Subject Classification (2000)

53C42 53A10 


  1. 1.
    Alencar H., do Carmo M.P.: Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120, 1223–1229 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Cartan E.: Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45, 335–367 (1939)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cheng S.Y., Yau S.T.: Hypersurfaces with constant scalar curvature. Math. Ann. 225, 195–204 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cheng Q-M.: Hypersurfaces in a unit sphere S n+1(1) with constant scalar curvature. J. Lond. Math. Soc. 64, 755–768 (2001)zbMATHCrossRefGoogle Scholar
  5. 5.
    Cheng Q-M.: Complete hypersurfaces in a Euclidean space R n+1 with constant scalar curvature. Indiana Univ. Math. J. 51, 53–68 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Hu Z., Zhai S.: Hypersurfaces of the hyperbolic space with constant salar curvature. Results Math. 48, 65–88 (2005)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Li H.: Hypersurfaces with constant scalar curvature in space forms. Math. Ann. 305, 665–672 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Otsuki T.: Minimal hypersurfaces in a Riemannian manifold of constant curvature. Amer. J. Math. 92, 145–173 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Shu S., Yi Han A.: Hypersurfaces in a hyperbolic space with constant k-th mean curvature. Bull. Math. Soc. Sci. Math. Roumanie 52(100), 65–78 (2009)MathSciNetGoogle Scholar
  10. 10.
    Wei G.: Complete hypersurfaces with constant mean curvature in a unit sphere. Monatsh. Math. 149, 251–258 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

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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