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Journal of Zhejiang University-SCIENCE A

, Volume 6, Issue 4, pp 322–328 | Cite as

A rigidity theorem for submanifolds inSn+p with constant scalar curvature

  • Zhang Jian-feng
Applied Mathematics and Engineering Management Science
  • 42 Downloads

Abstract

LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.

Key words

Scalar curvature Mean curvature vector The second fundamental form 

Document code

CLC number

O186 

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References

  1. Alencar, H., de Carmo, M.P., 1994. Hypersurfaces with constant mean curvature in sphere.Proc Amer Math Soc,120:1223–1229.MathSciNetCrossRefMATHGoogle Scholar
  2. Cheng, S.Y., Yau, S.T., 1977. Hypersufaces with constant scalar curvature.Math. Ann.,225:195–204.MathSciNetCrossRefMATHGoogle Scholar
  3. Chern, S.S., de Carmo, M., Kobayashi, S., 1970. Minimal Submanifolds of A Sphere with Second Fundamental Form of Constant Length. Functional A Analysis and Related Fields, p.59–75.Google Scholar
  4. Hou, Z.H., 1997. Hypersurfaces in sphere with constant mean curvature.Proc Amer Soc,125(4):1193–1196.MathSciNetCrossRefMATHGoogle Scholar
  5. Hou, Z.H., 1998. Submanifolds of constant scalar curvature in a space form.Kyun Math J,38:439–458.MathSciNetMATHGoogle Scholar
  6. Li, A.M., Li, J.M., 1992. An intrinsic rigidity theorem for minimal submanifolds in a sphere.Arch Math,58:582–594.MathSciNetCrossRefMATHGoogle Scholar
  7. Li, H.Z., 1994. Hypersurfaces with parallel mean curvature in a space forms.Math Ann,305:403–415.Google Scholar
  8. Okumura, M., 1974. Hypersurfaces and a pinching problem on the second fundamental tensor.Amer J Math,96:207–213.MathSciNetCrossRefMATHGoogle Scholar
  9. Simons, J., 1968. Minimal varieties in Riemannian manifolds.Ann of Math,88(2):62–105.MathSciNetCrossRefMATHGoogle Scholar
  10. Zhang, J.F., 1999. On submanifolds with parallel mean curvature vector in a locally symmetric conformally flat riemannian manifold.J Zhejiang Univ (Engineering Science),26(4):26–34 (in Chinese).MathSciNetGoogle Scholar

Copyright information

© Zhejiang University Press 2005

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsLishui Teachers’ CollegeLishuiChina

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