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Journal of Zhejiang University SCIENCE A 2005 Vol.6 No.4 P.322-328

http://doi.org/10.1631/jzus.2005.A0322


A rigidity theorem for submanifolds in Sn+p with constant scalar curvature


Author(s):  ZHANG Jian-feng

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310028, China; more

Corresponding email(s):   zjf7212@163.com

Key Words:  Scalar curvature, Mean curvature vector, The second fundamental form


ZHANG Jian-feng. A rigidity theorem for submanifolds in Sn+p with constant scalar curvature[J]. Journal of Zhejiang University Science A, 2005, 6(4): 322-328.

@article{title="A rigidity theorem for submanifolds in Sn+p with constant scalar curvature",
author="ZHANG Jian-feng",
journal="Journal of Zhejiang University Science A",
volume="6",
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pages="322-328",
year="2005",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2005.A0322"
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%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2005.A0322

TY - JOUR
T1 - A rigidity theorem for submanifolds in Sn+p with constant scalar curvature
A1 - ZHANG Jian-feng
J0 - Journal of Zhejiang University Science A
VL - 6
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SP - 322
EP - 328
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Y1 - 2005
PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2005.A0322


Abstract: 
Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn+p. Denote by R, H and S, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn+p with parallel normalized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.

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Reference

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[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.

[4] Hou, Z.H., 1997. Hypersurfaces in sphere with constant mean curvature. Proc Amer Soc, 125(4):1193-1196.

[5] Hou, Z.H., 1998. Submanifolds of constant scalar curvature in a space form. Kyun Math J, 38:439-458.

[6] Li, A.M., Li, J.M, 1992. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch Math, 58:582-594.

[7] Li, H.Z., 1994. Hypersurfaces with parallel mean curvature in a space forms. Math Ann, 305:403-415.

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[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).

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