Full Text:   <1154>

CLC number: O186

On-line Access: 

Received: 2004-06-06

Revision Accepted: 2004-12-01

Crosschecked: 0000-00-00

Cited: 0

Clicked: 2865

Citations:  Bibtex RefMan EndNote GB/T7714

-   Go to

Article info.
1. Reference List
Open peer comments

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",
number="4",
pages="322~328",
year="2005",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2005.A0322"
}

%0 Journal Article
%T A rigidity theorem for submanifolds in Sn+p with constant scalar curvature
%A ZHANG Jian-feng
%J Journal of Zhejiang University SCIENCE A
%V 6
%N 4
%P 322~328
%@ 1673-565X
%D 2005
%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
IS - 4
SP - 322
EP - 328
%@ 1673-565X
Y1 - 2005
PB - Zhejiang University Press & Springer
ER -
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.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

[1] Alencar, H., de Carmo, M.P., 1994. Hypersurfaces with constant mean curvature in sphere. Proc Amer Math Soc, 120:1223-1229.

[2] Cheng, S.Y., Yau, S.T., 1977. Hypersurfaces with constant scalar curvature. Math. Ann., 225:195-204.

[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.

[8] Okumura, M., 1974. Hypersurfaces and a pinching problem on the second fundamental tensor. Amer J Math, 96:207-213.

[9] Simons, J., 1968. Minimal varieties in Riemannian manifolds. Ann of Math, 88(2):62-105.

[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).

Open peer comments: Debate/Discuss/Question/Opinion

<1>

Please provide your name, email address and a comment





Journal of Zhejiang University-SCIENCE, 38 Zheda Road, Hangzhou 310027, China
Tel: +86-571-87952783; E-mail: cjzhang@zju.edu.cn
Copyright © 2000 - Journal of Zhejiang University-SCIENCE