scholarly journals RIGIDITY OF HYPERSURFACES IN A EUCLIDEAN SPHERE

2006 ◽  
Vol 49 (1) ◽  
pp. 241-249 ◽  
Author(s):  
Qiaoling Wang ◽  
Changyu Xia
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Closed Geodesic ◽  
Constant Scalar Curvature ◽  
Geodesic Ball ◽  
Euclidean Sphere ◽  
Gauss Image ◽  
Closed Hypersurface ◽  
Rigidity Theorems ◽  
Orientable Hypersurface

AbstractThis paper studies topological and metric rigidity theorems for hypersurfaces in a Euclidean sphere. We first show that an $n({\geq}\,2)$-dimensional complete connected oriented closed hypersurface with non-vanishing Gauss–Kronecker curvature immersed in a Euclidean open hemisphere is diffeomorphic to a Euclidean $n$-sphere. We also show that an $n({\geq}\,2)$-dimensional complete connected orientable hypersurface immersed in a unit sphere $S^{n+1}$ whose Gauss image is contained in a closed geodesic ball of radius less than $\pi/2$ in $S^{n+1}$ is diffeomorphic to a sphere. Finally, we prove that an $n({\geq}\,2)$-dimensional connected closed orientable hypersurface in $S^{n+1}$ with constant scalar curvature greater than $n(n-1)$ and Gauss image contained in an open hemisphere is totally umbilic.

scholarly journals Rigidity theorems of Clifford Torus

2001 ◽  
Vol 73 (3) ◽  
pp. 327-332 ◽  
Author(s):  
LUIZ A. M. SOUSA JR.
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Principal Curvatures ◽  
Clifford Torus ◽  
Rigidity Theorems

Let M be an n-dimensional closed minimally immersed hypersurface in the unit sphere Sn + 1. Assume in addition that M has constant scalar curvature or constant Gauss-Kronecker curvature. In this note we announce that if M has (n - 1) principal curvatures with the same sign everywhere, then M is isometric to a Clifford Torus <img src="http:/img/fbpe/aabc/v73n3/03ab.gif" alt="03ab.gif (725 bytes)" align="middle">.

scholarly journals RIGIDITY THEOREMS FOR HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN A UNIT SPHERE

2007 ◽  
Vol 49 (2) ◽  
pp. 235-241 ◽  
Author(s):  
GUOXIN WEI ◽  
YOUNG JIN SUH
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Clifford Tori ◽  
Rigidity Theorems

AbstractIn this paper, we give a characterization of Clifford tori $S^1(\sqrt{\textstyle\frac{nr+2-n}{nr}})\times S^{n-1}(\sqrt{\frac{n-2}{nr}})$ and $S^m(a)\times S^{n-m}(\sqrt{1-a^2}) (2 \le m \le n - 2, 0 \lt a^2 \lt 1)$ in a unit sphere Sn+1 (1). Our results extend the results due to Cheng and Yau [4], and Wang and Xia [11].

EMBEDDED HYPERSURFACES WITH CONSTANT mTH MEAN CURVATURE IN A UNIT SPHERE

2010 ◽  
Vol 12 (06) ◽  
pp. 997-1013 ◽  
Author(s):  
GUOXIN WEI ◽  
QING-MING CHENG ◽  
HAIZHONG LI
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Scalar Curvature

In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere Sn+1(1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature Hm > 0 in Sn+1(1), for 1 ≤ m ≤ n-1. Moreover, if the 2nd mean curvature H2 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 2 and n ≥ 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H2 (i.e. constant scalar curvature) in Sn+1(1); If the 4th mean curvature H4 takes value between [Formula: see text] and [Formula: see text] for any integer k ≥ 3 and n ≥ 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H4 in Sn+1(1).

THE SCALAR CURVATURE OF MINIMAL HYPERSURFACES IN A UNIT SPHERE

2007 ◽  
Vol 09 (02) ◽  
pp. 183-200 ◽  
Author(s):  
YOUNG JIN SUH ◽  
HAE YOUNG YANG
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Minimal Hypersurface ◽  
Second Fundamental Form ◽  
Geodesic Sphere ◽  
Clifford Torus ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces ◽  
The Second Fundamental Form

In this paper, we study n-dimensional compact minimal hypersurfaces in a unit sphere Sn+1(1) and give an answer for S. S. Chern's conjecture. We have shown that [Formula: see text] if S > n, and prove that an n-dimensional compact minimal hypersurface with constant scalar curvature in Sn+1(1) is a totally geodesic sphere or a Clifford torus if [Formula: see text], where S denotes the squared norm of the second fundamental form of this hypersurface.

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

2011 ◽  
Vol 54 (1) ◽  
pp. 67-75 ◽  
Author(s):  
QIN ZHANG
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface ◽  
Small Constant

AbstractLet Mn be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1(1), n ≤ 8 and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

Submanifolds with constant scalar curvature

2002 ◽  
Vol 132 (5) ◽  
pp. 1163-1183 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Euclidean Space ◽  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Totally Geodesic ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Image Position ◽  
Generalized Cylinder

In this paper, we study n-dimensional complete submanifolds with constant scalar curvature in the Euclidean space En+p and n-dimensional compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1). We prove that the totally umbilical sphere Sn(r), totally geodesic Euclidean space En and generalized cylinder Sn−1(c) × E1 are the only n-dimensional (n > 2) complete submanifolds Mn with constant scalar curvature n(n − 1)r in the Euclidean space En+p, which satisfy the following condition: where S denotes the squared norm of the second fundamental form of Mn. For compact submanifolds with constant scalar curvature in the unit sphere Sn+p(1), we also obtain a corresponding result (see theorem 1.3).

SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN UNIT SPHERES

2011 ◽  
Vol 22 (01) ◽  
pp. 131-143 ◽  
Author(s):  
GANGYI CHEN ◽  
HAIZHONG LI
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Unit Sphere ◽  
Positive Constant ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Closed Hypersurface

Let M be an n-dimensional closed hypersurface with constant mean curvature H in a unit sphere Sn+1, n ≤ 8, and S the squared length of the second fundamental form of M. If |H| ≤ ε(n), then there exists a positive constant α(n, H), which depends only on n and H, such that if S0 ≤ S ≤ S0 + α(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ε(n) is a positive constant depending only on n and [Formula: see text].

HYPERSURFACES IN A UNIT SPHERE Sn+1(1) WITH CONSTANT SCALAR CURVATURE

2001 ◽  
Vol 64 (3) ◽  
pp. 755-768 ◽  
Author(s):  
QING-MING CHENG
Keyword(s):  
Scalar Curvature ◽  
Unit Sphere ◽  
Constant Scalar Curvature ◽  
Second Fundamental Form ◽  
Conformally Flat ◽  
Principal Curvatures ◽  
The Second Fundamental Form ◽  
Locally Conformally Flat ◽  
Complete Hypersurfaces ◽  
Conformally Flat Hypersurface

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and(1) when r ≠ (n−2)/(n−1), ifthen M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

Sign in / Sign up

Export Citation Format

Share Document