scholarly journals Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues inSn+1(1)

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Junfeng Chen ◽  
Shichang Shu
Keyword(s):  
Fundamental Form ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Möbius Form ◽  
Umbilical Points ◽  
Blaschke Tensor ◽  
Möbius Second Fundamental Form

Letx:M↦Sn+1(1)be ann  (n≥3)-dimensional immersed hypersurface without umbilical points and with vanishing Möbius form in a unit sphereSn+1(1), and letAandBbe the Blaschke tensor and the Möbius second fundamental form ofx, respectively. We define a symmetric(0,2)tensorD=A+λBwhich is called the para-Blaschke tensor ofx, whereλis a constant. An eigenvalue of the para-Blaschke tensor is calleda para-Blaschke eigenvalueofx. The aim of this paper is to classify the oriented hypersurfaces inSn+1(1)with two distinct para-Blaschke eigenvalues under some rigidity conditions.

scholarly journals PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

2012 ◽  
Vol 54 (3) ◽  
pp. 579-597 ◽  
Author(s):  
SHICHANG SHU ◽  
BIANPING SU
Keyword(s):  
Fundamental Form ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Isoparametric Hypersurfaces ◽  
Blaschke Tensor ◽  
Blaschke Isoparametric Hypersurface ◽  
Full Classification ◽  
Möbius Second Fundamental Form

AbstractLet A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.

scholarly journals Classification of Möbius Isoparametric Hypersurfaces in 4

2005 ◽  
Vol 179 ◽  
pp. 147-162 ◽  
Author(s):  
Zejun Hu ◽  
Haizhong Li
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Isoparametric Hypersurface ◽  
Isoparametric Hypersurfaces ◽  
Möbius Form ◽  
Möbius Geometry ◽  
Dimensional Unit ◽  
Dupin Hypersurfaces ◽  
Möbius Second Fundamental Form

AbstractLet Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

On submanifolds with parallel Möbius second fundamental form in the unit sphere

2014 ◽  
Vol 25 (06) ◽  
pp. 1450062 ◽  
Author(s):  
Shujie Zhai ◽  
Zejun Hu ◽  
Changping Wang
Keyword(s):  
Fundamental Form ◽  
Unit Sphere ◽  
Additional Condition ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Codimension Two ◽  
Möbius Second Fundamental Form

In this paper, we study umbilic-free submanifolds of the unit sphere with parallel Möbius second fundamental form. As one of our main results, we establish a complete classification for such submanifolds under the additional condition of codimension two.

scholarly journals A characterization of complete Riemannian manifolds minimally immersed in the unit sphere

1993 ◽  
Vol 131 ◽  
pp. 127-133 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Second Fundamental Form ◽  
Clifford Torus ◽  
Veronese Surface ◽  
The Second Fundamental Form ◽  
Complete Riemannian Manifolds

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totallygeodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

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 $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

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].

Complete hypersurfaces with constant Möbius scalar curvature

2016 ◽  
Vol 27 (08) ◽  
pp. 1650063
Author(s):  
Feng Jiang Li ◽  
Jian Bo Fang
Keyword(s):  
Scalar Curvature ◽  
Transformation Group ◽  
Minimal Hypersurface ◽  
Second Fundamental Form ◽  
Projection Formula ◽  
Möbius Form ◽  
Blaschke Tensor ◽  
Generalized Maximum Principle ◽  
Tensor Formula ◽  
Complete Hypersurfaces

Let [Formula: see text] be an umbilical free hypersurface in the unit sphere [Formula: see text]. Four basic invariants of [Formula: see text], under the Möbius transformation group of [Formula: see text] are the Möbius metric [Formula: see text], the Möbius second fundamental form [Formula: see text], the Blaschke tensor [Formula: see text] and the Möbius form [Formula: see text]. In this paper, we study complete hypersurfaces with constant normalized Möbius scalar curvature [Formula: see text] and vanishing Möbius form [Formula: see text]. By computing the Laplacian of the funtion [Formula: see text], where the trace-free Blaschke tensor [Formula: see text], and applying the well known generalized maximum principle of Omori–Yau, we obtain the following result: [Formula: see text] must be either Möbius equivalent to a minimal hypersurface with constant Möbius scalar curvature, when [Formula: see text]; [Formula: see text] in [Formula: see text], when [Formula: see text]; the pre-image of the stereographic projection [Formula: see text] of the circular cylinder [Formula: see text] in [Formula: see text], when [Formula: see text]; or the pre-image of the projection [Formula: see text] of the hypersurface [Formula: see text] in [Formula: see text], when [Formula: see text].

VANISHING THEOREMS FOR HYPERSURFACES IN THE UNIT SPHERE

2018 ◽  
Vol 60 (3) ◽  
pp. 661-671
Author(s):  
HEZI LIN
Keyword(s):  
Fundamental Form ◽  
Unit Sphere ◽  
Positive Constant ◽  
Total Curvature ◽  
Second Fundamental Form ◽  
Homology Sphere ◽  
Vanishing Theorems

AbstractLet Mn, n ≥ 3, be a complete hypersurface in $\mathbb{S}$n+1. When Mn is compact, we show that Mn is a homology sphere if the squared norm of its traceless second fundamental form is less than $\frac{2(n-1)}{n}$. When Mn is non-compact, we show that there are no non-trivial L2 harmonic p-forms, 1 ≤ p ≤ n − 1, on Mn under pointwise condition. We also show the non-existence of L2 harmonic 1-forms on Mn provided that Mn is minimal and $\frac{n-1}{n}$-stable. This implies that Mn has only one end. Finally, we prove that there exists an explicit positive constant C such that if the total curvature of Mn is less than C, then there are no non-trivial L2 harmonic p-forms on Mn for all 1 ≤ p ≤ n − 1.

scholarly journals SCALAR CURVATURE OF HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A SPHERE

2009 ◽  
Vol 51 (2) ◽  
pp. 413-423 ◽  
Author(s):  
QING-MING CHENG ◽  
YIJUN HE ◽  
HAIZHONG LI
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 M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, 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 $S_0=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4+4(n-1)H^2}$.

Sign in / Sign up

Export Citation Format

Share Document