Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

1998 ◽  
Vol 97 (3) ◽  
pp. 335-342 ◽  
Author(s):  
Sergio Console ◽  
Carlos Olmos
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Isoparametric Hypersurfaces ◽  
Clifford Systems

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.

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 First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

Second Fundamental Form and Mean Curvature

1988 ◽  
pp. 62-73
Author(s):  
Philippe Tondeur
Keyword(s):  
Fundamental Form ◽  
Mean Curvature ◽  
Second Fundamental Form

scholarly journals On the inner geometry of the second fundamental form.

1972 ◽  
Vol 19 (2) ◽  
pp. 129-132 ◽  
Author(s):  
Udo Simon
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
The Second Fundamental Form

scholarly journals Convex hypersurfaces with pinched second fundamental form

1994 ◽  
Vol 2 (1) ◽  
pp. 167-172 ◽  
Author(s):  
Richard S. Hamilton
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Convex Hypersurfaces

Complete 3-dimensional λ-translators in the Euclidean space ℝ4

2021 ◽  
pp. 1-54
Author(s):  
Zhi Li ◽  
Guoxin Wei ◽  
Gangyi Chen
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
3 Dimensional ◽  
The Second Fundamental Form

In this paper, we obtain the classification theorems for 3-dimensional complete [Formula: see text]-translators [Formula: see text] with constant squared norm [Formula: see text] of the second fundamental form and constant [Formula: see text] in the Euclidean space [Formula: see text].

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