A note on Blaschke isoparametric hypersurfaces

2014 ◽  
Vol 25 (12) ◽  
pp. 1450117 ◽  
Author(s):  
Tongzhu Li ◽  
Changping Wang
Keyword(s):  
Isoparametric Hypersurface ◽  
Isoparametric Hypersurfaces ◽  
Blaschke Tensor ◽  
Blaschke Isoparametric Hypersurface

In this paper, we prove that a Möbius isoparametric hypersurface is a Blaschke isoparametric hypersurface, and a Blaschke isoparametric hypersurface is a Möbius isoparametric hypersurface provided that the Blaschke tensor has more than two distinct eigenvalues.

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.

Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Fengyun Zhang ◽  
Huafei Sun
Keyword(s):  
Conformal Transformation ◽  
Transformation Group ◽  
Second Fundamental Form ◽  
Conformal Metric ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Lorentzian Space ◽  
Blaschke Tensor ◽  
Lorentzian Space Forms

AbstractIn this paper, we study regular immersed hypersurfaces in Lorentzian space forms with a conformal metric, a conformal second fundamental form, the conformal Blaschke tensor and a conformal form, which are invariants under the conformal transformation group. We classify all the immersed hypersurfaces in Lorentzian space forms with two distinct constant Blaschke eigenvalues and vanishing conformal form.

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 Isoparametric hypersurfaces with four principal curvatures, II

2011 ◽  
Vol 204 ◽  
pp. 1-18 ◽  
Author(s):  
Quo-Shin Chi
Keyword(s):  
Commutative Algebra ◽  
Quaternion Algebra ◽  
Type A ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Octonion Algebra ◽  
Isoparametric Hypersurfaces ◽  
New Approach ◽  
Condition B

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) inS15is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively asof OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraintm2≥2m1-1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

scholarly journals Isoparametric hypersurfaces with four principal curvatures, II

2011 ◽  
Vol 204 ◽  
pp. 1-18 ◽  
Author(s):  
Quo-Shin Chi
Keyword(s):  
Commutative Algebra ◽  
Quaternion Algebra ◽  
Type A ◽  
Isoparametric Hypersurface ◽  
Principal Curvatures ◽  
Octonion Algebra ◽  
Isoparametric Hypersurfaces ◽  
New Approach ◽  
Condition B

AbstractIn this sequel to an earlier article, employing more commutative algebra than previously, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki and Takeuchi and Ferus, Karcher, and Münzner, referred to collectively as of OT-FKM type. In fact, this new approach also gives a considerably simpler proof, both structurally and technically, that an isoparametric hypersurface with four principal curvatures in spheres with the multiplicity constraint m2≥ 2m1 -1 is of OT-FKM type, which left unsettled exactly the four anomalous multiplicity pairs (4,5),(3,4),(7,8), and (6, 9), where the last three are closely tied, respectively, with the quaternion algebra, the octonion algebra, and the complexified octonion algebra, whereas the first stands alone in that it cannot be of OT-FKM type. A by-product of this new approach is that we see that Condition B, introduced by Ozeki and Takeuchi in their construction of inhomogeneous isoparametric hypersurfaces, naturally arises. The cases for the multiplicity pairs (4,5), (6, 9), and (7,8) remain open now.

scholarly journals Isoparametric Hypersurfaces with four Principal Curvatures Revisited

2009 ◽  
Vol 193 ◽  
pp. 129-154 ◽  
Author(s):  
Quo-Shin Chi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fundamental Form ◽  
Isoparametric Hypersurface ◽  
Geometric Meaning ◽  
Principal Curvatures ◽  
Expansion Formula ◽  
Isoparametric Hypersurfaces ◽  
Focal Submanifold

AbstractThe classification of isoparametric hypersurfaces with four principal curvatures in spheres in [2] hinges on a crucial characterization, in terms of four sets of equations of the 2nd fundamental form tensors of a focal submanifold, of an isoparametric hypersurface of the type constructed by Ferus, Karcher and Münzner. The proof of the characterization in [2] is an extremely long calculation by exterior derivatives with remarkable cancellations, which is motivated by the idea that an isoparametric hypersurface is defined by an over-determined system of partial differential equations. Therefore, exterior differentiating sufficiently many times should gather us enough information for the conclusion. In spite of its elementary nature, the magnitude of the calculation and the surprisingly pleasant cancellations make it desirable to understand the underlying geometric principles.In this paper, we give a conceptual, and considerably shorter, proof of the characterization based on Ozeki and Takeuchi’s expansion formula for the Cartan-Münzner polynomial. Along the way the geometric meaning of these four sets of equations also becomes clear.

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