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On the blaschke Isoparametric Hypersurfaces in the Unit Sphere

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Acta Mathematica Sinica,English Series 2009年第4期25卷 657-678页

作者： Xing Xiao LI Feng Yun ZHANG Department of Mathematics. He'nan Normal University Xinxiang 453007 P. R. China

Given an immersed submanifold x ： M^M → S^n in the unit sphere S^n without umbilics, a blaschke eigenvalue of x is by definition an eigenvalue of the blaschke tensor of x. x is called blaschke isoparametric if its Mobius form vanishes identically and all of its blaschke eigenvalues are constant. Then the classification of blaschke isoparametric hypersurfaces is natural and interesting in the MSbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the blaschke isoparametric hypersurfaces in S^5 with more than two distinct blaschke eigenvalues are necessarily Mobius isoparametric.

关键词： Mobius form blaschke eigenvalues blaschke tensor Mobius metric Mobius second fundamental form

Immersed Hypersurfaces in the Unit Sphere S^(m+1) with Constant blaschke Eigenvalues

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Acta Mathematica Sinica,English Series 2007年第3期23卷 533-548页

作者： Xing Xiao LI Feng Yun ZHANG Department of Mathematics He'nan Normal University Xinxiang 453007 He'nan P. R. China

For an immersed submanifold x ： M^m→ Sn in the unit sphere S^n without umbilics, an eigenvalue of the blaschke tensor of x is called a blaschke eigenvalue of x. It is interesting to determine all hypersurfaces in Sn with constant blaschke eigenvalues. In this paper, we are able to classify all immersed hypersurfaces in S^m＋1 with vanishing MSbius form and constant blaschke eigenvalues, in case （1） x has exact two distinct blaschke eigenvalues, or （2） m = 3. With these classifications, some interesting examples are also presented.

关键词： Moebius form blaschke eigenvalues blaschke tensor MSbius metric MSbius second fundamental form

On the blaschke isoparametric hypersurfaces in the unit sphere with three distinct blaschke eigenvalues

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Science China Mathematics 2011年第10期54卷 2171-2194页

作者： HU ZeJun LI XingXiao ZHAI ShuJie Department of Mathematics Zhengzhou University Zhengzhou 450052 China Department of Mathematics Henan Normal University Xinxiang 453007 China

An immersed umbilic-free submanifold in the unit sphere is called blaschke isoparametric if its Mobius form vanishes identically and all of its blaschke eigenvalues are constant. In this paper, we give a complete clas... 详细信息

关键词： blaschke isoparametric hypersurface Mobius metric MSbius form blaschke tensor MSbius second fundamental form

A Formula for Submanifolds in S^n and Its Applications in Moebius Geometry

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Northeastern Mathematical Journal 2001年第3期17卷 361-370页

作者：钟定兴 Department of Mathematics Gannan Teachers College Ganzhou Jiangxi 341000

In this paper, we obtain a formula for submanifolds in Sn+p by calculating the Laplacian of the Moebius second fundamental form. Using this formula, we obtain some pinching theorems about the minimal eigenvalue of the... 详细信息

关键词： Moebius metric Moebius second fundamental form Moebius form blaschke tensor eigenvalue

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

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Northeastern Mathematical Journal 2007年第1期23卷 15-23页

作者：钟定兴孙弘安 Department of Mathematics Gannan Teachers＇ College Ganzhou 341000

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces：（i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.

关键词： Mobius sectional curvature Mobius form Mobius second fundamental form blaschke tensor