M¨obius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1?

Tongzhu LI

1 Introduction

A diffeomorphism φ :is said to be a M¨obius transformation if φ takes the set of round n-spheres into the set of round n-spheres.All M¨obius transformations form a transformation group,which is called the M¨obius transformation group ofand denoted by.It is well-known that,for n≥2,the M¨obius transformation groupcoincides with the conformal transformation groupIn[11],Wang introduced complete M¨obius invariants for a submanifold x:,and obtained a congruent theorem of hypersurfaces in(also see[1]).Recently some special hypersurfaces,including the M¨obius isoparametric hypersurfaces,the Blaschke isoparametric hypersurfaces and so on,have been extensively studied in the context of M¨obius geometry(see[4–5,7]).

Another special hypersurface is the M¨obius homogeneous hypersurface.A hypersurface x:is called a M¨obius homogeneous hypersurface if for any two pointsthere exists a M¨obius transformationsuch thatLet x:be a M¨obius homogeneous hypersurface,we de fi ne

Then Π is a subgroup of the M¨obius group M,and the hypersurface x is the orbit of the subgroup Π.Thus the M¨obius scalar invariants on the hypersurface are constant.

Standard examples of M¨obius homogeneous hypersurfaces inare the image of homogeneous hypersurfaces innder the M¨obius transformations.The homogeneous hypersurface inis the isoparametric hypersurface,which was systematically studied(see[2–3]).Another standard examples come from homogeneous hypersurfaces in.The inverse of the stereographic projection σ :is defined by

The conformal map σ assigns any hypersurface into a hypersurface in.In[8],authors proved that the M¨obius invariants on f:1are the same as the M¨obius invariants on σ ? f:.If f:is a homogeneous hypersurface,then the hypersurface x= σ ? f is a M¨obius homogeneous hypersurface.

Next we give a method to construct the M¨obius homogeneous hypersurface in.

Proposition 1.1 Let u:be an immersed hypersurface.We define the cone over u as

If u:be a homogeneous hypersurface,then the image of σ of the cone hypersurface f over u is a M¨obius homogeneous hypersurface in.

These examples above come from homogeneous hypersurfaces in.But there are some examples of M¨obius homogeneous hypersurfaces which can not be obtained in this way.In[10],Sulanke constructed a M¨obius homogeneous surface,which is a cylinder over a logarithmic spiral in,and classified the M¨obius homogeneous surfaces in R3.In[6],authors constructed a M¨obius homogeneous hypersurface,a logarithmic spiral cylinder,which is a high dimensional version of Sulanke’s example and classi fi ed the M¨obius homogeneous hypersurfaces inwith two distinct principal curvatures.In addition,in[6],authors also classi fi ed the M¨obius homogeneous hypersurfaces in.

In this paper,the M¨obius homogeneous hypersurfaces with three distinct principal curvatures are classi fi ed,and the main results are as follows.

Theorem 1.1 Let x:be a M¨obius homogeneous hypersurface with three distinct principal curvatures.Then x is M¨obius equivalent to one of the following hypersurfaces:

(1)The isoparametric hypersurfaces inwith three distinct principal curvatures;(2)the image of σ of the cone over a standard torus

(3)the image of σ of the cone over the cartan’s minimal isoparametric hypersurface u:with three distinct principal curvatures.

Remark 1.1 Two hypersurfaces x,:are M¨obius equivalent,if there exists a M¨obius transformationsuch that

According to the classification results in[5],combining Proposition 1.1 and our main Theorem 1.1,we can derive the following corollary.

Corollary 1.1 A M¨obius homogeneous hypersurface with three distinct principal curvatures is a M¨obius isoparametric hypersurface.Conversely,a M¨obius isoparametric hypersurface with three distinct principal curvatures is a M¨obius homogeneous hypersurface.

We organize the paper as follows.In Section 2,we give the elementary facts about M¨obius geometry for hypersurfaces in Sn+1.In Section 3,we prove Proposition 1.1 and give a characterization of the cone hypersurfaces.In Section 4,we prove that the M¨obius form of the M¨obius homogeneous hypersurfaces with three distinct principal curvatures vanishes.In Section 5,we give the proof of Theorem 1.1.

2 M¨obius Invariants for Hypersurfaces in Sn+1

In this section,we recall some facts about the M¨obius transformation group and define M¨obius invariants of hypersurfaces in.For details we refer to[11].

Theorem 2.1(see[11])Two hypersurfaces x,are M¨obius equivalent if and only if there existssuch that

It follows immediately from Theorem 2.1 that respectively.The integrability conditions for the structure equations are given by

where Rijkldenote the curvature tensor of g,is the normalized M¨obius scalar curvature.When n≥3,we know that all coefficients in the structure equations are determined by{g,B},and we have the following theorem.

Theorem 2.2(see[11])Two hypersurfaces x:andare M¨obius equivalent if and only if there exists a diffeomorphism ? :,which preserves the M¨obius metric g and the M¨obius second fundamental form B.

The coefficients of the M¨obius second fundamental form can be written by

Clearly the number of distinct M¨obius principal curvatures is the same as that of its distinct principal curvatures.

3 A Method to Construct the M¨obius Homogeneous Hypersurface

Proposition 3.1 Let f:be an immersed submanifold without umbilical points.If there existand a submanifold u:such that the M¨obius position vector of f is

Then f is a cone over u.

Proof of Proposition 1.1 If u:is a homogeneous hypersurface,from(3.2),we know that Y=ρ0(u,id):is homogeneous.Thus the cone f is M¨obius homogeneous,and we finish the proof of Proposition 1.1.

Next,we give the proof of Theorem 1.1 for the case n=3.Let x:be a M¨obius homogeneous hypersurface with three distinct principal curvatures.From[6,12],we know that x is M¨obius equivalent to the two classes of hypersurfaces.One is the 1-parameter family of isoparametric hypersurfaces with three principal curvatures.Another is the images of σ of the cone over the 1-parameter family of isoparametric torus in S3.Thus,Theorem 1.1 holds for the hypersurfaces in S4.

4 The M¨obius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in Sn+1

Proposition 4.1 Let x:(n≥4)be a M¨obius homogeneous hypersurface with three distinct principal curvatures.Then,the M¨obius form of x vanishes,i.e.,C=0.

The proof of Proposition 4.1 is divided into the following three lemmas.

Lemma 4.1 Let x:(n≥4)be a M¨obius homogeneous hypersurface with three distinct principal curvatures.If,then the M¨obius form of x vanishes,i.e.,C=0.

The distributionsdetermined by eigenvectors of the M¨obius second fundamental form,are invariant under the subgroup Π.Since dimV1=1,the eigenvectors E1are invariant under the subgroup Π.Therefore the data,are constants.Since m2,m3≥2,we can choosesuch that i 6=j.From(4.4),we have

5 The Proof of the Main Theorem 1.1

When the dimension of the hypersurfaces n≥4.From Proposition 4.1,we know that the M¨obius form of the M¨obius homogeneous hypersurfaces vanishes when x has three distinct principal curvatures.On the other hand,the M¨obius principal curvatures of the M¨obius homogeneous hypersurfaces are constant.Thus the M¨obius homogeneous hypersurfaces with three distinct principal curvatures are M¨obius isoparametric hypersurfaces.In[5],the authors classified the M¨obius isoparametric hypersurfaces inwith three distinct principal curvatures.

Theorem 5.1(see[5])Let x:be a M¨obius isoparametric hypersurface with three distinct principal curvatures.Then x is M¨obius equivalent to an open part of one of the following hypersurfaces:

(1)The image of σ of the warped product embedding with p≥1,q≥1,p+q≤n?1 defined by

(2)The image of σ of the conedefined by ex(x,t)=tx,whereand x:is the Cartan’s minimal isoparametric hypersurface in Snwith three distinct principal curvatures.

(3)The Euclidean isoparametric hypersurfaces inwith three distinct principal curvatures.

Using Proposition 1.1,we know that the isoparametric hypersurfaces in Theorem 5.1 is M¨obius homogeneous.Thus when the M¨obius homogeneous hypersurfaces have three distinct principal curvatures,the main Theorem 1.1 holds.Combining the result in Section 3,we finish the proof of the main Theorem 1.1.

AcknowledgementsThe author would like to thank Professor Jie Qing for his hospitality and help.The author would also like to thank the referee for some valuable suggestions.

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