Hung-Lin Chiu

Department of Mathematics,Tsing Hua University,Hsinchu 300,China.

Abstract.In this paper,we study some basic geometric properties of pseudohermitian submanifolds of the Heisenberg groups.In particular,we obtain the uniqueness and existence theorems,and some rigidity theorems.

Key words:Motion equations,structure equations,Darboux frame,Darboux derivative.

1 Introduction

In this paper,for m≤n,we specify the ranges of indices as follows

1.1 The Heisenberg groups

The origin of pseudohermitian geometry came from the construction of a pseudohermitian connection,independently by N.Tanaka[15]and S.Webster[16].In this paper,the Heisenberg group is a pseudohermitian manifold and it plays the role of the model in pseudohermitian geometry.That is,any pseudohermitian manifold with vanishing curvature and torsion locally is part of the Heisenberg group.Let Hnbe the Heisenberg group,with coordinates(xβ,yβ,t).The group multiplication is defined by

The associated standard CR structure J and contact form Θ are defined respectively by

The contact bundle is ξ=kerΘ.We refer the reader to[2,3,5]for the details about the Heisenberg groups,and to[6,11,12,15,16]for pseudohermitian geometry.

The symmetry group PSH(n)of Hnis the group consisting of all pseudohermitian transformations.Left translations Lpare symmetries.Another kind of examples are a rotation ΦRaround the t-axis which is defined by

1.2 Pseudohermitian submanifolds

We now give the definition of pseudohermitian submanifold.

Definition 1.1.A(2m+1)-dimensional pseudohermitian manifoldis called a pseudohermitian??In[6],S.Dragomir and G.Tomassini call it isopseudo-hermitian,instead of pseudohermitian.submanifold of Hn,1≤m≤n,if

Example 1.1.Suppose M,→ Hnis an embedded submanifold with CR dimension n?1.Then it is not hard to see that

?In general,dim(TpM∩ξp)≥2n?2,for all p∈M.

?dim(TpM∩ξp)=2n?2,for a generic point p∈M.

All the generic points constitute the regular part of M,and those points p such that dim(TpM∩ξp)=2n?1 are called the singular points.On the regular part Mre,assume that TpM∩ξ is invariant under J,then it inherits a pseudohermitian structurefrom Hnsuch thatis a pseudohermitian submanifold of Hn.

In Section 3,we define some local invariants for pseudohermitian submanifolds,including the second fundamental form,the normal connection and the fundamental vector field ν(see the definition after Proposition 3.1).In addition,from Proposition 3.1,we see that the fundamental vector field ν actually describes the difference between the two Reeb vector fields T and,which are,respectively,associated with Hnand the pseudohermitian submanifold M.Hence if ν=0,then=T.That means thatis always tangent to M at each point.Therefore,for such a submanifold,we call it vertical??In[6],S.Dragomir and G.Tomassini call it pseuo-Hermitian,instead of vertical pseudohermitian..On the other hand,if ν6=0 at each point,we call it completely non-vertical.

Example 1.2.The subspace Hm={(z,t)∈Hn|za=0}?Hnis a pseudohermitian submanifold of Hn.It is easy to see that Hmis vertical.

Example 1.3.Let S2n?1?Hnbe the sphere defined by

There are two pseudohermitian structures induced on S2n?1,one is from Hnand the other is from Cn.In Subsection 4.2,we show that these two induced pseudohermitian structures coincide.In addition,S2n?1is completely non-vertical.

1.3 Main theorems

There are many literatures which were given for the problem about CR embeddability of CR manifolds into spheres[4,9,10,17].In this paper,we obtain the fundamental theorems and rigidity theorems for pseudohermitian submanifolds in the Heisenberg groups.We have

Theorem 1.1(Theorem A).The induced pseudohermitian structure,the second fundamental form,the normal connection,as well as the fundamental vector field constitute a complete set of invariants for pseudohermitian submanifolds of the Heisenberg groups.

Theorem A is shown in Section 5.It specifies that there are only four invariants for pseudohermitian submanifolds.That is,if two pseudohermitian submanifolds have the same such four invariants,then they are locally congruent with each other in the sense that they differ from each other by nothing more than an action of a symmetry.

Any pseudohermitian submanifold M?Hnautomatically satisfies a natural geometric condition which we call the integrability condition(defined in Subsection 5.2).Conversely,we will show that it is also a condition for an arbitrary pseudohermitian manifold to be(locally)embedded as a pseudohermitian submanifold of Hn.

Theorem 1.2(Theorem B).Let(M2m+1,JM,θM)be a simply connected pseudohermitian manifold satisfying the integrability condition for some n≥m.Then M can be embedded as a pseudohermitian submanifold of the Heisenberg group Hn.

Theorem B is shown in Section 6.In[9],S.-Y.Kim and J.-W.Oh also studied the problem of characterizing pseudohermitian manifolds which are pseudohermitian embeddable into the Heisenberg groups§§In their paper,they used the pseudohermitian flat sphere as the ambient space,instead of the Heisenberg group.But after a Cayley transformation,this two spaces are isomorphic as pseudohermitian manifolds.In the case that M is nondegenerate,S.-Y.Kim and J.-W.Oh used Cartan’s prolongation method to show that the induced pseudohermitian structure constitutes a complete set of invariants.In addition,they gave a necessary and sufficient condition,in terms of Webster curvature and torsion tensor,for pseudohermitian manifolds to be embeddable into the Heisenberg groups nondegenerately.This condition is just equivalent to the integrability condition which we define in Subsection 5.2.However,S.-Y.Kim and J.-W.Oh did not deal with the degenerate cases.

In the case of CR codimension one,the nondegeneracy just means that the second fundamental form does not vanish at each point.In such a case,we basically recover the results of S.-Y.Kim and J.-W.Oh.Moreover,we give the rigidity theorems for pseudohermitian degenerate submanifolds,which are shown in Section 7.We have

(i)if the second fundamental form II6=0 at each point,then the induced pseudohermitian structureconstitutes a complete set of invariants.

(ii)if II=0,then M is an open part of Hn?1={zn=0},after a Heisenberg rigid motion.

(i)the induced pseudohermitian structureconstitutes a complete set of invariants.

(ii)if the second fundamental form II=0(or,equivalently,the pseudohermitian torsion Ajk=0,1≤j,k≤m),then M is an open part of the standard sphere S2m+1?Hn,after a Heisenberg rigid motion.

Finally,in subsection 4.1,we study the general properties of vertical pseudohermitian submanifolds and obtain

For the fundamental theorems,we used Cartan’s method of moving frame as well as calculus on Lie groups.And we prove(ii)of Theorem D by means of the motions equation of the Darboux frame.Therefore,in Section 2,we give a brief review of Cartan’s method of moving frame,which includes the motion equations and the structure equations.We would like to end the introduction by pointing out that,in[4],Curry and Gover recently addressed the so called CR Bonnet theorem.They formulated and proved the theorem inspired by the conformal Bonnet theorem formulated and proved in terms of standard conformal tractors.

2 Cartan’s method of moving frame

In this section,we give a brief review of Cartan’s method of moving frame and Calculus on Lie groups.For the details,we refer the reader to[5].Let(X,G)be a Klein geometry.The philosophy of Elie Cartan is that in many cases,the symmetry group G may be identified with a set of frame on X.Then to investigate the geometry of a submanifold M of X,one associates the submanifold with a natural set of frames.In this situation,the infinitesimal motion of this natural frame should contain all the geometric information of the submanifold M.Now we will go along the idea of Elie Cartan to get a complete set of invariants for M.

2.1 The frames on Hn

An frame for Hnis a set of vectors of the form

where p∈ Hn,eβ∈ξ(p)and en+β=Jeβ,for 1≤β≤n.In addition{eβ,en+β,T}is an orthonormal frame with respect to the adapted metric gθ,which is defined by viewing the basis?eβ,?en+β,T as an orthonormal basis.

2.2 Identifying PSH(n)with a set of frames

We identify a symmetry Φ with a frame(p;eβ,en+β,T),provided that Φ is the unique transformation on Hnmapping the frame(0;?eβ,?en+β,T)to the given frame(p;eβ,en+β,T).That is

2.3 The matrix group representation of PSH(n)

2.4 The motion equations

Let ω be the(left)Maurer Cartan form of PSH(n).This is a psh(n)-valued one form defined by

for each v∈TgG,where G=PSH(n),g∈G.That is,the Maurer Cartan form moves each vector v to the identity element by the left translations.It is a natural way for us to identify each vector v with a vector tangent to the identity.Since PSH(n)has a matrix group representation,The Maurer Cartan form has the simple elegant expression

where A∈PSH(n)is the moving point.This formula(2.3)is equivalent to

which is called the motion equations of the Heisenberg group.Taking the exterior derivative of the motion equations,we get the structure equations

2.5 The Darboux frames for pseudohermitian submanifolds

2.6 The Darboux derivative

2.7 The complex version

2.8 Calculus on Lie groups

Let M be a simply connected smooth manifold,f:M→PSH(n)be a smooth map.Recall that The(left)Darboux derivative ωfof f is the psh(n)-valued 1-form defined by ωf=ω? f?=f?ω.The Darboux derivative plays an important role in the theory of calculus on Lie groups.The fundamental theorems are Theorem 2.1 and Theorem 2.2.

Theorem 2.1(The uniqueness theorem).Let f1,f2:M→PSH(n)be smooth maps.Then ωf1=ωf2if and only if there exists g∈PSH(n)such that f2(x)=g·f1(x)for all x∈M.

Theorem 2.1 says that two maps from M into PSH(n)are congruent to each other if and only if they have the same infinitesimal motions.Recall that ωfsatisfies the integrability conditions

Conversely,one has

Theorem 2.2(The existence theorem).Let η be a psh(n)-valued one form on M satisfying dη+η∧η=0.Then there is a smooth map f:U→PSH(n)such that η|U=ωf.

Theorem 2.2 totally depends on Frobenius Theorem.We will apply theorem 2.1 to the Darboux frames of pseudohermitian submanifolds.Then,to prove Theorem A,we are reduced to compute the Darboux derivatives of the Darboux frames.And using Theorem 2.2,we obtain Theorem B.For the details about calculus on Lie groups,we refer the reader to[1,7,8,13,14].

3 Local invariants of Pseudohermitian submanifolds

In this section,we define some geometric invariants for pseudohermitian submanifolds.

3.1 The fundamental vector field ν

3.2 The normal connection

The normal connection?⊥which is defined,on the normal complex bundle?C spanned by Za,by

which is the orthogonal projection of the pseudohermitian connection?Zaonto the normal bundle.

3.3 The tangential connection

The tangential connection?twhich is defined,on the complex bundlespanned by Zj,by

which is the orthogonal projection of the pseudohermitian connection?Zjonto the contact bundle.

Then,from(5.11),we have

Therefore,in general,?t6=?p.h.,the associated pseudohermitian connection of M.

?If M is vertical,then?t=?p.h..

3.4 The second fundamental form

4 General properties

4.1 Pseudohermitian submanifolds with ν≡0

4.2 Pseudohermitian submanifolds with ν nowhere zero

where the norm|·|is measured by the levi metric.Next,there are two pseudohermitian structures induced on S2n?1,one is from the Heisenberg group Hn,denoted by,and the other is from Cn.It is easy to see that these two induced pseudohermitian structures

5 The uniqueness theorem

Corollary 5.1.If M and N are vertical,then the(induced)pseudohermitian structures,the second fundamental forms and the normal connections constitute a complete set of invariants.

5.1 The proof of Theorem 5.1

5.2 The integrability condition

Definition 5.2.The restriction to M of the structure equations of Hn,

is defined to be the integrability condition of M.Note that the restrictions of θβand θβγto M have the expressions of the forms as(5.11)specifies..

6 The existence theorem

In this section,we would like to show Theorem B.Let(M,JM,θM)be a pseudohermitian manifold with CR dimension m.Since the existence theory is local,we assume that M is simply connected.Putting ξM=kerθMand η=θM.

7 Rigidity theorems for submanifolds with CR co-dimension one

In this section,we prove some rigidity theorems for pseudohermitian submanifolds,including both the nondegenerate and degenerate cases.

Proof.In the case m=n?1,we write θjn=hjkθk,here hjkare the coefficients of the second fundamental form II.If II=0,then θjn=0.On the other hand,ν=0 implies θn=0.Hence the structure equations of Hn,restricting to M,are reduced to

The last equation of(7.1)says that θnnis closed,and hence locally is exact.By the transformation law of the normal connection,we can choose a normal frame Znsuch that the corresponding connection form θnnvanishes.On the other hand,the first three equations of(7.1)is just the structure equations of Hn?1.This means that M is an open part U of Hn?1? Hn,up to a pseudohermitian transformatio ? from M to U.Define F by F(x,Zn(x))=(?(x),?Zn).Then F defines the normal bundle isomorphism covering ? which preserving the induced pseudohermitian structures,the second fundamental forms and the normal connections of M and U,respectively.Hence ? is just the restriction of a Heisenberg rigid motion.

For a vertical pseudohermitian submanifold of Hn,we define a flat point of M to be a point such that II=0 at that point.Theorem 7.4 says that the induced pseudohermitian structure is the only invariant for vertical pseudohermitian submanifolds without flat points.

Acknowledgments

The author’s research is supported in part by NCTS and in part by MOST 106-2115-M-007-017-MY3.He would like to thank Prof.Jih-Hsin Cheng,Prof.Jenn-Fang Hwang,and Prof.Paul Yang for regular kind encouragement and advising in his research.