Wenjuan ZHANG

School of Science,East China University of Technology,Nanchang 330013,China

E-mail:zwj19891213@126.com

Jie FEI

Department of Mathematical Sciences,Xi’an Jiaotong-Liverpool University,Suzhou 215123,China

E-mail:jie.fei@xjtlu.edu.cn

Xiaoxiang JIAO(

School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 101408,China

E-mail:xxjiao@ucas.ac.cn

Abstract In this article,we determine all homogeneous two-spheres in the complex Grassmann manifold G(2,5;C)by theory of unitary representations of the 3-dimensional special unitary group SU(2).

Key words homogeneous immersion;Gauss curvature;K?hler angle;rigidity

1 Introduction

Let G(k+1,n+1;C)be a complex Grassmann manifold,which is the set of all k+1-dimensional complex subspaces in the n+1-dimensional complex vector space Cn+1.It is isomorphic to a Hermitian symmetric space U(n+1)/(U(k+1)×U(n?k)).We equip G(k+1,n+1;C)with a canonical K?hler metric which is U(n+1)-invariant and has Einstein constant 2(n+1).Particularly,when k=0,G(1,n+1;C)is the complex projective space CPn,which is a complex space form of constant holomorphic sectional curvature 4.

It is one of fundamental problems in di ff erential geometry to study the rigidity and homogeneity of special surfaces and submanifolds in a given Riemannian manifold.The most important and interesting case is that the ambient manifold is a space form.For example,minimal surfaces with constant Gaussian curvature in real space forms were classi fi ed completely(see[2,4,12]),and minimal two-spheres with constant curvature in the complex projective space CPnalso were classi fi ed completely(see[1,3]).We wish to study two-spheres with constant curvature immersed in the complex Grassmannians which is a generalization of the complex projective space CPn.However,the geometric structure of G(k+1,n+1;C)is much more complicated when k≥1.For example,when k≥1,G(k+1,n+1;C)does not have constant holomorphic sectional curvature,and the rigidity of holomorphic curves in G(k+1,n+1;C)fails[7].For this reason,it is hard to generalize some perfect results of submanifolds in CPnto the ones of submanifolds in a general complex Grassmannians.However,when the integers k,n are small,there are some results about minimal two-spheres in G(k+1,n+1;C).Minimal two-spheres with constant curvature in G(2,4;C)were determined by Li and Yu[15],and holomorphic two-spheres with constant curvature in G(2,5;C)were also investigated by Jiao and Peng[10].Furthermore,in[9],He,Jiao and Zhou studied rigidity of holomorphic curves of constant curvature in G(2,5;C).Besides,Li,Jiao and He[14]gave a classi fi cation theorem of linearly full totally unrami fi ed conformal minimal immersions of constant curvature from S2to G(2,5;R).

In[3],Bando and Ohnita proved that all minimal two-spheres with constant curvature in CPnare homogeneous,and later,Li,Wang and Wu classi fi ed homogeneous two-spheres in CPnin[13]by the method of harmonic sequence.For G(k+1,n+1;C),k≥1,Fei and Jiao[8,9]discussed a classi fi cation problem of homogeneous two-spheres in G(k+1,n+1;C)and describe explicitly all homogeneous two-spheres in G(2,4;C).The purpose of this article is to classify homogeneous two-spheres in G(2,5;C).Arrangement is as follows.

In Section 2,we recall some basic facts of unitary representations of SU(2),and give some known conclusions.In Section 3,we describe explicitly all homogeneous two-spheres in G(2,5;C).

2 Preliminaries

An immersion ? :S2→ G(k+1,n+1;C)is said to be homogeneous,if for any two points p,q∈S2there exists an isometry?σ of S2and a holomorphic isometry σ of G(k+1,n+1;C)such that?σ(p)=q and the following diagram communicates

The 3-dimensional special unitary group SU(2)is de fi ned by

and its Lie algebra su(2)is given by

with a natural basis{ε1,ε2,ε3},which is given by

Set T={exp(tε1);t∈ R},then,we have the homeomorphism S2? SU(2)/T.

For each nonnegative integer n,let Vnbe the representation space of SU(2),which is an(n+1)-dimensional complex vector space of all complex homogeneous polynomials of degree n in two variables z0and z1.The standard irreducible representation ρnof SU(2)on Vnis de fi ned by

where g∈SU(2)and f∈Vn.

For ρn,we have a Lie group homomorphism(cf.[8])

where g∈SU(2)and

Obviously,we can obtain a Lie algebra homomorphism(cf.[8])

where X∈su(2)and

From article[8],we know that the classi fi cation of equivalent classes of homogeneous twospheres in G(k+1,n+1)is reduced to the following two problems:(i)Classifying the equivalence classes of unitary representations of SU(2);(ii)Determining all(k+1)-dimensional vector subspaces invariant by T.

For the two problems,there are some results as follows.

Let ρ :SU(2)→ U(n+1)be a unitary representation of SU(2)and ρ|T:T → U(n+1)be the restriction of ρ from SU(2)to T.

If ρ is irreducible,that is,ρ = ρnfor some nonnegative integer n,by(2.1),it is easy to have

In general,a(k+1)-dimensional vector subspace invariant by T can be spanned by k+1 complex vectors{vi|i=1,···,k+1},where each vihas form(2.4)and they satisfy hvi,vji= δijwith respect to the standard Hermitian inner product h,i on Cn+1.

3 Homogeneous Two-Spheres in G(2,5;C)

3.1 The Construction of All the Homogeneous Two-Spheres

Let ? :S2→ G(2,5;C)be a homogeneous immersion,and ρ be a unitary representation of SU(2).In this section,we will describe explicitly all such immersions ?.To do it,we should consider irreducible case and reducible case respectively.

I.If ρ is irreducible,that is,ρ = ρ4,then ρ :SU(2)→ U(5)and ρ?:su(2)→ u(5)can be given by(2.1)and(2.2)as follows

By the above arguments,we know that[Ek],0≤k≤4 are all 1-dimensional vector subspaces invariant by T.Then,spanC{Ek,El},0≤k

II.If ρ is reducible,then ρ = ρn1⊕ ···⊕ ρnr,n1+ ···+nr=5 ? r.Evidently,we have some cases as follows

(1)when r=4,n1+n2+n3+n4=1,then we have ρ = ρ1⊕ ρ0⊕ ρ0⊕ ρ0;

(2)when r=3, n1+n2+n3=2,then we have ρ = ρ2⊕ ρ0⊕ ρ0or ρ = ρ1⊕ ρ1⊕ ρ0;

(3)when r=2,n1+n2=3,then we have ρ = ρ3⊕ ρ0or ρ = ρ2⊕ ρ1.

Now,we consider these cases respectively.

(IIi)If ρ = ρ1⊕ ρ0⊕ ρ0⊕ ρ0,then ρ :SU(2)→ U(5)and ρ?:su(2)→ u(5)can be written explicitly as follows

by(2.1)and(2.2).Then the restriction representation ρ|T:T → U(5)is given by

Hence,all 1-dimensional vector subspaces invariant by T are[E0],[E1]and[v],where v=c1E2+c2E3+c3E4with|c1|2+|c2|2+|c3|2=1.Then,up to U(5)-equivalent,we have

by(2.1)and(2.2).The restriction representation ρ|T:T → U(5)is given by

Hence,all 1-dimensional vector subspaces invariant by T are[E4],[v1]and[v2],where v1=c1E0+c2E2and v2=d1E1+d2E3with|c1|2+|c2|2=|d1|2+|d2|2=1.Then,up to U(5)-equivalent,we have

(IIiv)If ρ = ρ3⊕ ρ0,similarly,ρ :SU(2)→ U(5)and ρ?:su(2)→ u(5)can be written explicitly as follows

by(2.1)and(2.2).The restriction representation ρ|T:T → U(5)is given by

So,we have[Ek],0≤k≤4 are all 1-dimensional vector subspaces invariant by T.Then,we can obtain the following ten homogeneous two-spheres.

by(2.1)and(2.2).The restriction representation ρ|T:T → U(5)is given by

It follows that[Ek],0≤k≤4 are all 1-dimensional vector subspaces invariant by T.Hence,we obtain the following nine homogeneous two-spheres.

By observation,we have

Lemma 3.1(1)The one given in(IIii2)(resp.(IIii3))with(α,β)=(,0)is U(5)-equivalent to the one given in(IIv1)(resp.(IIv5)).

(2)The one given in(IIii2)(resp.(IIii3))with sinαcosβ =is U(5)-equivalent to the one given in(IIiii4)(resp.(IIiii5)).

(3)The one given in(IIi1)with c1=c2=0,c3=1(resp.(IIi2)with c1=c2=0,c3=1)is U(5)-equivalent to the one given in(IIiii1)with c1=1,c2=0(resp.(IIiii2)with d1=1,d2=0).

(4)Up to isometric transformations of S2and G(2,5;C),the one given in(I1)(resp.(I2),(I3),(I5),(IIi1),(IIii2),(IIiii1),(IIiii4),(IIiv1),(IIiv2),(IIiv4),(IIiv7),(IIv1),(IIv3),(IIv4),(IIv6))is equivalent to the one given in(I10)(resp.(I9),(I7),(I8),(IIi2),(IIii3),(IIiii2),(IIiii5),(IIiv8),(IIiv6),(IIiv10),(IIiv9),(IIv5),(IIv9),(IIv8),(IIv7)).

Thus,we have further

Theorem 3.2Up to U(5)-equivalent,the one given in(I1),(I2),(I3),(I4),(I5),(I6),(IIi1),(IIii1),(IIiii1),(IIiii3),(IIiii4),(IIiv1),(IIiv2),(IIiv3),(IIiv4),(IIiv5),(IIiv7),(IIv1),(IIv3),(IIv4)and the one given in(IIv6)are all di ff erent homogeneous two-spheres in G(2,5;C).

3.2 Some Geometric Properties

Next,we will give some geometrical descriptions of these homogeneous two-spheres in G(2,5;C).We only compute the geometric quantities of case(I1).For other cases,we omit the details of calculations and just list the results in Table 1.

Table 1

Let ?=(?AˉB),0≤A,B≤4 be the u(5)-valued right-invariant Maurer-Cartan form of U(5).The Maurer-Cartan structure equations of U(5)are

Then,the canonical K?hler metric of G(2,5;C)and its K?hler form can be written as

where the range of the indices are α=0,1 and i=2,3,4,respectively.

We choose a unitary frame fi eld e=(e0,e1,e2,e3,e4)along ?,where eA=EA·ρ(g),A=0,1,2,3,4.It is easily see from(3.2)that the pull back of Maurer-Cartan form can be written as

with ω2ˉ2=0, ω1ˉ1+ ω3ˉ3=0, ω0ˉ0+ ω4ˉ4=0 and ω0ˉ0=2ω1ˉ1,where φ is a complex-valued(1,0)form of S2,which de fi ned up to a factor of absolute value 1,and the induced metric is ??ds2=.

If we write

and note

It follows from(3.14)that

Thus,the K?hler angle(cf.[6])of ? is cosθ=tr(AA?? BB?)=1.

The structure equations of S2with respect to the induced metric can be written as

Using the Maurer-Cartan structure equations(3.13),we can obtain

By(3.17),it follows

Making use of(3.13)again,we obtain

Taking the exterior derivative of(3.15)and using(3.13),(3.17),we can obtain

where

By(3.14),(3.16),(3.19),(3.20)and(3.21),we have

The second identities qαˉi=0 imply that ? is a minimal immersion(cf.[5]).By identity(1’.15)in[17],the square length of the second fundamental form B is

Through some similar straightforward computations,we can obtain the following lemma and theorem.

Lemma 3.3In case(IIiii3),when|μ|=1,? are totally geodesic with K=2.They are all U(5)-equivalent to

The others in the case(IIiii3)are non-minimal.

Theorem 3.4The di ff erential geometric quantities of homogeneous two-spheres in G(2,5;C)are given in Table 1,where α,β ∈,μ =c1d2? c2d16=0,and K is its Gaussian curvature,θ is the K?hler angle and kBk2is the square length of the second fundamental form.

Thus,we have further

Corollary 3.5Up to U(5)-equivalent,the one given in(I1),(I2),(I3),(I4),(I5),(I6),(IIi1),(IIii1),(IIiii1),(IIiii4),(IIiv1),(IIiv2),(IIiv3),(IIiv4),(IIiv5),(IIiv7),(IIv1),(IIv3),(IIv4),(IIv6)and the one given in(IIiii3’)are all minimal homogeneous two-spheresin G(2,5;C).

Because every closed totally geodesic submanifold of a homogeneous Riemannian manifolds is homogeneous(see[13]),we obtain

Theorem 3.6Up to U(5)-equivalent,the one given in(I6),(IIi1),(IIi2),(IIii1),(IIiii4),(IIiii5),(IIiv2),(IIiv6),(IIv6),(IIv7)and the one given in(IIiii3)′are all totally geodesic two-spheres in G(2,5;C).