Harmonic curvature for real hypersurfaces in complex hyperbolic two plane Grassmannians

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Abstract

In this paper we first introduce the full expression of the curvature tensor of a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,mS(U2Um) from the equation of Gauss and derive a new formula for the Ricci tensor of M in SU2,mS(U2Um). Finally we give a complete classification for Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,mS(U2Um) with harmonic curvature or harmonic Weyl tensor.

Introduction

For a Riemannian manifold (N,gN) the Ricci tensor can be regarded as a 1-form with values in the cotangent bundle TN. Then a Riemannian manifold N is said to have harmonic curvature or harmonic Weyl tensor, if RicN or RicNrNgN2(n1) for the scalar curvature rN is a tensor of Codazzi type, that is, it satisfies dRicN=0ord{RicNrNgN2(n1)}=0,where d denotes the exterior differential. Then it is equivalent to the following δRN=0orδWN=0for the curvature tensor RN and Weyl curvature tensor WN respectively. Here the operator δ=d denotes an adjoint operator of the exterior derivative d on the Riemannian manifold (N,gN).

For the harmonic tensor, it is seen that in the case of n4 the Riemannian curvature tensor RN which can be regarded as a 2-form with values in the bundle Λ2TN is closed and coclosed, that is, dRN=0 and δRN=0. It means, together with the second Bianchi identity dRN=0, the Laplacian RN=(dδ+δd)RN=0. So it is said to be harmonic (see Besse [6]).

In the geometry of real hypersurfaces in complex space forms or in quaternionic space forms it can be easily checked that there does not exist any real hypersurface with parallel shape operator A by virtue of the equation of Codazzi.

From this point of a view many differential geometers have considered a notion weaker than the parallel Ricci tensor, that is, Ric=0. In particular, Kwon and Nakagawa [12] have proved that there are no real hypersurfaces M in a complex projective space Pm with harmonic curvature (XRic)Y=(YRic)X for any X,Y in M. Moreover, Ki, Nakagawa and Suh [9] have also proved that there are no real hypersurface with harmonic Weyl tensor in non-flat complex space forms Mn(c), c0, n3.

Now let us consider the transitive group SU2,m which denotes the set of (m+2)×(m+2)-indefinite special unitary matrices and Um the set of m×m-unitary matrices. Accordingly, the Riemannian symmetric space SU2,mS(U2Um), m2, which consists of complex two-dimensional subspaces in indefinite complex Euclidean space 2m+2, has remarkable geometric properties that it is both a Hermitian symmetric space and a quaternionic Kähler symmetric space. Among all Riemannian symmetric spaces of noncompact type, the symmetric spaces SU2,mS(U2Um), m2, are the only ones which are Hermitian symmetric and quaternionic Kähler symmetric. It is said to be complex hyperbolic two-plane Grassmannian whose two structures give us several kinds of interesting geometric problems on SU2,mS(U2Um).

Moreover, we denote by J the Kähler structure and by J the quaternionic Kähler structure on SU2,mS(U2Um), where J=Span{J1,J2,J3} not containing J (see Berndt and Suh [2], [4], [5], Pérez and Suh [14], [15]). Let M be a connected hypersurface in SU2,mS(U2Um) and denote by TM the tangent bundle of M. The maximal complex subbundle C of TM is defined by C={XTMJXTM}, and the maximal quaternionic subbundle Q of TM is defined by Q={XTMJXTM}.

Now we want to give some mentions about real hypersurfaces in SU2,mS(U2Um) for which both C and Q are invariant under the shape operator of M. Now we introduce a few hypersurfaces in SU2,mS(U2Um) with these two properties. We denote by oSU2,mS(U2Um) the unique fixed point of the action of the isotropy group S(U2Um) on SU2,mS(U2Um).

First, let us consider the conic (or geodesic) compactification of SU2,mS(U2Um). The points in the boundary of this compactification correspond to equivalence classes of asymptotic geodesics in SU2,mS(U2Um). Every geodesic in SU2,mS(U2Um) lies in a maximal flat, that is, a two-dimensional Euclidean space embedded in SU2,mS(U2Um) as a totally geodesic submanifold. A geodesic in SU2,mS(U2Um) is called singular if it lies in more than one maximal flat in SU2,mS(U2Um). A singular point at infinity is the equivalence class of a singular geodesic in SU2,mS(U2Um). Up to isometry, there are exactly two singular points at infinity for SU2,mS(U2Um). The singular points at infinity correspond to the geodesics in SU2,mS(U2Um) which are determined by nonzero tangent vectors X with the property JXJX and JXJX respectively.

As a second, we consider the standard embedding of SU2,m1 in SU2,m. Then the orbit SU2,m1o of SU2,m1 through o is the Riemannian symmetric space SU2,m1S(U2Um1) embedded in SU2,mS(U2Um) as a totally geodesic submanifold. Every tube around SU2,m1S(U2Um1) in SU2,mS(U2Um) has the property that both C and Q are invariant under the shape operator.

Finally, let m be even, say m=2n, and consider the standard embedding of Sp1,n in SU2,2n. Then the orbit Sp1,no of Sp1,n through o is the quaternionic hyperbolic space HHn embedded in SU2,2nS(U2U2n) as a totally geodesic submanifold. Any tube around HHn in SU2,mS(U2Um) has the property that both C and Q are invariant under its shape operator (see Berndt and Suh [5], Suh [16], [19]).

As a converse problem of the results mentioned above, we want to assert that there are no other such real hypersurfaces except only one kind of exceptional case. Related to such a result, we introduce a well known theorem due to Berndt and Suh [4], [5] as follows:

Theorem A

Let M be a connected hypersurface in SU2,mS(U2Um), m2. Then the maximal complex subbundle C of TM and the maximal quaternionic subbundle Q of TM are both invariant under the shape operator of M if and only if M is congruent to an open part of one of the following hypersurfaces:

(A) a tube around a totally geodesic SU2,m1S(U2Um1) in SU2,mS(U2Um);

(B) a tube around a totally geodesic HHn in G2(2n+2), m=2n;

(C) a horosphere in SU2,mS(U2Um) whose center at infinity is singular; or the following exceptional case holds:

(D) The normal bundle νM of M consists of singular tangent vectors of type JXJX.

Moreover, M has at least four distinct principal curvatures, three of which are given by α=2,γ=0,λ=12with corresponding principal curvature spaces Tα=TM(CQ),Tγ=J(TMQ),TλCQJQ.If μ is another (possibly nonconstant) principal curvature function, then we have TμCQJQ, JTμTλ and JTμTλ.

On the other hand, Suh [18] has considered a real hypersurface with harmonic curvature in the complex two-plane Grassmannian G2(m+2) under the conditions of constant scalar and mean curvatures and gave a classification as follows:

Theorem B

Let M be a Hopf real hypersurface of harmonic curvature in G2(m+2) with constant scalar and mean curvatures. If the shape operator commutes with the structure tensor on the distribution Q, then M is locally congruent to a tube over a totally geodesic G2(m+1) in G2(m+2) with radius r such that cot22r=43(m1).

Now in this paper we consider a real hypersurface M in complex hyperbolic two-plane Grassmanians SU2,mS(U2Um) with harmonic curvature. The notion of harmonic curvature in complex hyperbolic two-plane Grassmannian SU2,mS(U2Um) is quite different and discriminating from the result of Theorem B in G2(m+2).

Accordingly, the main result of this paper is to give some classifications of all Hopf real hypersurfaces in SU2,mS(U2Um) with harmonic or Weyl harmonic tensor. First for a real hypersurface in SU2,mS(U2Um) with harmonic curvature tensor, that is, (XRic)Y=(YRic)X on M, we assert a non-existence theorem as follows:

Main Theorem

There does not exist a Hopf real hypersurface of harmonic curvature in SU2,mS(U2Um) with constant scalar and mean curvatures, provided that the shape operator commutes with the structure tensor on the distribution Q.

Remark 0.1

Compared to Theorem B in G2(m+2) with our Main Theorem in SU2,mS(U2Um), the results are remarkable and quite different from each other. Though the conditions assumed in our Main Theorem and Theorem B are being the same, but they are important and valuable to get the corresponding results.

Next let us consider a real hypersurface M in SU2,mS(U2Um) with harmonic Weyl tensor. Here M is said to be with harmonic Weyl tensor if the Laplacian of Weyl curvature tensor W of M defined by W=Ricr4(2m1)g is identically vanishing, that is, W=0, where Ric and r denote the Ricci tensor and the scalar curvature of M in SU2,mS(U2Um) respectively. Then by using the second Bianchi identity it can be easily verified that the vanishing Laplacian W=0 is equivalent to (XRic)Y(YRic)X={dr(X)Ydr(Y)X}4(2m1).Then by virtue of our Main Theorem we get the following

Corollary

There does not exist a Hopf real hypersurface of Weyl harmonic curvature in SU2,mS(U2Um) with constant scalar and mean curvatures, provided that the shape operator commutes with the structure tensor on the distribution Q.

In Section 1 we recall Riemannian geometry of complex hyperbolic two-plane Grassmannians SU2,mS(U2Um) and in Section 2 we will show some fundamental properties of real hypersurfaces in SU2,mS(U2Um). The formula for the Ricci tensor Ric and its covariant derivative Ric will be shown explicitly in this section.

In order to prove our Main Theorem in the introduction, in Section 4 we will give some formulas from the contraction of harmonic curvature tensors related to constant scalar and mean curvatures and prove that the Reeb vector field ξ=JN either belongs to the distribution Q or Q. In Sections 5 Real hypersurfaces in, 6 Real hypersurfaces in we will give a complete proof of the Main Theorem according to the Reeb vector field ξ=JN satisfying either JNJN or JNJN respectively.

Section snippets

Complex hyperbolic two-plane Grassmannian SU2,mS(U2Um)

In this section we summarize basic material about complex hyperbolic two-plane Grassmannian manifold SU2,mS(U2Um), for details we refer to [1], [4], [7], [10], [11], [13], [17], [19], and [20].

The Riemannian symmetric space SU2,mS(U2Um), which consists of all complex two-dimensional linear subspaces in indefinite complex Euclidean space 2m+2, becomes a connected, simply connected, irreducible Riemannian symmetric space of noncompact type and with rank two. Let G=SU2,m and K=S(U2Um), and

Real hypersurfaces in complex hyperbolic two-plane Grassmannian

Let M be a real hypersurface in SU2,mS(U2Um), that is, a hypersurface in SU2,mS(U2Um) with real codimension one. The induced Riemannian metric on M will also be denoted by g, and denotes the Levi Civita covariant derivative of (M,g). We denote by C and Q the maximal complex and quaternionic subbundle of the tangent bundle TM of M, respectively. Now let us put JX=ϕX+η(X)N,JνX=ϕνX+ην(X)Nfor any tangent vector field X of a real hypersurface M in SU2,mS(U2Um), where ϕX denotes the

Proof of the main theorem

Let M be a real hypersurfaces in SU2,mS(U2Um) with harmonic curvature, that is, (XRic)Y=(YRic)X for any vector fields X,YTxM,xM. This condition comes from the 2nd Bianchi identity and δR=llRijkl=0, where δ=d denotes an adjoint coderivative of the exterior derivative d on M (see Besse [6]). Then the harmonic curvature of M in SU2,mS(U2Um) satisfies 0=(XRic)Y(YRic)X=32g((ϕA+Aϕ)X,Y)ξ+32{η(Y)ϕAXη(X)ϕAY}+32ν=13{qν+2(X)ην+1(Y)qν+1(X)ην+2(Y)qν+2(Y)ην+1(X)+qν+1(Y)ην+1(X)}ξν+32ν=13g

Contraction of codazzi type

In this section, we contract (3.1) with X=Ei. Then it follows that 0=i=14m1g((EiRic)Y(YRic)Ei,Ei)=32ν=13{qν+2(ξν)ην+1(Y)qν+1(ξν)ην+2(Y)}+32ν=13ην(Y){qν+2(ξν+1)qν+1(ξν+2)}+32ν=13g(ϕνAξν,Y)12[ν=13{ϕνϕY(ην(ξ))+ην(ξ){qν+1(ϕν+2ϕY)+qν+2(ϕν+1ϕY)+hην(ϕY)g(Aξν,ϕY)}}ν=13{(Trϕνϕ)Y(ην(ξ))+ην(ξ){qν+1(Y)Trϕν+2ϕ+qν+2(Y)Trϕν+1ϕ+ην(ϕAY)g(AY,ϕξν)}}ν=13ην(ξ)g(ϕνAY,ξ)ν=13{g(ϕAϕνξ,ϕνY)g(ϕAY,ϕν2ξ)}+ν=13{qν+1(ϕνξ)η(ϕν+2Y)qν+2(ϕνξ)η(ϕν+1Y)qν+1(Y)η(ϕν+2ϕνξ)+qν+2(Y)η(ϕν+1ϕνξ)}ν=13{ην(Y)η(Aϕνξ)

Real hypersurfaces in SU2,mS(U2Um) with geodesic Reeb flow satisfying ξQ

Let us put ξ=ξ1. Then grad α=(ξα)ξ+2ν=13ην(ξ)ϕξν=2η1(ξ)ϕξ1=0. So this means that the Reeb function α is constant on M.

In this section we could prove an important part of our main theorem in the introduction for ξQ. Then for the case ξQ in (3.5) it gives the following 0=32g((Aϕ+ϕA)X,Y)+32g((ϕ1A+Aϕ1)X,Y)+3η2(Y)η3(AX)3η3(Y)η2(AX)+2η2(X)η3(AY)2η3(X)η2(AY)+(hα){g(ϕX,Y)2η2(X)η3(Y)+2η2(Y)η3(X)+g(ϕ1X,Y)}2αg(AϕAX,Y)+g((Aϕ+ϕA)AX,AY).Now let us take a symmetric part from (5.1). Then we have 5η2(Y

Real hypersurfaces in SU2,mS(U2Um) with geodesic Reeb flow satisfying ξQ

In this section we consider a Hopf real hypersurface M in SU2,mS(U2Um) with harmonic curvature for ξQ.

The Reeb vector ξ is said to be a Hopf vector if it is a principal vector for the shape operator A of M in SU2,mS(U2Um), that is, the Reeb vector ξ is invariant under the shape operator A.

On the other hand, first Suh [19] has shown that the Reeb vector ξ of M belongs to the distribution Q if M is locally congruent to a hypersurface of type (B) or a horosphere of type (C2) with JNJN in

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This work was supported by grant Proj. No. NRF 2018-R1D1A1B-05040381 from National Research Foundation of Korea .

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