KYUNGPOOK Math. J. 2019; 59(3): 525-535  
Hopf Hypersurfaces in Complex Two–plane Grassmannians with Generalized Tanaka–Webster Reeb–parallel Structure Jacobi Operator
Byung Hak Kim, Hyunjin Lee, Eunmi Pak∗
Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin-si, Gyeonggi-do 17104, Korea
e-mail : bhkim@khu.ac.kr
The Research Institute of Real and Complex Manifolds, Kyungpook National University, Daegu 41566, Korea
e-mail : lhjibis@hanmail.net
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
e-mail : empak@hanmail.net
* Corresponding Author.
Received: December 28, 2017; Accepted: February 27, 2018; Published online: September 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

In relation to the generalized Tanaka-Webster connection, we consider a new notion of parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and prove the non-existence of real hypersurfaces in G2(ℂm+2) with generalized Tanaka-Webster parallel structure Jacobi operator.

Keywords: real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, generalized Tanaka-Webster connection, structure Jacobi operator.
Introduction
roduction

In complex projective spaces or in quaternionic projective spaces, many differential geometers studied real hypersurfaces with parallel curvature tensor [8, 9, 10, 14, 15, 16]. Taking a new perspective, we look to classify real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator; that is, having ∇Rξ = 0 [6, 7, 12, 14].

As an ambient space, a complex two-plane Grassmannian G2(ℂm+2) consists of all complex two-dimensional linear subspaces in ℂm+2. This Riemannian symmetric space is the unique compact irreducible Riemannian manifold being equipped with both a Kähler structure J and a quaternionic Kähler structure not containing J. There are two natural geometric conditions to consider for hypersurfaces M in G2(ℂm+2). The first is that a 1-dimensional distribution [ξ] = Span{ξ} and a 3-dimensional distribution are both invariant under the shape operator A of M [2], where the Reeb vector field ξ is defined by ξ = −JN, and N denotes a local unit normal vector field of M in G2(ℂm+2). The second is that the almost contact 3-structure vector fields {ξ1, ξ2, ξ3} are defined by ξν = −JνN (ν = 1, 2, 3).

Using a result from Alekseevskii [1], Berndt and Suh [2] proved the following:

Theorem A

Let M be a connected orientable real hypersurface in G2(ℂm+2), m ≥ 3. Then both [ξ] andare invariant under the shape operator of M if and only if

  • M is an open part of a tube around a totally geodesic G2(ℂm+1) in G2(ℂm+2), or

  • m is even, say m = 2n, and M is an open part of a tube around a totally geodesicin G2(ℂm+2).

  • The Reeb vector field ξ is said to be Hopf if it is invariant under the shape operator A. The one dimensional foliation of M by the integral manifolds of the Reeb vector field ξ is said to be a Hopf foliation of M. We say that M is a Hopf hypersurface in G2(ℂm+2) if and only if the Hopf foliation of M is totally geodesic. By the formulas in Section 2 [11] it can be easily checked that M is Hopf if and only if the Reeb vector field ξ is Hopf.

    Now, instead of the Levi-Civita connection, we consider the generalized Tanaka-Webster connection ∇̂ for contact Riemannian manifolds introduced by Tanno [18]. The original Tanaka-Webster connection [17, 19] is given as a unique affine connection on a non-degenerate, pseudo-Hermitian CR manifolds which associated with the almost contact structure. Cho [4, 5] defined the generalized Tanaka-Webster connection for a real hypersurface of a Kähler manifold as ^X(k)Y=XY+g(φAX,Y)ξη(Y)ϕAXkη(X)ϕY,where k ∈ ℝ {0}.

    We put the Reeb vector field ξ into the curvature tensor R of a real hypersurface M in G2(Cm+2). Then for any tangent vector field X on M, the structure Jacobi operator Rξ is defined by Rξ(X)=R(X,ξ)ξ.Using this structure Jacobi operator Rξ, in [6] and [7] the authors proved non-existence theorems. On the other hand, using the generalized Tanaka-Webster connection ∇̂(k), we considered the notion of -parallel structure Jacobi operator in the generalized Tanaka-Webster connection, that is, (^X(k)Rξ)Y=0 for any and any tangent vector field Y in M. We gave a classification theorem as follows (see [13]):

    Theorem B

    Let M be a connected orientable Hopf hypersurface in a complex two-plane Grassmannian G2(ℂm+2), m ≥ 3. If the structure Jacobi operator Rξis-parallel in the generalized Tanaka-Webster connection, M is an open part of a tube around a totally geodesicin G2(ℂm+2), where m = 2n.

    In the present paper, motivated by Theorem B, we consider another new notion for generalized Tanaka-Webster parallelism of the structure Jacobi operator on a real hypersurface M in G2(ℂm+2), when the structure Jacobi operator Rξ of M satisfies (^ξ(k)Rξ)Y=0 for any tangent vector field Y in M. In this case, the stucture Jacobi operator is said to be a Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. We can give a non-existence theorem as follows:

    Main Theorem

    There does not exist any Hopf hypersurface in a complex two-plane Grassmannian G2(ℂm+2), m ≥ 3, with Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection.

    On the other hand, we consider another new notion for generalized Tanaka-Webster parallelism of the structure Jacobi operator on a real hypersurface M in G2(ℂm+2). If the structure Jacobi operator Rξ of M satisfies (^X(k)Rξ)Y=0 for any tangent vector fields X and Y in M, then the the structure Jacobi operator is said to be parallel structure Jacobi operator in the generalized Tanaka-Webster connection. Naturally, we see that this notion of parallel structure Jacobi operator in the generalized Tanaka-Webster connection is stronger than Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. Related to this notion, we have the following corollary.

    Corollary

    There does not exist any Hopf hypersurface in a complex two-plane Grassmannian G2(ℂm+2), m ≥ 3, with parallel structure Jacobi operator in the generalized Tanaka-Webster connection.

    We refer to [1, 2, 3] and [11, section 1] for Riemannian geometric structures of G2(ℂm+2), m ≥ 3 and [11, section 2] for basic formulas of tangent space at pM of real hypersufaces M in G2(ℂm+2).

    Key Lemma
    Key Lemma

    Let us denote by R(X, Y)Z the curvature tensor of M in G2(ℂm+2). Then the structure Jacobi operator Rξ of M in G2(ℂm+2) can be defined by RξX = R(X, ξ)ξ for any vector field , xM. In [6] and [7], by using the structure Jacobi operator Rξ, the authors obtained (XRξ)Y=g(ϕAX,Y)ξη(Y)ϕAXν=13[g(ϕνAX,Y)ξν2η(Y)ην(ϕAX)ξν+ην(Y)ϕνAX+3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ+ην(ϕY)(ϕνϕAXαη(X)ξν)}+4ην(ξ){ην(ϕY)AXg(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY]+η((XA)ξ)AY+α(XA)Yη((XA)Y)Aξg(AY,ϕAX)Aξη(AY)(XA)ξη(AY)AϕAX.On the other hand, by using the generalized Tanaka-Webster connection, we have (^X(k)Rξ)Y=^X(k)(RξY)Rξ(^X(k)Y)=X(RξY)+g(ϕAX,RξY)ξη(RξY)ϕAXkη(X)ϕRξYRξ(XY+g(ϕAX,Y)ξη(Y)ϕAXkη(X)ϕY).From this, together with the fact that M is Hopf, it becomes (^X(k)Rξ)Y=ν=13[g(ϕνAX,Y)ξνη(Y)ην(ϕAX)ξν+ην(Y)ϕνAX+3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ+ην(ϕY)(ϕνϕAXαη(X)ξν)}+4ην(ξ){ην(ϕY)AXg(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY+ην(Y)ην(ϕAX)ξην(ξ)η(Y)ην(ϕAX)ξ+3η(ϕνY)g(ϕAX,ϕνξ)ξ+ην(ξ)g(ϕAX,ϕνϕY)ξην(Y)ην(ξ)ϕAX+ην2(ξ)η(Y)ϕAXην(ξ)η(ϕνϕY)ϕAXkη(X)ην(Y)ϕξν4kη(X)η(ϕνY)ην(ξ)ξ4kη(X)η(ϕνY)ξν+3η(Y)η(ϕνϕAX)ϕνξη(Y)ην(ξ)ϕνAX+αη(X)η(Y)ην(ξ)ϕνξ+3kη(X)η(ϕνϕY)ϕνξ+kη(x)η(Y)ην(ξ)ϕνξ]+η((XA)ξ)AY+α(XA)Yαη((XA)Y)ξαη(Y)(XA)ξαkη(X)ϕAY+αkη(X)AϕYfor any tangent vector fields X and Y on M. Let us assume that the structure Jacobi operator Rξ on a Hopf hypersurface M in a complex two-plane Grassmann manifold G2(ℂm+2) is Reeb-parallel in the generalized Tanaka-Webster connection, that is, (^ξ(k)Rξ)Y=0for any tangent vector field Y on M.

    Here, it is a main goal to show that the Reeb vector field ξ belongs to either the distribution or orthogonal complement of (i.e., ) such that in G2(ℂm+2) when the structure Jacobi operator is Reeb-parallel in the generalized Tanaka-Webster connection.

    From now on, unless otherwise stated in the present section, we may put the Reeb vector field ξ as follows : ξ=η(X0)X0+η(ξ1)ξ1for some unit vector fields and .

    Putting X = ξ in (2.3) and using the condition (*), we have 0=(^ξ(k)Rξ)Y=ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ+3{αg(ϕνξ,ϕY)ϕνξ+αη(Y)ην(ξ)ϕνξαην(ϕY)ξν}+4ην(ξ){αην(ϕY)ξαg(ξ,Y)ϕνξ}kην(Y)ϕξν4kη(ϕνY)ην(ξ)ξ4kη(ϕνY)ξν+3kη(ϕνϕY)ϕνξ+kη(Y)ην(ξ)ϕνξ]+η((ξA)ξ)AY+α(ξA)Yαη((ξA)Y)ξαη(Y)(ξA)ξαkϕAY+αkAϕYfor any tangent vector field Y on M.

    Now, using these facts, we prove the following Lemma.

    Lemma 2.1

    Let M be a Hopf hypersurface in a complex two-plane Grassmannian G2(ℂm+2), m ≥ 3, with Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. Then the Reeb vector field ξ belongs to either the distributionor the distribution.

    Proof

    By taking the inner product with ξ in (2.4), it becomes 0=ν=13{αg(ϕνξ,Y)ην(ξ)3αην(ϕY)ην(ξ)+4αην(ξ)ην(ϕY)4kην(ϕY)ην(ξ)4kη(ϕνY)ην(ξ)}+αη((ξA)ξ)η(Y)+αη((ξA)Y)αη((ξA)Y)αη(Y)η((ξA)ξ)=8kη(ϕ1Y)η1(ξ)=8kg(Y,ϕ1ξ)η1(ξ)=8kη(X0)η(ξ1)g(Y,ϕ1X0)for any tangent vector field Y on M, since ϕξ1 = η(X0)ϕ1X0.

    Thus substituting Y with ϕ1X0, it follows kη(X0)η(ξ1)=0.Since k is a nonzero real number, we get η(X0)η1(ξ) = 0, that is, η(X0) = 0 or η1(ξ) = 0. It means that ξ belongs to either the distribution or the distribution . Accordingly, it completes the proof of our Lemma.

    Proof of The Main Theorem
    n Theorem

    Let us consider a Hopf hypersurface M in G2(ℂm+2) with Reeb-parallel structure Jacobi operator Rξ in the generalized Tanaka-Webster connection, that is, (^ξ(k)Rξ)Y=0 for any vector field Y on M. Then by Lemma 2.1 we shall divide our consideration in two cases of which the Reeb vector field ξ belongs to either the distribution or the distribution .

    First of all, we consider the case . Without loss of generality, we may put ξ = ξ1.

    Lemma 3.1

    If the Reeb vector field ξ belongs to the distribution , then there does not exist any Hopf hypersurface M in a complex two-plane Grassmannian G2(ℂm+2), m ≥ 3, with Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection.

    Proof

    Since our assumption ξ belongs to the distribution , using (2.4), we have 0={αg(ϕ2ξ,Y)ξ2+αg(ϕ3ξ,Y)ξ3+αη2(Y)ϕ2ξ+αη3(Y)ϕ3ξ+3αg(ϕ2ξ,ϕY)ϕ2ξ+3αg(ϕ3ξ,ϕY)ϕ3ξ3αη2(ϕY)ξ23αη3(ϕY)ξ3kη2(Y)ϕξ2kη3(Y)ϕξ34kη(ϕ2Y)ξ24kη(ϕ3Y)ξ3+3kη(ϕ2ϕY)ϕ2ξ+3kη(ϕ3ϕY)ϕ3ξ}+η((ξA)ξ)AY+α(ξA)Yαη((ξA)Y)ξαη(Y)(ξA)ξαkϕAY+αkAϕY=8kη2(Y)ξ3+8kη3(Y)ξ2+η((ξA)ξ)AY+α(ξA)Yαη((ξA)Y)ξαη(Y)(ξA)ξαkϕAY+αkAϕYfor any tangent vector fields X and Y on M. Taking the inner product with X, we have 0=g((^ξ(k)Rξ)Y,X)=8kη2(Y)η3(X)+8kη3(Y)η2(X)+η((ξA)ξ)g(AY,X)+αg((ξA)Y,X)αη(X)η((ξA)Y)αη(Y)g((ξA)ξ,X)αkg(ϕAY,X)+αkg(AϕY,X)for any tangent vector fields X and Y on M. Interchanging X with Y in above equation, we get 0=g((^ξ(k)Rξ)X,Y)=8kη2(X)η3(Y)+8kη3(X)η2(Y)+η((ξA)ξ)g(AX,Y)+αg((ξA)X,Y)αη(Y)η((ξA)X)αη(X)g((ξA)ξ,Y)αkg(ϕAX,Y)+αkg(AϕX,Y)for any tangent vector fields X and Y on M. Thus subtracting (3.6) from (3.5), we obtain 0=g((^ξ(k)Rξ)Y,X)g((^ξ(k)Rξ)X,Y)=16kη3(Y)η2(X)16kη2(Y)η3(X)for any tangent vector fields X and Y on M. Since k is a nonzero real number, the equation (3.7) reduces to η3(Y)η2(X)η2(Y)η3(X)=0for any tangent vector fields X and Y on M. Replacing X with ξ2 and Y with ξ3, we have η3(ξ3)=0.Let {e1, e2, · · ·, e4m−4, e4m−3, e4m−2, e4m−1} be an orthonormal basis for a tangent vector space TxM at any point xM. Without loss of generality, we may put e4m−3 = ξ1, e4m−2 = ξ2 and e4m−1 = ξ3. Since the dimension of M is equal to 4m − 1, above equation (3.9) gives a contradiction. So, we can assert our Lemma 3.1.

    Next we consider the case . Using Theorem A, Lee and Suh [11] gave a characterization of real hypersurfaces of type (B) in G2(ℂm+2) in terms of the Reeb vector field ξ as follows:

    Lemma 3.2

    Let M be a Hopf hypersurface in G2(ℂm+2) with Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. If the Reeb vector field ξ belongs to the distribution, then M is locally congruent to an open part of a tube around a totally geodesicin G2(ℂm+2), m = 2n.

    From the above two Lemmas 3.1 and 3.2 and the classification theorem given by Theorem A in this paper, we see that M is locally congruent to a model space of type (B) in Theorem A under the assumption of our Main Theorem given in the introduction.

    Hence it remains to check that whether the stucture Jacobi operator Rξ of real hypersurfaces of type (B) satisfies the condition (*) for any tangent vector field Y on M or not. In order to do this, we introduce a proposition related to eigenspaces of the model space of type (B) with respect to the shape operator. As the following proposition [2] is well known, a real hypersurface M of type (B) has five distinct constant principal curvatures as follows:

    Proposition 3.3

    Let M be a connected real hypersurface in G2(ℂm+2). Suppose that , = αξ, and ξ is tangent to. Then the quaternionic dimension m of G2(ℂm+2) is even, say m = 2n, and M has five distinct constant principal curvaturesα=2tan(2r),β=2cot(2r),γ=0,λ=cot(r),μ=tan(r)with some r ∈ (0, π/4). The corresponding multiplicities arem(α)=1,m(β)=3=m(γ),m(λ)=4n4=m(μ)and the corresponding eigenspaces areTα=ξ=Span{ξ},Tβ=Jξ=Span{ξν|ν=1,2,3},Tγ=ξ=Span{ϕνξ|ν=1,2,3},Tλ,Tμ,whereTλTμ=(ξ),Tλ=Tλ,Tμ=Tμ,JTλ=Tμ.The distributionis the orthogonal complement ofwhereξ=ξJξξJξ.

    To check this problem, we suppose that M has a Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. Putting in (2.4), it becomes ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ+3{αg(ϕvξ,ϕY)ϕνξαην(ϕY)ξν}kην(Y)ϕξν4kη(ϕνY)ξν+3kη(ϕνϕY)ϕνξ]+η((ξA)ξ)AY+α(ξA)Yαη((ξA)Y)ξαη(Y)(ξA)ξαkϕAY+αkAϕY=0for any tangent vector field Y on M. Replacing Y into ξ2Tβ, we get 0=ν=13[αην(ξ2)ϕνξ+3αg(ϕνξ,ϕξ2)ϕνξkην(ξ2)ϕξν3kην(ξ2)ϕνξ]+α(ξA)ξ2αη((ξA)ξ2)ξαkϕAξ2=4αϕξ2+4kϕξ2+α2βϕξ2αβkϕξ2because of (∇ξA)ξ = 0, (∇ξA)ξ2 = αβϕξ2, γ = 0 and equations [13, (1.4) and (1.6)]. Taking the inner product with ϕ2ξ, we have (αk)(4+αβ)=0.Since αβ = −4 by virtue of Proposition, it follows that α=k.

    On the other hand, putting YTλ in (3.10), we get α(ξA)Yαη((ξA)Y)ξαkϕAY+αkAϕY=0Using the equation of Codazzi [13, (1.10)], we know (ξA)Y=(ξA)ξ+ϕY=αϕAYAϕAY+ϕY.Thus the equation (3.12) can be written as α2λϕYαλμϕY+αϕYαλkϕY+αμkϕY=0,because of ϕYTµ. Therefore, inserting (3.11) in (3.13) we have αλμϕY+αϕY+α2μϕY=0.Taking the inner product with ϕY , we obtain αλμ+α+α2μ=0.Since α = −2 tan(2r), λ = cot(r), µ = − tan(r) with some r ∈ (0, π/4), from Proposition, we get tan2(r) = −1. This gives a contradiction. So this case can not occur.

    Hence summing up these assertions, we give a complete proof of our main theorem in the introduction.

    On the other hand, we consider a new notion which is different from Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection. The parallel stucture Jacobi operator in the generalized Tanaka-Webster connection can be defined in such a way that (^X(k)Rξ)Y=0for any tangent vector fields X and Y on M. From this notion, together with Lemmas 2.1, 3.1, 3.2 and the classification theorem given by Theorem A in the introduction, we see that M is locally congruent to a model space of type (B) in Theorem A. Hence we can check that whether the stucture Jacobi operator Rξ of real hypersurfaces of type (B) satisfies the condition (*) for any tangent vector fields X and Y in M or not.

    To check this problem, we suppose that M has a parallel structure Jacobi operator in the generalized Tanaka-Webster connection. Putting X = ξ2Tβ and in (2.3), it becomes 0=(^ξ2(k)Rξ)ξ=ν=13[βg(ϕνξ2,ξ)ξνβην(ϕξ2)ξν+3βην(ξ2)ϕνξ+3βη(ϕνϕξ2)ϕνξ]=6βϕ2ξ.By taking the inner product with ϕ2ξ, we have β = 0. It gives a contradiction. Accordingly, we give a complete proof of our Corollary in the introduction.

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