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 ∇R_ξ = 0 [6, 7, 12, 14].

As an ambient space, a complex two-plane Grassmannian G₂(ℂ^m+2) consists of all complex two-dimensional linear subspaces in ℂ^m+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(ℂ^m+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(ℂ^m+2). The second is that the almost contact 3-structure vector fields {ξ₁, ξ₂, ξ₃} are defined by ξ_ν = −J_νN (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(ℂ^m+2), m ≥ 3. Then both [ξ] andare invariant under the shape operator of M if and only if

Let us denote by R(X, Y)Z the curvature tensor of M in G₂(ℂ^m+2). Then the structure Jacobi operator R_ξ of M in G₂(ℂ^m+2) can be defined by R_ξX = R(X, ξ)ξ for any vector field , x ∈ M. In [6] and [7], by using the structure Jacobi operator R_ξ, the authors obtained (2.1)(∇XRξ)Y    =−g(ϕAX,Y)ξ−η(Y)ϕAX         −∑ν=13[g(ϕνAX,Y)ξν−2η(Y)ην(ϕAX)ξν+ην(Y)ϕνAX                    +3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ                           +ην(ϕY)(ϕνϕAX−αη(X)ξν)}                    +4ην(ξ){ην(ϕY)AX−g(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY]     +η((∇XA)ξ)AY+α(∇XA)Y−η((∇XA)Y)Aξ−g(AY,ϕAX)Aξ−η(AY)(∇XA)ξ−η(AY)AϕAX.On the other hand, by using the generalized Tanaka-Webster connection, we have (2.2)(∇^X(k)Rξ)Y=∇^X(k)(RξY)−Rξ(∇^X(k)Y)=∇X(RξY)+g(ϕAX,RξY)ξ−η(RξY)ϕAX−kη(X)ϕRξY    −Rξ(∇XY+g(ϕAX,Y)ξ−η(Y)ϕAX−kη(X)ϕY).From this, together with the fact that M is Hopf, it becomes (2.3)(∇^X(k)Rξ)Y=−∑ν=13[g(ϕνAX,Y)ξν−η(Y)ην(ϕAX)ξν+ην(Y)ϕνAX              +3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ              +ην(ϕY)(ϕνϕAX−αη(X)ξν)}              +4ην(ξ){ην(ϕY)AX−g(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY              +ην(Y)ην(ϕAX)ξ−ην(ξ)η(Y)ην(ϕAX)ξ             +3η(ϕνY)g(ϕAX,ϕνξ)ξ+ην(ξ)g(ϕAX,ϕνϕY)ξ            −ην(Y)ην(ξ)ϕAX+ην2(ξ)η(Y)ϕAX−ην(ξ)η(ϕνϕY)ϕAX           −kη(X)ην(Y)ϕξν−4kη(X)η(ϕνY)ην(ξ)ξ−4kη(X)η(ϕνY)ξν           +3η(Y)η(ϕνϕAX)ϕνξ−η(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)Y−αη((∇XA)Y)ξ−αη(Y)(∇XA)ξ−αkη(X)ϕAY+αkη(X)AϕYfor any tangent vector fields X and Y on M. Let us assume that the structure Jacobi operator R_ξ on a Hopf hypersurface M in a complex two-plane Grassmann manifold G₂(ℂ^m+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(ℂ^m+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 (2.4)0=(∇^ξ(k)Rξ)Y   =−∑ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ                 +3{αg(ϕνξ,ϕY)ϕνξ+αη(Y)ην(ξ)ϕνξ−αην(ϕY)ξν}                 +4ην(ξ){αην(ϕY)ξ−αg(ξ,Y)ϕνξ}                 −kην(Y)ϕξν−4kη(ϕνY)ην(ξ)ξ−4kη(ϕνY)ξν             +3kη(ϕνϕY)ϕνξ+kη(Y)ην(ξ)ϕνξ]+η((∇ξA)ξ)AY+α(∇ξA)Y−αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+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(ℂ^m+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.

Let us consider a Hopf hypersurface M in G₂(ℂ^m+2) with Reeb-parallel structure Jacobi operator R_ξ 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(ℂ^m+2), m ≥ 3, with Reeb-parallel structure Jacobi operator in the generalized Tanaka-Webster connection.

Lemma 3.2

Let M be a Hopf hypersurface in G₂(ℂ^m+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(ℂ^m+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 R_ξ 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(ℂ^m+2). Suppose that , Aξ = αξ, and ξ is tangent to. Then the quaternionic dimension m of G₂(ℂ^m+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)=4n−4=m(μ)and the corresponding eigenspaces areTα=ℝξ=Span{ξ},Tβ=ℑJξ=Span{ξν|ν=1,2,3},Tγ=ℑξ=Span{ϕνξ|ν=1,2,3},Tλ,Tμ,whereTλ⊕Tμ=(ℍℂξ)⊥,      ℑTλ=Tλ,        ℑTμ=Tμ,        JTλ=Tμ.The distributionis the orthogonal complement ofwhereℍℂξ=ℝξ⊕ℝJξ⊕ℑξ⊗ℑJξ.

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 (3.10)−∑ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ+3{αg(ϕvξ,ϕY)ϕνξ−αην(ϕY)ξν}                       −kην(Y)ϕξν−4kη(ϕνY)ξν+3kη(ϕνϕY)ϕνξ]+η((∇ξA)ξ)AY+α(∇ξA)Y−αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+αkAϕY=0for any tangent vector field Y on M. Replacing Y into ξ₂ ∈ T_β, we get 0=−∑ν=13[αην(ξ2)ϕνξ+3αg(ϕνξ,ϕξ2)ϕνξ−kην(ξ2)ϕξν−3kην(ξ2)ϕνξ]      +α(∇ξA)ξ2−αη((∇ξA)ξ2)ξ−αkϕAξ2=−4αϕξ2+4kϕξ2+α2βϕξ2−αβkϕξ2because 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 (α−k)(−4+αβ)=0.Since αβ = −4 by virtue of Proposition, it follows that (3.11)α=k.

On the other hand, putting Y ∈ T_λ in (3.10), we get (3.12)α(∇ξA)Y−αη((∇ξA)Y)ξ−αkϕAY+αkAϕY=0Using the equation of Codazzi [13, (1.10)], we know (∇ξA)Y=(∇ξA)ξ+ϕY=αϕAY−AϕAY+ϕY.Thus the equation (3.12) can be written as (3.13)α2λϕY−αλμϕY+αϕY−αλkϕY+αμkϕY=0,because of ϕY ∈ T_µ. 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 R_ξ 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 = ξ₂ ∈ T_β and in (2.3), it becomes 0=(∇^ξ2(k)Rξ)ξ=−∑ν=13[βg(ϕνξ2,ξ)ξν−βην(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.