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.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 $∇^X(k)Y=∇XY+g(φAX,Y)ξ−η(Y)ϕAX−kη(X)ϕY,$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 $(∇XRξ)Y =−g(ϕAX,Y)ξ−η(Y)ϕAX −∑ν=13[g(ϕνAX,Y)ξν−2η(Y)ην(ϕAX)ξν+ην(Y)ϕνAX +3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ +ην(ϕY)(ϕνϕAX−αη(X)ξν)} +4ην(ξ){ην(ϕY)AX−g(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY] +η((∇XA)ξ)AY+α(∇XA)Y−η((∇XA)Y)Aξ−g(AY,ϕAX)Aξ−η(AY)(∇XA)ξ−η(AY)AϕAX.$On the other hand, by using the generalized Tanaka-Webster connection, we have $(∇^X(k)Rξ)Y=∇^X(k)(RξY)−Rξ(∇^X(k)Y)=∇X(RξY)+g(ϕAX,RξY)ξ−η(RξY)ϕAX−kη(X)ϕRξY −Rξ(∇XY+g(ϕAX,Y)ξ−η(Y)ϕAX−kη(X)ϕY).$From this, together with the fact that M is Hopf, it becomes $(∇^X(k)Rξ)Y=−∑ν=13[g(ϕνAX,Y)ξν−η(Y)ην(ϕAX)ξν+ην(Y)ϕνAX +3{g(ϕνAX,ϕY)ϕνξ+η(Y)ην(AX)ϕνξ +ην(ϕY)(ϕνϕAX−αη(X)ξν)} +4ην(ξ){ην(ϕY)AX−g(AX,Y)ϕνξ}+2ην(ϕAX)ϕνϕY +ην(Y)ην(ϕAX)ξ−ην(ξ)η(Y)ην(ϕAX)ξ +3η(ϕνY)g(ϕAX,ϕνξ)ξ+ην(ξ)g(ϕAX,ϕνϕY)ξ −ην(Y)ην(ξ)ϕAX+ην2(ξ)η(Y)ϕAX−ην(ξ)η(ϕνϕY)ϕAX −kη(X)ην(Y)ϕξν−4kη(X)η(ϕνY)ην(ξ)ξ−4kη(X)η(ϕνY)ξν +3η(Y)η(ϕνϕAX)ϕνξ−η(Y)ην(ξ)ϕνAX+αη(X)η(Y)ην(ξ)ϕνξ +3kη(X)η(ϕνϕY)ϕνξ+kη(x)η(Y)ην(ξ)ϕνξ]+η((∇XA)ξ)AY+α(∇XA)Y−αη((∇XA)Y)ξ−αη(Y)(∇XA)ξ−αkη(X)ϕAY+αkη(X)AϕY$for 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, $(∇^ξ(k)Rξ)Y=0$for 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 : $ξ=η(X0)X0+η(ξ1)ξ1$for some unit vector fields and .

Putting X = ξ in (2.3) and using the condition (*), we have $0=(∇^ξ(k)Rξ)Y =−∑ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ +3{αg(ϕνξ,ϕY)ϕνξ+αη(Y)ην(ξ)ϕνξ−αην(ϕY)ξν} +4ην(ξ){αην(ϕY)ξ−αg(ξ,Y)ϕνξ} −kην(Y)ϕξν−4kη(ϕνY)ην(ξ)ξ−4kη(ϕνY)ξν +3kη(ϕνϕY)ϕνξ+kη(Y)ην(ξ)ϕνξ]+η((∇ξA)ξ)AY+α(∇ξA)Y−αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+αkAϕY$for 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 $0=−{αg(ϕ2ξ,Y)ξ2+αg(ϕ3ξ,Y)ξ3+αη2(Y)ϕ2ξ+αη3(Y)ϕ3ξ +3αg(ϕ2ξ,ϕY)ϕ2ξ+3αg(ϕ3ξ,ϕY)ϕ3ξ−3αη2(ϕY)ξ2 −3αη3(ϕY)ξ3−kη2(Y)ϕξ2−kη3(Y)ϕξ3−4kη(ϕ2Y)ξ2 −4kη(ϕ3Y)ξ3+3kη(ϕ2ϕY)ϕ2ξ+3kη(ϕ3ϕY)ϕ3ξ}+η((∇ξA)ξ)AY+α(∇ξA)Y−αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+αkAϕY=−8kη2(Y)ξ3+8kη3(Y)ξ2+η((∇ξA)ξ)AY+α(∇ξA)Y −αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+αkAϕY$for 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 $η3(Y)η2(X)−η2(Y)η3(X)=0$for 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 curvatures$α=−2tan(2r), β=2cot(2r), γ=0, λ=cot(r), μ=−tan(r)$with some r ∈ (0, π/4). The corresponding multiplicities are$m(α)=1, m(β)=3=m(γ), m(λ)=4n−4=m(μ)$and the corresponding eigenspaces are$Tα=ℝξ=Span{ξ},Tβ=ℑJξ=Span{ξν|ν=1,2,3},Tγ=ℑξ=Span{ϕνξ|ν=1,2,3},Tλ,Tμ,$where$Tλ⊕Tμ=(ℍℂξ)⊥, ℑTλ=Tλ, ℑTμ=Tμ, JTλ=Tμ.$The distributionis the orthogonal complement ofwhere$ℍℂξ=ℝξ⊕ℝJξ⊕ℑξ⊗ℑJξ.$

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 $−∑ν=13[αg(ϕνξ,Y)ξν+αην(Y)ϕνξ+3{αg(ϕvξ,ϕY)ϕνξ−αην(ϕY)ξν} −kην(Y)ϕξν−4kη(ϕνY)ξν+3kη(ϕνϕY)ϕνξ]+η((∇ξA)ξ)AY+α(∇ξA)Y−αη((∇ξA)Y)ξ−αη(Y)(∇ξA)ξ−αkϕAY+αkAϕY=0$for any tangent vector field Y on M. Replacing Y into ξ2Tβ, we get $0=−∑ν=13[αην(ξ2)ϕνξ+3αg(ϕνξ,ϕξ2)ϕνξ−kην(ξ2)ϕξν−3kην(ξ2)ϕνξ] +α(∇ξA)ξ2−αη((∇ξA)ξ2)ξ−αkϕAξ2=−4αϕξ2+4kϕξ2+α2βϕξ2−αβkϕξ2$because 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 $α=k.$

On the other hand, putting YTλ in (3.10), we get $α(∇ξA)Y−αη((∇ξA)Y)ξ−αkϕAY+αkAϕY=0$Using 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 $α2λϕY−αλμϕY+αϕY−αλkϕY+αμkϕY=0,$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 $(∇^X(k)Rξ)Y=0$for 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 $0=(∇^ξ2(k)Rξ)ξ=−∑ν=13[βg(ϕνξ2,ξ)ξν−βην(ϕξ2)ξν +3βην(ξ2)ϕνξ+3βη(ϕνϕξ2)ϕνξ]=−6βϕ2ξ.$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.

References
1. DV. Alekseevskii. Compact quaternion spaces. Funct Anal Appl., 2(1968), 106-114.
2. J. Berndt, and YJ. Suh. Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math.., 127(1999), 1-14.
3. J. Berndt, and YJ. Suh. Isometric flows on real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math.., 137(2002), 87-98.
4. JT. Cho. CR-structures on real hypersurfaces of a complex space form. Publ. Math. Debrecen., 54(1999), 473-487.
5. JT. Cho. Levi parallel hypersurfaces in a complex space form. Tsukuba J. Math.., 30(2006), 329-343.
6. I. Jeong, CJG. Machado, JD. Pérez, and YJ. Suh. Real hypersurfaces in complex two-plane Grassmannians with -parallel structure Jacobi operator. Internat. J. Math.., 22(5)(2011), 655-673.
7. I. Jeong, JD. Pérez, and YJ. Suh. Real hypersurfaces in complex two-plane Grassmannians with parallel structure Jacobi operator. Acta Math. Hungar.., 122(2009), 173-186.
8. U-H. Ki, JD. Pérez, FG. Santos, and YJ. Suh. Real hypersurfaces in complex space forms with ξ-parallel Ricci tensor and structure Jacobi operator. J. Korean Math. Soc.., 44(2007), 307-326.
9. H. Lee, JD. Pérez, and YJ. Suh. On the structure Jacobi operator of a real hypersurface in complex projective space. Monatsh. Math.., 158(2)(2009), 187-194.
10. H. Lee, JD. Pérez, and YJ. Suh. Real hypersurfaces in a complex projective space with pseudo- -parallel structure Jacobi operator. Czechoslovak Math. J.., 60(4)(2010), 1025-1036.
11. H. Lee, and YJ. Suh. Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc.., 47(3)(2010), 551-561.
12. CJG. Machado, and JD. Pérez. Real hypersurfaces in complex two-plane Grassmannians some of whose Jacobi operators are ξ-invariant. Internat. J. Math.., 23(3)(2012), Array.
13. E. Pak, and YJ. Suh. Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster -parallel structure Jacobi operator. Cent. Eur. J. Math.., 12(2014), 1840-1851.
14. JD. Pérez, and YJ. Suh. Real hypersurfaces of quaternionic projective space satisfying ∇UiR = 0. Differential Geom. Appl.., 7(1997), 211-217.
15. JD. Pérez, and YJ. Suh. Two conditions on the structure Jacobi operator for real hypersurfaces in complex projective space. Canad. Math. Bull.., 54(3)(2011), 422-429.
16. JD. Pérez, FG. Santos, and YJ. Suh. Real hypersurfaces in complex projective space whose structure Jacobi operator is -parallel. Bull. Belg. Math. Soc. Simon Stevin., 13(2006), 459-469.
17. N. Tanaka. On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math.., 20(1976), 131-190.
18. S. Tanno. Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc.., 314(1)(1989), 349-379.
19. SM. Webster. Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom.., 13(1978), 25-41.