Article
KYUNGPOOK Math. J. 2019; 59(3): 525-535
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
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
Received: December 28, 2017; Accepted: February 27, 2018
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
Keywords: real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, generalized Tanaka-Webster connection, structure Jacobi operator.
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 ∇
As an ambient space, a complex two-plane Grassmannian
Using a result from Alekseevskii [1], Berndt and Suh [2] proved the following:
Theorem A
The Reeb vector field
Now, instead of the Levi-Civita connection, we consider the
We put the Reeb vector field
Theorem B
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
Main Theorem
On the other hand, we consider another new notion for generalized Tanaka-Webster parallelism of the structure Jacobi operator on a real hypersurface
Corollary
We refer to [1, 2, 3] and [11, section 1] for Riemannian geometric structures of
Let us denote by
Here, it is a main goal to show that the Reeb vector field
From now on, unless otherwise stated in the present section, we may put the Reeb vector field
Putting
Now, using these facts, we prove the following Lemma.
Lemma 2.1
Proof
By taking the inner product with
Thus substituting
Proof of The Main Theorem
Let us consider a Hopf hypersurface
First of all, we consider the case . Without loss of generality, we may put
Lemma 3.1
Since our assumption
Next we consider the case . Using Theorem A, Lee and Suh [11] gave a characterization of real hypersurfaces of type (B) in
Lemma 3.2
From the above two Lemmas 3.1 and 3.2 and the classification theorem given by Theorem A in this paper, we see that
Hence it remains to check that whether the stucture Jacobi operator
Proposition 3.3
To check this problem, we suppose that
On the other hand, putting
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
To check this problem, we suppose that
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