### Article

Kyungpook Mathematical Journal 2021; 61(1): 181-190

**Published online** March 31, 2021

Copyright © Kyungpook Mathematical Journal.

### Certain Characterization of Real Hypersurfaces of type A in a Nonflat Complex Space Form

U-Hang Ki

The National Academy of Siences, Seoul 06579, Korea

e-mail : uhangki2005@naver.com

**Received**: April 8, 2020; **Revised**: July 3, 2020; **Accepted**: August 18, 2020

### Abstract

Let *M* be a real hypersurface with almost contact metric structure *S* and *M* and by the structure Jacobi operator with respect to the vector field ξ respectively. In this paper, we prove that *M* is a Hopf hypersurface of type *A* in

**Keywords**: real hypersurface, structure Jacobi operator, Ricci tensor, Hopf hypersurface.

### 1. Introduction

An

In this paper we consider a real hypersurface

In the study of real hypersurfaces in _{1}_{2}

In the case of real hypersurfaces in _{0}_{1}_{2}

for any vector fields

(A a hyperplane_{1})${P}_{n-1}\u2102$ , where$0<r<\pi /2$ ,

(A a totally geodesic_{2})${P}_{k}\u2102$ $(1\le k\le n-2)$ , where$0<r<\pi /2$ .

(A a horosphere in_{0})${H}_{n}\u2102$ , i.e., a Montiel tube,

(A a geodesic hypersphere, or a tube over a hyperplane_{1})${H}_{n-1}\u2102$ ,

(A a tube over a totally geodesic_{2})${H}_{k}\u2102$ $(1\le k\le n-2)$ .

Characterization problems for a real hypersurface of type A in a complex space form were studied by many authors (cf. [3, 4, 5], etc.).

We denote by _{ξ}

Under the condition

The purpose of this paper is, using the hypothesis concerned with the Ricci tensor _{}ξ

All manifolds in the present paper are assume to be connected and of class

### 2. Preliminaries

Let _{n} (c)_{n} (c)

where _{n} (c)

For any vector field

We call ξ the structure vector field (or the Reeb vector field) and its flow also denoted by the same latter ξ. The Reeb vector field ξ is said to be

A real hypersurface

for any vector fields

for any vector fields

Since we consider that the ambient space is of constant holomorphic sectional curvature

for any vector fields

In what follows, to write our formulas in convention forms, we denote by

From the Gauss equation (2.3), the Ricci tensor

for any vector field

Now, we put

where

which shows that

Making use of (2.1), (2.7) and (2.8), it is verified that

because

Now, differentiating (2.8) covariantly and taking account of (2.1) and (2.2), we find

which together with (2.4) implies that

Applying (2.12) by ϕ and making use of (2.11), we obtain

which connected to (2.1) and (2.13) gives

Using (2.3), the structure Jacobi operator

for any vector field

From (2.5) we obtain

Because of (2.5) and (2.7), we also have

### 3. The Structure Jacobi Operator of Real Hypersurfaces

Let _{n} (c)_{ξ}

We set

Applying by

which together with (2.7) yields

Since

If we replace

then we find

which enables us to obtain

Further, we assume in the sequel that

for any vector field

Applying this by

by virtue of (3.6). Because of (2.18) we can write (3.9) as

since

If we combine this to (3.6), then we get

where we have used (3.7), which tells us that

Tranforming this to

Combining (2.19) to (3.10), we find

Now, we see from (2.21) and (3.12) that

which connected to (3.6) implies that

In the next step, if we apply by

If we use (3.8) to this, then we obtain

which together with (3.7) yields

Now, suppose that

because of (2.8), where we have put

In fact, from (2.16) we have

which together with (3.18) gives the requied relationship. According to Theorem CK, we conclude that

We are now going to prove that

Using (3.19), we also have from (3.13)

Because of (2.5) and (3.2) we have

which connected to (3.8) gives

On the other hand, we have from (2.5)

which together with (3.2), (3.12), (3.19) and (3.21) yields

We also, using (3.2) and (3.6) with

or, using (3.20)

Combining (3.23) to this, it follows that

Using (3.23) and this, (3.22) reformed as

In fact, if not, then we have

Therefore (3.8) tells us that

on

We are now going to prove

and taking account of (2.1), we find

which together with (2.13) and (3.24) yields

or, using (2.4),

Putting

which together with (3.24) implies that

Thus, it follows that

by virtue of

On the other hand, if we put

Taking the inner product with

which together with (3.28) gives

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