﻿The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form

Vol.3 No.2(2013), Article ID:28584,5 pages DOI:10.4236/apm.2013.32036

The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form

Dong Ho Lim1, Woon Ha Sohn2, Hyunjung Song1

1Department of Mathematics, Hankuk University of Foreign Studies, Seoul, Republic of Korea

2Department of Mathematics, Yeungnam University, Kyongbuk, Republic of Korea

Email: dhlnys@hufs.ac.kr, mathsohn@ynu.ac.kr, hsong@hufs.ac.kr

Received November 8, 2012; revised December 15, 2012; accepted January 2, 2013

Keywords: real hypersurface; η-parallel shape operator; η-parallel Ricci operator; Hopf hypersurface; ruled real hypersurfaces

ABSTRACT

In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M2(c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.

1. Introduction

A complex -dimensional Kaeherian manifold of constant holomorphic sectional curvature is called a  complex space form, which is denoted by. As is well-known, a complete and simply connected complex space form is complex analytically isometric to a complex projective space, a complex Euclidean space or a complex hyperbolic space, according to or

In this paper we consider a real hypersurface in a complex space form. Then has an almost contact metric structure induced from the Kaehler metric and complex structure on. The structure vector field is said to be principal if is satisfied, where is the shape operator of and. In this case, it is known that is locally constant ([1]) and that is called a Hopf hypersurface.

Typical examples of Hopf hypersurfaces in are homogeneous ones, R. Takagi [2] and M. Kimura [3] completely classified such hypersurfaces as six model spaces which are said to be and. On the other hand, real hypersurfaces in have been investigated by J. Berndt [4], S. Montiel and A. Romero [5] and so on. J. Berndt [4] classified all homogeneous Hopf hyersurfaces in as four model spaces which are said to be and. Further, Hopf hypersurfaces with constant principal curvatures in a complex space form have been completely classified as follows:

Theorem 1.1. ([2]) Let be a homogeneous real hypersurface of. Then is tube of radius over one of the following Kaeherian submanifolds:

(A1) a hyperplane, where;

(A2) a totally geodesic, where;

(C), where and is odd;

(D) a complex Grassmann, where and;

(E) a Hermitian symmetric space, where and.

Theorem 1.2. ([4]) Let be a real hypersurface in. Then has constant principal curvatures and is principal if and only if is locally congruent to one of the followings:

(A0) a self-tube, that is, a horosphere;

(A1) a geodesic hypersphere;

(A2) a tube over a totally geodesic ;

(B) a tube over a totally real hyperbolic space.

A real hypersurface of type or in or type or in, then is said to be of type for simplicity. As a typical characterization of real hypersurfaces of type, in a complex space form was given under the condition

, (1.1)

for any tangent vector fields and on by M. Okumura [5] for and S. Montiel and A. Romero [6] for. Namely Theorem 1.3. ([5,6]) Let be a real hypersurface in. It satisfies (1.1) on if and only if is locally congruent to one of the model spaces of type A.

The holomorphic distribution of a real hypersurface in is defined by

(1.2)

The following theorem characterizes ruled real hypersurfaces in.

Theorem 1.4. ([7]) Let be a real hypersurface in. Then is a ruled real hypersurfaces if and only if or equivalently for any.

A (1,1) type tensor field of is said to be -parallel if

(1.3)

for any vector fields and in. Real hypersurfaces with -parallel shape operator or Ricci operator have been studied by many authors (see [13]). Nevertheless, the classification of real hypersurfaces with -parallel shape operator or Ricci operator in remains open up to this point. Recently, S.H. Kon and T.H. Loo ([9]) investigated the conditions -parallel shape operator.

Theorem 1.5. ([9]) Let be a real hypersurface of. Then the shape operator is -parallel if and only if is locally congruent to a ruled real hypersurface, or a real hypersurface of type or.

Also, M. Kimura and S. Maeda ([10]) and Y.J. Suh ([11]) investigated the conditions -parallel Ricci operator.

Theorem 1.6. ([10,11]) Let be a real hypersurface in a complex space form. Then the Ricci operator of is -parallel and the structure vector field is princial if and only if is locally congruent to one of the model spaces of type or type.

As for the structure tensor field, shape operator and -parallel, I.-B. Kim, K. H. Kim and one of the present authors ([12]) have proved the following.

Theorem 1.7. ([12]) Let be a real hypersurface in a complex space form. If has the cyclic -parallel shape operator (resp. Ricci operator) and satisfies

(1.4)

for any and in, then is locally congruent to either a real hypersurface of type or a ruled real hypersurface (resp. is locally congruent to a real hypersurface of type).

The purpose of this paper is to give some characterizations of real hypersurface satisfying (1.4) and having the -parallel shape operator or Ricci operator in. We shall prove the following.

Theorem 1.8. Let be a real hypersurface in a complex space form, If has the - parallel shape operator and satisfies (1.4), then is locally congruent a ruled real hypersurface.

Theorem 1.9. Let be a real hypersurface in a complex space form, If has the - parallel Ricci operator and satisfies (1.4), then is locally congruent to a real hypersurface of type.

All manifolds in the present paper are assumed to be connected and of class and the real hypersurfaces are supposed to be orientable.

2. Preliminaries

Let be a real hypersurface immersed in a complex space form, and be a unit normal vector field of. By we denote the Levi-Civita connection with respect to the Fubini-Study metric tensor of. Then the Gauss and Weingarten formulas are given respectively by

for any vector fields and tangent to, where denotes the Riemannian metric tensor of induced from, and is the shape operator of in. For any vector field on we put

where is the almost complex structure of. Then we see that induces an almost contact metric structure, that is,

(2.1)

for any vector fields and on. Since the almost complex structure is parallel, we can verify from the Gauss and Weingarten formulas the followings:

(2.2)

(2.3)

Since the ambient manifold is of constant holomorphic sectional curvature, we have the following Gauss and Codazzi equations respectively:

(2.4)

(2.5)

for any vector fields and on, where denotes the Riemannian curvature tensor of. From (1.3), the Ricci operator of is expressed by

(2.6)

where is the mean curvature of, and the covariant derivative of (2.5) is given by

(2.7)

Let U be a unit vector field on with the same direction of the vector field, and let be the length of the vector field if it does not vanish, and zero (constant function) if it vanishes. Then it is easily seen from (1.1) that

(2.8)

where. We notice here that is orthogonal to. We put

(2.9)

Then is an open subset of.

3. Some Lemmas

In this section, we assume that is not empty, then there are sclar fields and and a unit vector field and orthogonal to such that

(3.1)

and

(3.2)

in We shall prove the following Lemmas.

Lemma 3.1. Let be a real hypersurface in a complex space form If satisfies (1.4), then we have and

Proof. If we put, or and into (1.4) and make use of (3.1), then we have

(3.3)

Therefore, it follows that is expressed in terms of and only and given by. □

It follows from (2.6), (2.8) and Lemma 3.1 that

(3.4)

Lemma 3.2. Under the assumptions of Lemma 3.1. If has the -parallel Ricci operator then we have and.

Proof. Differentiating the second of (3.4) covariantly along vector field in, we obtain

(3.5)

Taking inner product of (3.5) with and and making use of (3.5) and Lemma 3.1, we have

(3.6)

and

(3.7)

If we put and into (3.6) then we have

(3.8)

and

(3.9)

Putting and into (3.7), then we obtain

(3.10)

If we differentiate the third of (3.4) covariantly along vector field in, we obtain

(3.11)

If we take inner product of and using (3.4), then we have

(3.12)

Substituting and into (3.12), we obtain

(3.13)

By comparing (3.8) and (3.9) with (3.13), we have and

Lemma 3.3. Under the assumptions of Lemma 3.2, we have.

Proof. Since we have and using (3.7), we get

(3.14)

Thus, it follows from (3.14) that

Lemma 3.4. Under the assumptions of Lemma 3.2, we have and

Proof. Differentiating the smooth function along any vector field on and using (2.2) and (2.5) and Lemma 3.1, we have

(3.15)

Since we have, we see from this equation above that the gradient vector field of is given by

If we put into Lemma 3.3, then we have

(3.16)

Thus, the above equation is reduced to

(3.17)

Taking inner product of this equation with and respectively, we obtain

(3.18)

If we differentiate the smooth function along any vector field on and using (2.2), (2.5) and (2.8) and Lemma 3.2, we have

(3.19)

Putting into Lemma 3.3, then we have

(3.20)

If we substitute (3.20) into (3.19), then we obtain

(3.21)

If we take inner product of this equation with and using in Lemma 3.2, then we have

(3.22)

As a similar argument as the above, we can verify that the gradient vector fields of the smooth function

is given respectively by

(3.23)

and

(3.24)

by virtue of (2.3) and Lemma 3.2.

If we substitute (3.24) into (3.23) and make use of (3.20) and Lemma 3.1, then we obtain

(3.25)

If we take inner product of this equation with and respectively, then we have

(3.26)

If we substitute (3.26) into (3.14) and take account of (3.21), then we have. Also, if we differentiate (3.21) along any vector field, then we have

(3.27)

Taking inner product of (3.23) with and using (3.18), we get. Since, we see from (3.27) and the first of (3.18) that and. □

4. Proofs of Theorems

Proof Theorem 1.8. If (1.4) is given in, then we see that Lemma 3.1 holds on. If we differentiate (1.3) along any vector field in and using (2.3) and (2.8), then we have

(4.1)

for any vector fields and on. Putting into (4.1) and using Lemma 3.1 and 3.3, then we have

(4.2)

Since is not empty, we have hold on. It follows from (2.8) and Lemma 3.1 that

Thus is locally congruent to ruled real hypersurface (see [7]). □

Proof Theorem 1.9. Assume that the open set is not empty. Then we consider from Lemma 3.2 and 3.3 that and

. If we differentiate the smooth function

along vector field on and (2.2), (2.5) and (2.8), we have

(4.3)

Since we have, we see from this equation above that gradient vector field of is given by

(4.4)

where indicates the identity transformation on. If we substitute (3.16) into (4.4) and using Lemma 3.4, then we obtain

(4.5)

Since we have, we get

(4.6)

By (4.6) and (3.22), we have and hence it is a contradiction. Thus the set is empty, and hence is a Hopf hypersurface. Since is a Hopf hypersurface, we see from (2.1) and (2.8) that, which together with our assumption (1.4) implies (1.1), that is on. Thus, Theorem 1.9 shows that is locally congruent to a real hypersurface of type. □

5. Acknowledgements

The authors would like to express their sincere gratitude to the refree who gave them valuable suggestions and comments.

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