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

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.


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 n , a complex Euclidean space or a complex hyperbolic space


A A A and .Further, Hopf hypersurfaces with constant principal curvatures in a complex space form have been completely classified as follows: , where , where , where

 
H  M is said to be of type A for simplicity.As a typical characterization of real hypersurfaces of type A , in a complex space form   n M c was given under the condition The holomorphic distribution 0 of a real hypersurface The following theorem characterizes ruled real hypersurfaces in or equivalently for any for any vector fields X Y and Z in 0 T .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 M be a real hypersurface of n .Then the shape operator A is  -parallel if and only if M is locally congruent to a ruled real hypersurface, or a real hypersurface of type A or .B Also, M. Kimura and S. Maeda ([10]) and Y.J. Suh ([11]) investigated the conditions  -parallel Ricci operator.
Theorem 1.6.([10,11]) Let M be a real hypersurface in a complex space form X and Y in 0 T , then for any M is locally congruent to either a real hypersurface of type A or a ruled real hypersurface (resp.M is locally congruent to a real hypersurface of type A ).
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    , shape operator 2 M c .We shall prove the following.
Theorem 1.8.Let M be a real hypersurface in a complex space form If M has the M parallel shape operator and satisfies (1.4), then M is locally congruent a ruled real hypersurface.
Theorem 1.9.Let M be a real hypersurface in a complex space form parallel Ricci operator and satisfies (1.4), then M is locally congruent to a real hypersurface of type A .
All manifolds in the present paper are assumed to be connected and of class C  and the real hypersurfaces are supposed to be orientable.

Preliminaries
Let M be a real hypersurface immersed in a complex space form   X and Y tangent to M , where for any vector fields g denotes the Riemannian metric tensor of M induced from g A is the shape operator of M in  , and (2.1)


X and on Y for any vector fields M .Since the almost complex structure J is parallel, we can verify from the Gauss and Weingarten formulas the followings: Since the ambient manifold is of constant holomorphic sectional curvature , we have the following Gauss and Codazzi equations respectively: (2.5) for any vector fields X Y and Z on M , where denotes the Riemannian curvature tensor of where is the mean curvature of M , and the covariant derivative of (2.5) is given by Let U be a unit vector field on M 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 We notice here that U is orthogonal to  .We put Then is an open subset of M .

Some Lemmas
In this section, we assume that is not empty, then there are sclar fields  ,   and  and a unit vector field and M c We shall prove the following Lemmas.

5
. 4 Differentiating the second of (3.4) covariantly along vector field X in , we obtain Taking inner product of (3.5) with and  and making use of (3.5) and Lemma 3.1, we have If we differentiate the third of (3.4) covariantly along vector field X in T , we obtain If we take inner product of  and using (3.4), then we have (3.12) and  into (3.12),we obtain By comparing (3.8) and (3.9) with (3.13), we have 0 Proof.Since we have A U U      and using (3.7), we get Thus, it follows from (3.14) that along any vector field X on  and using (2.2) and (2.5) and Lemma 3.1, we have , we see from this equation above that the gradient vector field   of  is given by into Lemma 3.3, then we have (3.16)Thus, the above equation is reduced to

 
 along any vector field M and using (2.2), (2.5) and (2.8) and Lemma 3.2, we have If we take inner product of this equation with  and using  

Proofs of Theorems
Proof Theorem 1.8.If (1.4) is given in M , then we see that Lemma 3.1 holds on M .If we differentiate (1.3) along any vector field X in T and using (2.3) and (2.8), then we have 0 metric and complex structure J on   n M c .The structure vector field  is said to be principal if A   is satisfied, where A is the shape operator of M and 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 0 M is  -parallel and the structure vector field  is princial if and only if M is locally congruent to one of the model spaces of type A or type .B A and  -parallel, I.-B.Kim, K. H. Kim and one of the present authors ([12]) have proved the following.As for the structure tensor field Theorem 1.7.([12])Let M be a real hypersurface in a complex space form n cyclic  -parallel shape operator (resp.Ricci operator) and satisfies normal vector field of M .By   we denote the Levi-Civita connection with respect to the Fubini-Study metric tensor g  of   2 M c .Then the Gauss and Weingarten formulas are given respectively by argument as the above, we can verify that the gradient vector fields of the smooth function (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 (