The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form ()
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
;
(B) a complex quadric
, 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
![](https://www.scirp.org/html/4-5300366\3962ccdd-d03f-483a-b284-1d1f80b9161b.jpg)
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
![](https://www.scirp.org/html/4-5300366\a19ad0c1-a485-42e8-9774-37cef0ff49f3.jpg)
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 ![](https://www.scirp.org/html/4-5300366\71d55c65-5742-4a6c-a9c5-0df0067e740e.jpg)
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 ![](https://www.scirp.org/html/4-5300366\d77e0a45-4e29-40bf-8b6d-919b8d3b03ae.jpg)
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
![](https://www.scirp.org/html/4-5300366\2c0759bb-97a1-46cd-940f-f41c7cdfb6f5.jpg)
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
![](https://www.scirp.org/html/4-5300366\8719496f-bde0-4ced-a252-94741fe0e383.jpg)
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.