1. Introduction
A Kahler manifold is a complex manifold of complex dimension n and real dimension 2n, which is equipped with
a complex structure J defined , where is the tangent space of , satisfying relations and , i.e., J is parallel with respect to the Levi-Civita connection of
and a Riemanian metric G that is compatible with J, i.e., for all tangent X, Y on .
The pair () is called Kahler structure. A Kahler manifold of constant holomorphic sectional curvature c is called complex space form. Complete and simply connected complex space forms depending on the value of holomorphic sectional curvature c are analytically isometric to complex projective space if , to complex hyperbolic space if or to complex Euclidean space if . This paper focuses on complex space forms with denoted by and called non-flat complex space forms. Furthermore, in the case of and in the case of .
A submanifold M in a non-flat complex space form of real codimension equal to 1 is called real hypersurface. Let N be a locally defined unit normal vector on M. The Kahler structure () of the ambient space induces on M an almost contact metric structure () defined in the following way
is the structure vector field,
is a skew-symmetric tensor field of type (1,1) called structure tensor field and defined to be the tangential component of , for all tangent vectors X to M,
is a 1-form and is given by the relation for all tangent vectors X to M,
g is the induced Riemannian metric on M.
A big class of real hypersurfaces in
are
Hopf hypersurfaces, which are real hypersurfaces whose structure vector field
is an eigenvector of the shape operator
A of
M, i.e.,
where
and is called
Hopf principal curvature.
Takagi classified homogeneous real hypersurfaces in complex projective space . The real hypersurfaces are divided into six types:
type (A) which are either geodesic hyperspheres of radius r, , or tubes of radius r, with over totally geodesic ,,
type (B) which are tubes of radius r, , over the complex quadric ,
type (C) which are tubes over the Serge embedding of , with and ,
type (D) which are tubes over the Plücker embedding of the Grassmann manifold and ,
type (
E) which are tubes over the canonical embedding of the Hermitian symmetric space
and
, where
is a subgroup of
of dimension
n, which consists of all the orthogonal matrices with determinant equal 1. (see [
1,
2,
3]).
The above real hypersurfaces are Hopf ones with constant principal curvatures (see [
4]).
In the case of the ambient space being the complex hyperbolic
, Montiel in [
5] studied real hypersurfaces with two constant principal curvatures. Additionally, he proved that such real hypersurfaces are Hopf ones. Berndt in [
6] classified Hopf hypersurfaces with constant principal curvatures in
. The following list includes the Hopf hypersurfaces with constant principal curvatures.
type (A) which are either horospheres, or geodesic hyperspheres, or tubes over totally geodesic complex hyperbolic hyperplane, or tubes over totally geodesic , ,
type (B) which are tubes over totally geodesic real hyperbolic space (type (B)).
All of them are homogeneous ones, but in contrast to the case of complex projective space, it is proved that there are also non-Hopf hypersurfaces in which are homogeneous.
Let
be a Riemannian manifold of dimension
m and
g its Riemannian metric. Then the
Weyl curvature tensor of
is given by
with
R being the Riemannian curvature tensor,
S being the Ricci tensor and
being the scalar curvature of
. If
then
and if
then
is locally conformal flat if and only if
. The condition of locally conformal flat holds for three dimensional Riemannian manifolds if and only if the Cotton tensor of
, which is given by
vanishes identically.
The Weyl curvature tensor of real hypersurfaces
M in
satisfies the relation
for all
tangent to
M, where
R is the Riemannian curvature tensor,
S is the Ricci tensor,
is the scalar curvature of
M and
g is the induced Riemannian metric on
M. In [
7] the non-existence of real hypersurfaces in
with harmonic Weyl curvature tensor, i.e.,
with
denoting the codifferential of the exterior differential
d is proved. Moreover, in [
8] the classification of real hypersurfaces in
with
-parallel Weyl curvature tensor, i.e.,
is provided. Finally, in [
9] real hypersurfaces in
,
satisfying the previous geometric condition are classified.
In 1959 Tachibana defined
-Ricci tensor
on almost Hermitian manifold. In [
10] Hamada gave the definition of
-Ricci tensor
on real hypersurfaces in
in the following way
for all
tangent to
M and
is the sum of elements of the main diagonal of the matrix, which corresponds to the above endomorphism. He also presented
- Einstein, i.e.,
, where
is a constant multiple of
and provided classification of
-Einstein hypersurfaces. Ivey and Ryan in [
11] extended the Hamada’s work and studied the equivalence of
- Einstein condition with other geometric conditions such as the pseudo-Einstein and the pseudo-Ryan condition.
Motivated by the revious results and work we define
-Weyl curvature tensor of real hypersurfaces in the following way
for all
tangent to
M and
is the
-Ricci tensor and
is the
-scalar curvature corresponding to
of
M.
First it is examined if there are real hypersurfaces of dimension equal to or greater than three with vanishing -Weyl curvature tensor. The following Theorem is proved
Theorem 1. Let M be a Hopf hypersurface in , , with vanishing -Weyl curvature tensor. Then M is an open subset of a real hypersurface of type (A) or of a Hopf hypersurface with .
Next it is examined if there are three-dimensional real hypersurface in
with vanishing
-Weyl curvature tensor and the following Theorem is obtained
Theorem 2. Every real hypersurface M in with vanishing -Weyl curvature tensor is a Hopf hypersurface. Furthermore, M is an open subset of a real hypersurface of type (A) or of a Hopf hypersurface with .
The paper has the following outline: In
Section 2 relations and Theorems concerning real hypersurfaces in non-flat complex space forms are provided. In
Section 3 Theorems 1 and 2 are proved.
Section 4 concerns discussion on the new tensor and ideas of further research and
Section 5 includes the conclusions of the paper.
2. Preliminaries
The manifolds, vector fields, etc., are considered of class
. We consider
M to be a connected real hypersurface without boundary in
equipped with a Kahler structure (
) and
is the Levi-Civita connection of
and
N a locally unit normal vector field on
M. Then the shape operator
A of
M with respect to
N is given by
and the Levi-Civita connection ∇ of the induced metric
g on
M satisfies
As mentioned in the Introduction, on
M an almost contact metric structure
is defined and the following relations are satisfied (see [
12])
for all tangent vectors
to
M. Relation (
3) implies
Due to the fact that the complex structure
J is parallel, i.e.,
we have
for all
tangent to
M. Moreover, the ambient space is of holomorphic sectional curvature
c and this results in the Gauss and Codazzi equations becoming respectively
and
for all tangent vectors
to
M, where
R is the Riemannian curvature tensor of
M.
Let
P be a point of
M, then the tangent space
is decomposed into
where
and is called (
maximal)
holomorphic distribution (
if ).
The following Theorem concerns the shape operator of
M and is proved by Maeda [
13] in the case of
and by Ki and Suh [
14] in the case of
(also Corollary 2.3 in [
15]).
Theorem 3. Let M be a Hopf hypersurface in , . Then
- (i)
α is constant.
- (ii)
If W is a vector field which belongs to such that , then - (iii)
If the vector field W satisfies and then
We consider
M a three dimensional real hypersurface in
and
P a point of
M such that in the neighborhood of
P relation
holds. Let
U be a unit vector lying in the
satisfying relation
. Then, we can consider the standard non-Hopf local orthonormal frame
in the neighborhood of
P (see [
16] p. 445). Therefore, the shape operator
A is given by
The following Lemma holds for three dimensional non-Hopf real hypersurfaces in
Lemma 1. Let M be a non-Hopf real hypersurface in . The following relations hold on Mwhere are smooth functions on M and . Lemma 1 is proved in page 92 [
17].
The Codazzi Equation (
6) for
X∈
and
owing to Lemma 1 results in the following relations
and for
and
In the case of three dimensional Hopf hypersurfaces we consider a point
P of
M and we define in the neighborhood of
P a local orthonormal frame as follows: since
M is a Hopf hypersurface the shape operator
A restricted to the holomorphic distribution
has distinct eigenvalues. Thus, we choose a vector
W as one of the eigenvectors fields. Moreover, due to the fact that
M is three dimensional, the shape operator satisfies the following relations:
and Thereom 3 holds.
Finally, the following Theorem concerns the classification of real hypersurfaces in
,
, whose shape operator
A satisfies a commuting condition. It is proved by Okumura in the case of
(see [
18]) and by Montiel and Romero in the case of
(see [
19]).
Theorem 4. Let M be a real hypersurface of , . Then , if and only if M is an open subset of a homogeneous real hypersurface of type (A).
We mention that type () hypersurfaces do not occur in the case of three dimensional real hypersurface in .
3. Proof of Theorems 1 and 2
The
-Ricci tensor of a real hypersurface
M in a non-flat complex space form is given by
for all
X tangent to
M.
Let
M be a Hopf hypersurface in
,
, with vanishing
-Weyl curvature tensor, i.e.,
Since
M is a Hopf hypersurface
is an eigenvector of the shape operator relation (
1) holds and relation (
15) for
yields
. Next, we consider
W a unit vector field which belongs to the (maximal) holomorphic distribution such that relation
holds at some point
P∈
M and relation (
7) is satisfied. We have two cases:
Case I: .
In this case
so relation (
7) implies
and relation (
8) holds.
Relation (
16) for
taking into account (
2) implies
for all
X,
Y tangent to
M.
The inner product of relation (
17) for
and
with
W because of (
3), (
5), (
15),
,
and
yields
Furthermore, the inner product of relation (
17) for
and
with
due to (
3), (
5) and (
15),
,
and
implies
Combination of relations (
18) and (
19) results in
So, either and M is an open subset of a Hopf hypersurface with or which implies that and because of Theorem 4 M is an open subset of a real hypersurface of type (A).
Case II: .
This case occurs only when the ambient space is the complex hyperbolic space
. Thus,
and this results in
. We consider
W a unit vector field, which belongs to the (maximal) holomorphi distribution such that relation
holds at some point
P∈
M. Therefore, relation (
7) due to
and
implies
First we suppose that
. Then the above relation implies
. So, the inner product of relation (
17) for
and
with
W because of (
3), (
5) and (
15) for
which implies
,
and
results in
Moreover, the inner product of relation (
17) for
and
with
due to (
3), (
5), (
15),
,
and
implies
Combination of relations (
20) and (
21) yields
, which is a contradiction.
Therefore, we have for any vector field W∈ and M is an open subset of a horosphere, which is a real hypersurface of type (A) and this completes the proof of Theorem 1.
Remark 1. Examples of Hopf hypersurfaces with are the following:
Next we examine non-Hopf three-dimensional real hypersurfaces
M in
whose *-Weyl tensor vanishes identically, i.e., relation (
16) holds. We consider
the open subset of
M such that
and
be the local orthonormal frame in the neighborhood of a point
P defined as in
Section 2.
Relation (
2) for
and due to
implies
for all
X,
Y tangent to
M. The inner product of relation (
22) for
and
with
and
U taking into account relations (
9), (
5) and (
15) yields respectively
Moreover, the inner product of relation (
22) for
and
with
because of relations (
9), (
5) and (
15) and the second of (
23) results in
Suppose that
then the first of (
23) gives
. Substitution of the latter in (
24) results in
, which is a contradiction. Thus, relation
holds.
Relation (
22) for
and
because of (
5) implies
. So, relation (
24) results in
. Differentiating the latter with respect to
taking into account relations (
10)–(
13) results in
.
So is empty and the following Proposition has been proved.
Proposition 1. Every real hypersurface in whose -Weyl curvature tensor vanishes identically is a Hopf hypersurface.
The above proposition with Theorem 1 for the case of completes the proof of Theorem 2.
4. Discussion
In literature it is known that there are no Einstein real hypersurfaces in non-flat complex space forms, i.e., real hypersurfaces whose Ricci tensor satisfies relation
, where
is constant (see [
15]). Therefore, new notions such as
-Einstein, i.e., the Ricci tensor satisfies relation
or
-Ricci Einstein, i.e., the
-Ricci tensor satisfies
, with
being constant, are introduced and the real hypersurfaces are studied with respect to the previous relations (see [
10,
11,
15]). Thus, the next step is to introduce new tensors on real hypersurfaces in non-flat complex space forms related to the
-Ricci tensor, since there are results concerning notions and tensors related to the Ricci tensor. In this paper, we introduced the
-Weyl curvature tensor and studied real hypersurfaces in non-flat complex space forms in terms of it. Further work can be done in this direction. So, at this point some ideas for further research are mentioned:
it is worthwhile to study if there are non-Hopf real hypersurfaces of dimension greater than three in non-flat complex space forms with vanishing -Weyl curvature tensor,
the -Weyl curvature tensor could also be defined on real hypersurfaces in other symmetric Hermitian space forms such as the complex two-plane Grassmannians or the complex hyperbolic two-plane Grassmannians and it could be examined if there are real hypersurfaces with vanishing -Weyl curvature tensor.
Overall, real hypersurfaces in non-flat complex space forms can be potentially applied to finding solutions of nonlinear dynamical differential equations. Ideas for research in this direction can be derived methods based on Lie algebra. For a first idea in this direction one could have a look in works (1) A Lie algebra approach to susceptible-infected-susceptible epidemics (see [
22]), (2) Lie algebraic discussion for affinity based information diffusion in social networks (see [
23]).