On an Almost Manifold Satisfying Certain Conditions on the Concircular Curvature Tensor
Mehmet Atçeken, Umit Yildirim
Gaziosmanpasa University, Faculty of Arts and Sciences, Department of Mathematics, Tokat, Turkey
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Mehmet Atçeken, Umit Yildirim. On an Almost Manifold Satisfying Certain Conditions on the Concircular Curvature Tensor. Pure and Applied Mathematics Journal. Special Issue: Applications of Geometry. Vol. 4, No. 1-2, 2015, pp. 31-34. doi: 10.11648/j.pamj.s.2015040102.18
Abstract: We classify almost manifolds, which satisfy the curvature conditions , , and , where is the concircular curvature tensor, is the Weyl projective curvature tensor, is the Ricci tensor and is Riemannian curvature tensor of manifold.
Keywords: Almost Manifold, Concircular Curvature Tensor, Projective Curvature Tensor
1. Introduction
An odd-dimensional Riemannian manifold is said
to be an almost co-Hermitian or almost contact metric manifold if there exist on a tensor field , a vector field (called the structure vector field) and 1-form such that
(1.1)
(1.2)
(1.3)
for any vector fields on
The Sasaki form (or fundamental 2-form) of an almost co-Hermitian manifold is defined by
for all on and this form satisfies This means that every almost co-Hermitian manifold is orientable and defines an almost cosymplectic structure on If this associated structure is cosymplectic , then is called an almost co-Kähler manifold. On the other hand, when , the associated almost cosymplectic structure is a contact structure and is an almost Sasakian manifold. It is well known every contact manifold has an almost Sasakian structure.
The Nijenhuis tensor of typetensor field is type defined by
(1.4)
where is the Lie bracket of
On the other hand, an almost co-complex structure is called integrable if and normal An integrable almost cocomplex structure is a cocomplex structure. A co-Kähler manifold (or normal cosymplectic manifold) is an integrable (or equivalently, a normal) almost co-Kähler manifold, while a Sasakian manifold is a normal almost contact metric manifold [3].
2. Preliminaries
In [2], contact metric manifolds satisfying were classified.
In [1], on Sasakian manifolds and obtained the some results.
M. M. Tripathi and J. S. Kim gave a classification of manifolds satisfying the conditions [7].
Definition 2.1. An almost manifold is an almost co-Hermitian manifold such that the Riemann curvature tensor satisfies the following property, such that
Moreover, if such a manifold has constant sectional curvature equal to , then its curvature tensor is given by
for any
A normal almost manifold is said to be a manifold. For example, Co-Kählerian, Sasakian and Kenmotsu manifolds are and manifolds, respectively [3].
Theorem 2.1.
(i) An almost co-Hermitian manifold is Sasakian if and only if
for all
(ii) If is Sasakian, then is Killing vector field and
(iii) An Sasakian manifold is a manifold [3].
Theorem 2.2.
(i) An almost co-Hermitian manifold is an Kenmotsu manifold if and only if
for all
(iii) An Kenmotsu manifold is a manifold.
3. An Almost Manifold Satisfying Certain Conditions on the Concircular Curvature Tensor
In this section, we will give the main results for this paper.
Let be a dimensional almost manifold and denote Riemannian curvature tensor of , then we have from (2.2), for
In the same way, choosing in (2.2), we have
In (3.2), choosing , we obtain
Also from (3.2), we obtain
From (2.2), we can state
for orthonormal basis of From (3.5), for we obtain
which is equivalent to
From (3.7) we can give the following corollary.
Corollary 3.1. An almost manifold is always an Einstein manifold.
Also, from (3.6), we can easily see
and
Definition 3.1. Let be an dimensional Riemannian manifold. Then the Weyl concircular curvature tensor is defined by
for all , where is the scalar curvature of [5].
In (3.11), choosing , we obtain
Theorem 3.1. Let be a dimensional an almost manifold. Then if and only if either has sectional curvature or the scalar curvature
Proof: Suppose that Then from (3.11), we have
Using (3.12) in (3.13), we obtain
Using (3.2), (3.3) and putting in (3.14), we get
=0.
Therefore, manifold has either sectional curvature or . This implies that
Theorem 3.2. Let be a dimensional an almost manifold. if and only if either has sectional curvature or the scalar curvature .
Proof: Suppose that , we have
Using the equations (3.12) and (3.2), (3.3) in (3.15), we have
Putting in (3.16), we get
=0.
This tell us that has either sectional curvature or the scalar curvature .
The converse is obvious.
Theorem 3.3. Let be a dimensional an almost manifold. Then if and only if reduce an Einstein manifold.
Proof: We suppose that , which implies that
Using (3.12) in (3.17), we get
(3.18)
Using (3.1), (3.9) in (3.18), we obtain
(3.19)
Putting in (3.19), we get
under the condition
.
Therefore, the manifold is Einstein manifold.
The converse is obvious.
If is an Einstein manifold, the scalar curvature of is
By corresponding (3.8) and (3.20) we obtain which implies that is of constant sectional curvature
Definition 3.2. Let be a dimensional Riemannian manifold. Then Weyl projective curvature tensor is defined by
where is Riemannian curvature tensor and is Ricci tensor [5].
Theorem 3.2. Let be a dimensional an almost manifold. Then, if and only if reduce an Einstein manifold.
Proof: Suppose that . Then we have,
for Using (3.12) in (3.22), we get
Taking inner product both sides of (3.23) by , we obtain
Also making use of (3.21), we obtain
(3.25)
Using (3.25) in (3.24) and choosing , we have provided that
So, the manifold is an Einstein manifold. The converse is obvious.
References