Pure and Applied Mathematics Journal
Volume 4, Issue 1-2, January 2015, Pages: 31-34

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

(M. Atçeken)
(U. Yildirim)

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

1. C. Özgür and M. M. Tripathi, On P-Sasakian manifolds satisfying certain conditions on the concircular curvature tensor, Turkish Journal of Math. , 31(2007), 171 – 179.
2. D. E. Blair, J. S. Kim and M. M. Tripathi, On concircular curvature tensor of a contact metric manifold, J. Korean Math. Soc. 42(2005), 883-892.
3. D. Janssens and L. Vanhecke, Almost contact structure and curvature tensors, Kodai Math.J., 4(1981), 1-27.
4. D. Perrone, Contact Riemannian manifolds satisfying R(X, ξ)·R = 0, Yokohama Math. J. 39 (1992), 2, 141-149.
5. K. Yano and M. Kon, Structures on manifolds, Series in Pure Math., Vol. 3, Word Sci., (1984).
6. K. Yano, Concircular geometry I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200.
7. M. M. Tripathi and J. S. Kim, On the concircular curvature tensor of a (κ, µ)-manifold, Balkan J. Geom. Appl. 9, no.1, 104 - 114 (2004).
8. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y).R = 0, the local version, Diff. Geom., 17(1982), 531-582.

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