Abstract

The object of the present paper is to characterize -contact metric manifolds satisfying certain curvature conditions on the conharmonic curvature tensor. In this paper we study conharmonically symmetric, -conharmonically flat, and -conharmonically flat -contact metric manifolds.

1. Introduction

Let and be two Riemannian manifolds with and being their respective metric tensors related through where is a real function. Then and are called conformally related manifolds and the correspondence between and is known as conformal transformation [1].

It is known that a harmonic function is defined as a function whose Laplacian vanishes. A harmonic function is not invariant, in general. The condition under which a harmonic function remains invariant have been studied by Ishii [2] who introduced the conharmonic transformation as a subgroup of the conformal transformation (1.1) satisfying the condition where denotes the covariant differentiation with respect to the metric .

A rank-four tensor that remains invariant under conharmonic transformation for a -dimensional Riemannian manifold is given by where denotes the Riemannian curvature tensor of type defined by where is the Riemannian curvature tensor of type and denotes Ricci tensor of type , respectively.

The curvature tensor defined by (1.3) is known as conharmonic curvature tensor. A manifold whose conharmonic curvature vanishes at every point of the manifold is called conharmonically flat manifold. Thus this tensor represents the deviation of the manifold from conharmonic flatness. It satisfies all the symmetric properties of the Riemannian curvature tensor . There are many physical applications of the tensor . For example, in [3], Abdussattar showed that sufficient condition for a space-time to be conharmonic to a flat space-time is that the tensor vanishes identically. A conharmonically flat space-time is either empty in which case it is flat or filled with a distribution represented by energy momentum tensor possessing the algebraic structure of an electromagnetic field and conformal to a flat space-time [3]. Also he described the gravitational field due to a distribution of pure radiation in presence of disordered radiation by means of spherically symmetric conharmonically flat space-time. Conharmonic curvature tensor have been studied by Siddiqui and Ahsan [4], Özgür [5], and many others.

Let be an almost contact metric manifold equipped with an almost contact metric structure . At each point , decompose the tangent space into direct sum , where is the 1-dimensional linear subspace of generated by . Thus the conformal curvature tensor is a map

It may be natural to consider the following particular cases:(1), that is, the projection of the image of in is zero;(2), that is, the projection of the image of in () is zero;(3), that is, when is restricted to , the projection of the image of in is zero. This condition is equivalent to

Here cases 1, 2, and 3 are synonymous to conformally symmetric, -conformally flat, and -conformally flat.

In [6], it is proved that a conformally symmetric -contact manifold is locally isometric to the unit sphere. In [7], it is proved that a -contact manifold is -conformally flat if and only if it is an -Einstein Sasakian manifold. In [8], some necessary conditions for a -contact manifold to be -conformally flat are proved. In [9], a necessary and sufficient condition for a Sasakian manifold to be -conformally flat is obtained. In [10], projective curvature tensor in -contact and Sasakian manifolds has been studied. Moreover, the author [11] considered some conditions on conharmonic curvature tensor , which has many applications in physics and mathematics, on a hypersurface in the semi-Euclidean space . He proved that every conharmonically Ricci-symmetric hypersurface satisfying the condition is pseudosymmetric. He also considered the condition on hypersurfaces of the semi-Euclidean space .

Motivated by the studies of conformal curvature tensor in (see [6–9]) and the studies of projective curvature tensor in -contact and Sasakian manifolds in [10] and Lorentzian para-Sasakian manifolds in [5], in this paper we study conharmonic curvature tensor in -contact metric manifolds.

Analogous to the considerations of conformal curvature tensor, we give following definitions.

Definition 1.1. A -dimensional -contact metric manifold is said to be conharmonically symmetric if , where .

Definition 1.2. A -dimensional -contact metric manifold is said to be -conharmonically flat if for .

Definition 1.3. A -dimensional -contact metric manifold is said to be -conharmonically flat if , where .

The paper is organized as follows. After preliminaries in Section 2, in Section 3 we consider conharmonically symmetric -contact metric manifolds. In this section we prove that if an -dimensional -contact metric manifold is conharmonically symmetric, then it is locally isometric to the product . Section 4 deals with -conharmonically flat -contact metric manifolds and we prove that an -dimensional -contact metric manifold is -conharmonically flat if and only if it is an -Einstein manifold. Besides these some important corollaries are given in this section. Finally, in Section 5, we prove that a -conharmonically flat -contact metric manifold is a Sasakian manifold with vanishing scalar curvature.

2. Preliminaries

A -dimensional differentiable manifold is said to admit an almost contact structure if it admits a tensor field of type , a vector field , and a 1-form satisfying (see [12, 13])

An almost contact metric structure is said to be normal if the almost induced complex structure on the product manifold defined by is integrable, where is tangent to , is the coordinate of , and is a smooth function on . Let be the compatible Riemannian metric with almost contact structure , that is, Then becomes an almost contact metric manifold equipped with an almost contact metric structure . From (2.1) it can be easily seen that for any vector fields on the manifold. An almost contact metric structure becomes a contact metric structure if , for all vector fields .

A contact metric manifold is said to be Einstein if , where is a constant and -Einstein if , where and are smooth functions.

A normal contact metric manifold is a Sasakian manifold. An almost contact metric manifold is Sasakian if and only if, where is the Levi-Civita connection of the Riemannian metric . A contact metric manifold for which is a Killing vector field is said to be a -contact metric manifold. A Sasakian manifold is -contact but not conversely. However a 3-dimensional -contact manifold is Sasakian [14].

It is well known that the tangent sphere bundle of a flat Riemannian manifold admits a contact metric structure satisfying [15]. Again on a Sasakian manifold [16] we have

As a generalization of both and the Sasakian case, Blair et al. [17] introduced the -nullity distribution on a contact metric manifold and gave several reasons for studying it. The -nullity distribution [17] of a contact metric manifold is defined by for all , where . A contact metric manifold with is called a -contact metric manifold. If , the -nullity distribution reduces to -nullity distribution [18]. The -nullity distribution of a Riemannian manifold is defined by [18] with being a constant. If the characteristic vector field , then we call a contact metric manifold as -contact metric manifold [19]. If , then the manifold is Sasakian, and if , then the manifold is locally isometric to the product for and flat for [15].

Given a non-Sasakian -contact manifold , Boeckx [20] introduced an invariant and showed that, for two non-Sasakian ()-manifolds and, we haveif and only if, up to a -homothetic deformation, the two manifolds are locally isometric as contact metric manifolds.

Thus we see that from all non-Sasakian -manifolds of dimension and for every possible value of the invariant , one -manifold can be obtained with . For such examples may be found from the standard contact metric structure on the tangent sphere bundle of a manifold of constant curvature , where we have . Boeckx also gives a Lie algebra construction for any odd dimension and value of .

Using this invariant, Blair et al. [19] constructed an example of a -dimensional -contact metric manifold, . The example is given in the following.

Since the Boeckx invariant for a -manifold is , we consider the tangent sphere bundle of an -dimensional manifold of constant curvature so choosing that the resulting -homothetic deformation will be a -manifold. That is, for and we solve for and . The result is and taking and to be these values we obtain -contact metric manifold.

However, for a -contact metric manifold of dimension , we have [19] where ,

In a -dimensional almost contact metric manifold, if is a local orthonormal basis of the tangent space of the manifold, then is also a local orthonormal basis. It is easy to verify that for . In particular in view of , we get

Here we state a lemma due to Baikoussis and Koufogiorgos [21] which will be used in this paper.

Lemma 2.1. Let be an -Einstein manifold of dimension . If belongs to the -nullity distribution, then and the structure is Sasakian.

3. Conharmonically Symmetric -Contact Metric Manifolds

In this section we study conharmonically symmetric -contact metric manifolds. Differentiating (1.3) covariantly with respect to , we obtain Therefore for conharmonically symmetric -contact metric manifolds we have

Differentiating (2.12) covariantly with respect to and using (2.15) we obtain

Again, differentiating (2.14) covariantly with respect to and using (2.16) and (2.17) we have Therefore we have Putting in (3.2) and using (3.3), (3.4), and (3.5) we obtain

Taking inner product of (3.6) with and using (2.1) we obtain From (3.7) we get, either or Putting instead of in (3.8) and using (2.12) we obtain Using (3.9) in (3.7) yields The relation (3.10) gives , since gives (by putting ), which is not the case for a -contact metric manifold, in general.

Therefore in either case we obtain .

Hence we have the following.

Theorem 3.1. A conharmonically symmetric -dimensional -contact metric manifold is locally isometric to the product .

Remark 3.2. The converse of the above theorem is not true in general. However if , then we get , and hence from the definition of the conharmonic curvature tensor we obtain , that is, the manifold under consideration is -conharmonically flat. Thus if an -contact manifold is locally isometric to , then the manifold is -conharmonically flat.

4. -Conharmonically Flat -Contact Metric Manifolds

In this section we consider a -dimensional -conharmonically flat -contact metric manifolds. Then from (1.3) we obtain

Using (2.1), (2.13), and (2.15) in (4.1) we obtain Putting in (4.2) and using (2.1) and (2.15) we get Taking inner product with of (4.3) yields

From relation (4.4), we conclude that the manifold is an -Einstein manifold.

Conversely, we assume that a -dimensional -contact manifold satisfies the relation (4.4). Then we easily obtain from (1.3) that .

In view of the above discussions we state the following.

Theorem 4.1. A -dimensional -contact metric manifold is -conharmonically flat if and only if it is an -Einstein manifold.

Hence in view of Lemma 2.1 we state the following.

Corollary 4.2. Let be a -dimensional -conharmonically flat -contact metric manifold, then and the structure is Sasakian.

Let be a local orthonormal basis of the tangent space of the manifold. Putting in (4.4) and summing up from 1 to we obtain in view of (2.18) and (2.19) that Therefore we have the following corollary.

Corollary 4.3. In a -dimensional -conharmonically flat -contact metric manifold, the scalar curvature vanishes.

5. -Conharmonically Flat -Contact Metric Manifolds

This section deals with a -dimensional -conharmonically flat -contact metric manifold. Then we have from (1.3) that

Let be a local orthonormal basis of the tangent space of the manifold. Then is also a local orthonormal basis of the tangent space. Putting in (5.1) and summing up from 1 to we have

Using (2.18), (2.19), (2.20), and (2.21) in (5.2) we obtain

Replacing and by and in (5.3) and using (2.1) we have

Putting in (5.4) and taking summation over to we get by using (2.18) and (2.19) that In view of the above discussions we have the following.

Proposition 5.1. A -dimensional -conharmonically flat -contact metric manifold is an -Einstein manifold with vanishing scalar curvature.

Therefore in view of the Lemma 2.1 we state the following theorem.

Theorem 5.2. A -dimensional -conharmonically flat -contact metric manifold is a Sasakian manifold with vanishing scalar curvature.

Definition 5.3. In a -dimensional -contact metric manifold, if the Ricci tensor satisfies , then the Ricci tensor is said to be -parallel.

The notion of -parallel Ricci tensor for Sasakian manifold was introduced by Kon [22].

Putting in (5.4) we have

Replacing and by and in (5.6) and using (2.1) we obtain

Relation (5.7) yields since is a constant. Therefore we have the following corollary.

Corollary 5.4. A -dimensional -conharmonically flat -contact metric manifold satisfies -parallel Ricci tensor.