On the Conharmonic Curvature Tensor of Generalized Sasakian-Space-Forms

Conformal transformations of a Riemannian structures are an important object of study in differential geometry. Of considerable interest in a special type of conformal transformations, conharmonic transformations, which are conformal transformations are preserving the harmonicity property of smooth functions. This type of transformation was introduced by Ishii 1 in 1957 and is now studied from various points of view. It is well known that such transformations have a tensor invariant, the so-called conharmonic curvature tensor. It is easy to verify that this tensor is an algebraic curvature tensor; that is, it possesses the classical symmetry properties of the Riemannian curvature tensor. LetM andM be two Riemannianmanifolds with g and g being their respective metric tensors related through


Introduction
Conformal transformations of a Riemannian structures are an important object of study in differential geometry.Of considerable interest in a special type of conformal transformations, conharmonic transformations, which are conformal transformations are preserving the harmonicity property of smooth functions.This type of transformation was introduced by Ishii 1 in 1957 and is now studied from various points of view.It is well known that such transformations have a tensor invariant, the so-called conharmonic curvature tensor.It is easy to verify that this tensor is an algebraic curvature tensor; that is, it possesses the classical symmetry properties of the Riemannian curvature tensor.
Let M and M be two Riemannian manifolds with g and g being their respective metric tensors related through g X, Y e 2σ g X, Y , 1.
where σ is a real function.Then M and M are called conformally related manifolds, and the correspondence between M and M is known as conformal transformation 2 .
It is known that a harmonic function is defined as a function whose Laplacian vanishes.A harmonic function is not invariant, in general.The conditions under which a harmonic function remains invariant have been studied by Ishii 1 who introduced the conharmonic transformation as a subgroup of the conformal transformation 1.1 satisfying the condition σ, i i σ, i σ, i 0, 1.2 where comma denotes the covariant differentiation with respect to metric g.A rank-four tensor C that remains invariant under conharmonic transformation for 2n 1 -dimensional Riemannian manifold is given by where R and S denote the Riemannian curvature tensor of type 0, 4 defined by R X, Y, Z, U g R X, Y Z, U and the Ricci tensor of type 0, 2 , respectively.The curvature tensor defined by 1.3 is known as conharmonic curvature tensor.A manifold whose conharmonic curvature tensor vanishes at every point of the manifold is called conharmonically flat manifold.Thus this tensor represents the deviation of the manifold from canharmonic flatness.Conharmonic curvature tensor has been studied by Abdussattar 3 , Siddiqui and Ahsan 2 , Özg ür 4 , and many others.
Let M be an almost contact metric manifold equipped with an almost contact metric structure φ, ξ, η, g .At each point p ∈ M, decompose the tangent space T p M into the direct sum T p M φ T p M ⊕{ξ p }, where {ξ p } is the 1-dimensional linear subspace of T p M generated by ξ p .Thus the conformal curvature tensor C is a map An almost contact metric manifold M is said to be Here cases 1 , 2 , and 3 are synonymous to conformally symmetric, ξ-conformally flat and φ-conformally flat.In 5 , it is proved that a conformally symmetric K-contact manifold is locally isometric to the unit sphere.In 6 , it is proved that a K-contact manifold is ξconformally flat if and only if it is an η-Einstein Sasakian manifold.In 7 , some necessary conditions for K-contact manifold to be φ-conformally flat are proved.Moreover, in 8 some conditions on conharmonic curvature tensor C are studied which has many applications in physics and mathematics on a hypersurfaces in the semi-Euclidean space E n 1 S .Also, it is shown that every conharmonically Ricci-semisymmetric hypersurface M satisfies the the condition C • R 0 is pseudosymmetric.
On the other hand a generalized Sasakian-space-form was defined by Alegre et al. 9 as the almost contact metric manifold M 2n 1 , φ, ξ, η, g whose curvature tensor R is given by where f 1 , f 2 , f 3 are some differential functions on M and for any vector fields X, Y, Z on M 2n 1 .In such a case we denote the manifold as M f 1 , f 2 , f 3 .This kind of manifold appears as a generalization of the well-known Sasakian-space-forms by taking It is known that any three-dimensional α, β -trans-Sasakian manifold with α, β depending on ξ is a generalized Sasakian-spaceform 10 .Alegre et al. give results in 11 about B. Y Chen's inequality on submanifolds of generalized complex space-forms and generalized Sasakian-space-forms. Al-Ghefari et al. analyse the CR submanifolds of generalized Sasakian-space-forms 12, 13 .In 14 , Kim studied conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms. De and Sarkar 15 have studied generalized Sasakianspace-forms regarding projective curvature tensor.Motivated by the above studies, in the present paper, we study flatness and symmetry property of generalized Sasakian-space-forms regarding conharmonic curvature tensor.The present paper is organized as follows.
In this paper, we study the conharmonic curvature tensor of generalized Sasakianspace-forms.In Section 2, some preliminary results are recalled.In Section 3, we study conharmonically semisymmetric generalized Sasakian-space-forms. Section 4 deals with conharmonically flat generalized Sasakian-space-forms. ξ-conharmonically flat generalized Sasakian-space-forms are studied in Section 5 and obtain necessary and sufficient condition for a generalized Sasakian-space-form to be ξ-conharmonically flat.In Section 6, conharmonically recurrent generalized Sasakian-space-forms are studied.Section 7 is devoted to study generalized Sasakian-space-forms satisfying C • S 0. The last section contains generalized Sasakian-space-forms satisfying C • R 0.

Preliminaries
If, on an odd-dimensional differentiable manifold M 2n 1 of differentiability class C r 1 , there exists a vector valued real linear function φ, a 1-form η, the associated vector field ξ, and the Riemannian metric g satisfying for arbitrary vector fields X and Y , then M 2n 1 , g is said to be an almost contact metric manifold 16 , and the structure φ, ξ, η, g is called an almost contact metric structure to M 2n 1 .In view of 2.1 , 2.2 and 2.3 , we have

2.4
Again we know 9 that in a 2n 1 -dimensional generalized Sasakian-space-form for all vector fields X, Y, Z on M 2n 1 , where R denotes the curvature tensor of M 2n 1 : We also have for a generalized Sasakian-space-forms where Q is the Ricci operator, that is, g QX, Y S X, Y .
A generalized Sasakian space-form is said to be η-Einstein if its Ricci tensor S is of the form: S X, Y ag X, Y bη X η Y , 2.14 for arbitrary vector fields X and Y , where a and b are smooth functions on M 2n 1 .For a 2n 1 -dimensional n > 1 almost contact metric manifold the conharmonic curvature tensor C is given by 17 :

2.15
The conharmonic curvature tensor C in a generalized Sasakian-space-form satisfies 2.20

ISRN Geometry
Proof.Let us suppose that the generalized Sasakian-space-form M f 1 , f 2 , f 3 is conharmonically semisymmetric.Then we can write The above equation can be written as In view of 2.10 the above equation reduces to

3.3
Now, taking the inner product of above equation with ξ and using 2.2 and 2.17 , we get From the above equation, we have either which by using 2.15 and 2.16 gives which is not possible in generalized Sasakian-space-form.Conversely, if f 1 f 3 , then from 2.10 , we have R ξ, U 0. Then obviously R• C 0 is satisfied.This completes the proof.

Conharmonically Flat Generalized Sasakian-Space-Forms
Theorem 4.1.A 2n 1 -dimensional n > 1 generalized Sasakian-space-form is conharmonically flat if and only if Proof.For a 2n 1 -dimensional n > 1 conharmonically flat generalized Sasakian-spaceform, we have from 2.15 In view of 2.6 and 2.7 the above equation takes the form

4.2
By virtue of 2.5 the above equation reduces to

4.3
Now, replacing Z by φZ in the above equation, we obtain Putting X ξ in the above equation, we get Since g Y, φZ / 0, in general, we obtain Again replacing X by φX in 4.  e i , e i , U S X, U .

4.15
In view of 2.5 and 4.14 , we have Now, putting Y Z e i in above equation and taking summation over i,

4.17
In view of 4.12 , 4.15 and 4.17 , we have Putting X W e i in above equation and taking summation over i, 1 ≤ i ≤ 2n 1, we get f 1 0. Then in view of 4.11 , f 2 f 3 0. Therefore, we obtain from 2.5 R X, Y Z 0.

4.19
Hence in view of 4.12 , 4.13 and 4.19 , we have C X, Y Z 0. This completes the proof.
Proof.Let us consider that a generalized Sasakian-space-form is ξ-conharmonically flat, that is, C X, Y ξ 0. Then in view of 2.15 , we have In virtue of 2.9 and 2.12 the above equation reduces to which by putting Y ξ gives Now, taking the inner product of the above equation with U, we get which shows that generalized Sasakian-space-form is an η-Einstein manifold.Conversely, suppose that 5.4 is satisfied.Then by virtue of 5.1 and 5.3 , we have C X, Y ξ 0. This completes the proof.Proof.We define a function f 2 g C, C on M 2n 1 , where the metric g is extended to the inner product between the tensor fields.Then we have

Conharmonically Recurrent
This can be written as From the above equation, we have Since the left hand side of the above equation is identically zero and f / 0 on M 2n 1 , then dA X, Y 0, 6.5 that is, 1-form A is closed.

Now from
we have In view of 6.5 and 6.7 , we have Thus in view of Theorem 3.2, we have f 1 f 3 .Converse follows from retreating the steps.
Proof.Proof follows from the above theorem.Proof.Let us consider generalized Sasakian-space-form M 2n 1 satisfying C ξ, X • S 0. In this case we can write In view of 2.18 the above equation reduces to Now, putting Z ξ in the above equation, we get In virtue of 2.6 the above equation takes the form: where K 3f 2 − 2 n − 1 f 3 .This completes the proof.

Generalized Sasakian-Space-Forms Satisfying
Proof.Let generalized Sasakian-space-form satisfying This can be written as which on using 2.18 takes the following form: Now taking the inner product of the above equation with ξ, we get In consequence of 2.5 , 2.9 , 2.10 , and 2.11 the above equation takes the form: − f 2 g Y, φU g X, φZ − g Z, φU g X, φY 2g Y, φZ g φU, X − f 2 g Y, φU S X, φZ − g Z, φU S X, φY 2g Y, φZ S φU, X Putting Z U e i in the above equation and taking summation over i, 1 ≤ i ≤ 2n 1, we get which shows that M 2n 1 is an η-Einstein manifold.This completes the proof.
Definition 3.1.A 2n 1 -dimensional n > 1 generalized Sasakian-space-form is said to be conharmonically semisymmetric 15 if it satisfies R • C 0, where R is the Riemannian curvature tensor, and C is the conharmonic curvature tensor of the space-forms.