On the Conharmonic Curvature Tensor of Generalized Sasakian-Space-Forms
The object of the present paper is to characterize generalized Sasakian-space-forms satisfying certain curvature conditions on conharmonic curvature tensor. In this paper we study conharmonically semisymmetric, conharmonically flat, -conharmonically flat, and conharmonically recurrent generalized Sasakian-space-forms. Also generalized Sasakian-space-forms satisfying and have been studied.
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  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 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 .
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  who introduced the conharmonic transformation as a subgroup of the conformal transformation (1.1) satisfying the condition where comma denotes the covariant differentiation with respect to metric . A rank-four tensor that remains invariant under conharmonic transformation for -dimensional Riemannian manifold is given by where and denote the Riemannian curvature tensor of type defined by and the Ricci tensor of type , 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 , Siddiqui and Ahsan , Özgür , 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 the direct sum , where is the 1-dimensional linear subspace of generated by . Thus the conformal curvature tensor is a map An almost contact metric manifold is said to be (1) conformally symmetric  if the projection of the image of in is zero, (2)-conformally flat  if the projection of the image of in is zero, (3)-conformally flat  if the projection of the image of in is zero.
Here cases (1), (2), and (3) are synonymous to conformally symmetric, conformally flat and conformally flat. In , it is proved that a conformally symmetric -contact manifold is locally isometric to the unit sphere. In , it is proved that a -contact manifold is -conformally flat if and only if it is an Einstein Sasakian manifold. In , some necessary conditions for -contact manifold to be -conformally flat are proved. Moreover, in  some conditions on conharmonic curvature tensor are studied which has many applications in physics and mathematics on a hypersurfaces in the semi-Euclidean space . Also, it is shown that every conharmonically Ricci-semisymmetric hypersurface satisfies the the condition is pseudosymmetric.
On the other hand a generalized Sasakian-space-form was defined by Alegre et al.  as the almost contact metric manifold whose curvature tensor is given by where are some differential functions on and for any vector fields on . In such a case we denote the manifold as . This kind of manifold appears as a generalization of the well-known Sasakian-space-forms by taking , . It is known that any three-dimensional trans-Sasakian manifold with depending on is a generalized Sasakian-space-form . Alegre et al. give results in  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 , Kim studied conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms. De and Sarkar  have studied generalized Sasakian-space-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 Sasakian-space-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. -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 -conharmonically flat. In Section 6, conharmonically recurrent generalized Sasakian-space-forms are studied. Section 7 is devoted to study generalized Sasakian-space-forms satisfying . The last section contains generalized Sasakian-space-forms satisfying .
If, on an odd-dimensional differentiable manifold of differentiability class , there exists a vector valued real linear function , a 1-form , the associated vector field , and the Riemannian metric satisfying for arbitrary vector fields and , then is said to be an almost contact metric manifold , and the structure is called an almost contact metric structure to . In view of (2.1), (2.2) and (2.3), we have Again we know  that in a -dimensional generalized Sasakian-space-form for all vector fields on , where denotes the curvature tensor of : We also have for a generalized Sasakian-space-forms where is the Ricci operator, that is, .
A generalized Sasakian space-form is said to be -Einstein if its Ricci tensor is of the form: for arbitrary vector fields and , where and are smooth functions on . For a -dimensional almost contact metric manifold the conharmonic curvature tensor is given by : The conharmonic curvature tensor in a generalized Sasakian-space-form satisfies
3. Conharmonically Semisymmetric Generalized Sasakian-Space-Forms
Definition 3.1. A dimensional generalized Sasakian-space-form is said to be conharmonically semisymmetric  if it satisfies , where is the Riemannian curvature tensor, and is the conharmonic curvature tensor of the space-forms.
Theorem 3.2. A dimensional generalized Sasakian-space-form is conharmonically semisymmetric if and only if .
Proof. Let us suppose that the generalized Sasakian-space-form is conharmonically semisymmetric. Then we can write
The above equation can be written as
In view of (2.10) the above equation reduces to
Now, taking the inner product of above equation with and using (2.2) and (2.17), we get From the above equation, we have either or which by using (2.15) and (2.16) gives which is not possible in generalized Sasakian-space-form. Conversely, if , then from (2.10), we have . Then obviously is satisfied. This completes the proof.
4. Conharmonically Flat Generalized Sasakian-Space-Forms
Theorem 4.1. A dimensional generalized Sasakian-space-form is conharmonically flat if and only if .
Proof. For a dimensional conharmonically flat generalized Sasakian-space-form, we have from (2.15) In view of (2.6) and (2.7) the above equation takes the form By virtue of (2.5) the above equation reduces to Now, replacing by in the above equation, we obtain Putting in the above equation, we get Since , in general, we obtain Again replacing by in (4.3), we get which, by putting , gives Since , in general, we obtain From (4.6) and (4.9), we have Thus in view of (4.9) and (4.10), we have Conversely, suppose that satisfies in generalized Sasakian-space-form, and then we have Also, in view of (2.15), we have where and . Putting in (4.14) and taking summation over , we get In view of (2.5) and (4.14), we have Now, putting in above equation and taking summation over , , we get In view of (4.12), (4.15) and (4.17), we have Putting in above equation and taking summation over , , we get . Then in view of (4.11), . Therefore, we obtain from (2.5) Hence in view of (4.12), (4.13) and (4.19), we have . This completes the proof.
5. -Conharmonically Flat Generalized Sasakian-Space-Forms
Definition 5.1. A dimensional generalized Sasakian-space-form is said to be -conharmonically flat  if for all .
Theorem 5.2. A dimensional generalized Sasakian-space-form is -conharmonically flat if and only if it is Einstein manifold.
Proof. Let us consider that a generalized Sasakian-space-form is -conharmonically flat, that is, . Then in view of (2.15), we have In virtue of (2.9) and (2.12) the above equation reduces to which by putting gives Now, taking the inner product of the above equation with , we get which shows that generalized Sasakian-space-form is an -Einstein manifold. Conversely, suppose that (5.4) is satisfied. Then by virtue of (5.1) and (5.3), we have . This completes the proof.
6. Conharmonically Recurrent Generalized Sasakian-Space-Forms
Definition 6.1. A nonflat Riemannian manifold is said to be conharmonically recurrent if its conharmonic curvature tensor satisfies the condition where is nonzero 1-form.
Theorem 6.2. A dimensional generalized Sasakian-space-form is conharmonically recurrent if and only if .
Proof. We define a function on , where the metric is extended to the inner product between the tensor fields. Then we have
This can be written as
From the above equation, we have
Since the left hand side of the above equation is identically zero and on , then
that is, 1-form 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 . Converse follows from retreating the steps.
Corollary 6.3. Conharmonically recurrent generalized Sasakian-space-form is conharmonically semisymmetric.
Proof. Proof follows from the above theorem.
7. Generalized Sasakian-Space-Forms Satisfying
Theorem 7.1. A dimensional generalized Sasakian-space-form satisfying is an -Einstein manifold.
Proof. Let us consider generalized Sasakian-space-form satisfying . In this case we can write In view of (2.18) the above equation reduces to Now, putting in the above equation, we get In virtue of (2.6) the above equation takes the form: where . This completes the proof.
8. Generalized Sasakian-Space-Forms Satisfying
Theorem 8.1. A -dimensional generalized Sasakian-space-form satisfying is an -Einstein manifold.
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 , we get In consequence of (2.5), (2.9), (2.10), and (2.11) the above equation takes the form: Putting in the above equation and taking summation over , , we get which shows that is an Einstein manifold. This completes the proof.
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