On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3)

In [1,2], Sato introduced the notions of an almost para contact Riemannian manifold. In 1977, Adati and Matsumoto defined para-Sasakian and special para-Sasakian manifolds, which are regarded as a special kind of an almost contact Riemannian manifolds [3]. Para-Sasakian manifolds have been studied by Adati and Miyazawa [4], De and Avijit [5], Matsumoto, Ianus and Mihai [6] and many others. Before Sato, Kenmotsu defined a class of almost contact Riemannian manifolds [7]. In 1995, Sinha and Sai Prasad defined a class of almost para contact metric manifolds namely para-Kenmotsu (briefly P-Kenmotsu) and special para-Kenmotsu (briefly SP-Kenmotsu) manifolds [8]. In 1970, Pokhariyal and Mishra introduced new tensor fields, called W2 and E tensor fields, on a Riemannian manifold [9]. Later, in [10], Pokhariyal studied some of the properties of these tensor fields on a Sasakian manifold. In 1986, Matsumoto, Ianus and Mihai have extended these concepts to almost para-contact structures and studied para-Sasakian manifolds admitting these tensor fields [6]. These results were further generalised by De and Sarkar, in [5]. Motivated by these studies, in 2015, Sai Prasad and Satyanarayana studied W2-tensor field in an SP-Kenmotsu manifold [11]. In the present work, we investigate a class of para-Kenmotsu manifolds admitting Weyl-projective curvature tensor W2 of type (1, 3). The present work is organised as follows: Section 2 is equipped with some prerequisites about P-Kenmotsu manifolds. In Section 3, we define W2-recurrent and semisymmetric para-Kenmotsu manifolds and shown that W2-recurrent para-Kenmotsu manifold is a semisymmetric manifold. Further, it is shown that the curvature of W2-semisymmetric para-Kenmotsu manifold is constant and hence we establish that a W2-recurrent para-Kenmotsu manifold is an SP-Kenmotsu manifold. Section 4 is devoted to study Ricci semisymmetric P-Kenmotsu manifold.


Introduction
In [1,2], Sato introduced the notions of an almost para contact Riemannian manifold. In 1977, Adati and Matsumoto defined para-Sasakian and special para-Sasakian manifolds, which are regarded as a special kind of an almost contact Riemannian manifolds [3]. Para-Sasakian manifolds have been studied by Adati and Miyazawa [4], De and Avijit [5], Matsumoto, Ianus and Mihai [6] and many others. Before Sato, Kenmotsu defined a class of almost contact Riemannian manifolds [7]. In 1995, Sinha and Sai Prasad defined a class of almost para contact metric manifolds namely para-Kenmotsu (briefly P-Kenmotsu) and special para-Kenmotsu (briefly SP-Kenmotsu) manifolds [8].
In 1970, Pokhariyal and Mishra introduced new tensor fields, called W 2 and E tensor fields, on a Riemannian manifold [9]. Later, in [10], Pokhariyal studied some of the properties of these tensor fields on a Sasakian manifold. In 1986, Matsumoto, Ianus and Mihai have extended these concepts to almost para-contact structures and studied para-Sasakian manifolds admitting these tensor fields [6]. These results were further generalised by De and Sarkar,in [5]. Motivated by these studies, in 2015, Sai Prasad and Satyanarayana studied W 2 -tensor field in an SP-Kenmotsu manifold [11]. In the present work, we investigate a class of para-Kenmotsu manifolds admitting Weyl-projective curvature tensor W 2 of type (1,3). The present work is organised as follows: Section 2 is equipped with some prerequisites about P-Kenmotsu manifolds. In Section 3, we define W 2 -recurrent and semisymmetric para-Kenmotsu manifolds and shown that W 2 -recurrent para-Kenmotsu manifold is a semisymmetric manifold. Further, it is shown that the curvature of W2-semisymmetric para-Kenmotsu manifold is constant and hence we establish that a W 2 -recurrent para-Kenmotsu manifold is an SP-Kenmotsu manifold. Section 4 is devoted to study Ricci semisymmetric P-Kenmotsu manifold.

Preliminaries
Let M n be an n-dimensional differentiable manifold equipped with structure tensors ( Φ , ξ, η) where Φ is a tensor of type (1, 1), ξ is a vector field, η is a 1-form such that Then the manifold M n is called an almost para-contact manifold.
Let g be a Riemannian metric such that, for all vector fields and on M n Then the manifold M n [1] is said to admit an almost para-contact Riemannian structure (Φ, , , ).
In addition, if (Φ, , , ) satisfies the conditions Turkish Journal of Analysis and Number Theory 156 then M n is called para-Kenmotsu manifold or briefly a P-Kenmotsu manifold [8].
A P-Kenmotsu manifold admitting a 1-form η satisfying where is an associate of Φ, is called special para-Kenmotsu manifold or briefly SP-Kenmotsu manifold [8]. Let ( , ) be an n-dimensional, ≥ 3, differentiable manifold of class ∞ and let ∇ be its Levi-Civita connection. Then the Riemannian Christoffel curvature tensor R of type (1, 3) is given by: , .
The Ricci operator S and the (0, 2)-tensor 2 are defined by It is known [8] that in a P-Kenmotsu manifold the following relations hold: when is orthogonal to . An n-dimensional (n > 2) Riemannian manifold is said to be Einstein manifold if the Ricci curvature tensor S(X, Y) of the Levi-Civita connection satisfies the condition where is a constant.

-Recurrent P-Kenmotsu Manifolds
The Weyl-projective curvature tensor 2 of type (1, 3) of a Riemannian manifold with respect to Riemannian connection is given by [9]: , , Now, we define a 2 -semisymmetric para-Kenmotsu manifold as: Definition 3.1: An n-dimensional para-Kenmotsu manifold is called 2 -semisymmetric if its 2 -curvature tensor satisfies the condition where ( , ) is considered to be a derivation of the tensor algebra at each point of the manifold for tangent vectors and . It can be easily shown that on a P-Kenmotsu manifold the 2 -curvature tensor satisfies the condition Further, we define a 2 -recurrent para-Kenmotsu manifold as: Definition 3.2: An n-dimensional para-Kenmotsu manifold with respect to the Levi-Civita connection is called 2 -recurrent manifold if its 2 -curvature tensor satisfies the condition where A is some non-zero 1-form. Now, let us establish a relation between 2 -recurrent and 2 -semisymmetric para-Kenmotsu manifolds.

Uf f A U
Then, from (3.6), we get and hence, we have Since the left hand side of (3.9) is zero and ≠ 0 , we deduce that ( , ) = 0 and it shows that the 1-form A is closed.
Then from (3.4), we get that and hence, we get that i.e., ( , ). 2 = 0, where ( , ) is considered to be a derivation of tensor algebra at each point of the manifold for the tangent vectors V and U. This shows that a 2 -recurrent P-Kenmotsu manifold is 2 -semisymmetric and hence we state that: Theorem 3.1: A 2 -recurrent para-Kenmotsu manifold is 2 -semisymmetric. Further we determine the curvature value of 2 -semisymmetric P-Kenmotsu manifold. From (3.2), we have Now, by using (2.8) and (3.3), the above equation reduces to: Again on using (3.3), we get that 2 ( , , , ) = 0. Therefore, from (3.1) we have On contracting the above equation, we get Then, from equations (3.16) and (3.17), we have This shows that the curvature of 2 -semisymmetric P-Kenmotsu manifold is constant.
As it is known [8] that a P-Kenmotsu manifold with constant curvature is an SP-Kenmotsu manifold and using the above shown result, we state that: Theorem 3.2: A W 2 -semisymmetric P-Kenmotsu manifold is an SP-Kenmotsu manifold.
Therefore, form theorems (3.1) and (3.2), we have the following result: Theorem 3.3: A 2 -recurrent P-Kenmotsu manifold is an SP-Kenmotsu manifold.  This proves that the manifold is an Einstein manifold.

Ricci Semisymmetric Para-Kenmotsu Manifolds
As an every Einstein manifold is Ricci semisymmetric, the converse of the theorem is trivial.
This completes the proof.

Statement of Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.