Abstract

We study a class of para-Kenmotsu manifolds admitting Weyl-projective curvature tensor of type (1, 3). At the end, it is shown that an n-dimensional (n > 2) P-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein manifold.

1. 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 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.

2. Preliminaries

Let M_n be an n-dimensional differentiable manifold equipped with structure tensors (, ξ, η) where is a tensor of type (1, 1), ξ is a vector field, η is a 1-form such that

(2.1)

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

(2.2)

Then the manifold M_n ¹ is said to admit an almost para-contact Riemannian structure

In addition, if satisfies the conditions

(2.3)

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

(2.4)

where is an associate of is called special para-Kenmotsu manifold or briefly SP- Kenmotsu manifold ⁸.

Let be an n-dimensional, 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:

(2.5)

The Ricci operator S and the (0, 2)- tensor are defined by

(2.6)

and

(2.7)

It is known ⁸ that in a P-Kenmotsu manifold the following relations hold:

(2.8)

when is orthogonal to 𝛏.

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

(2.9)

where is a constant.

3. - Recurrent P-Kenmotsu Manifolds

The Weyl-projective curvature tensor of type (1, 3) of a Riemannian manifold with respect to Riemannian connection is given by ⁹:

(3.1)

Now, we define a -semisymmetric para-Kenmotsu manifold as:

Definition 3.1: An n-dimensional para-Kenmotsu manifold is called -semisymmetric if its -curvature tensor satisfies the condition

(3.2)

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 -curvature tensor satisfies the condition

(3.3)

Further, we define a -recurrent para-Kenmotsu manifold as:

Definition 3.2: An n-dimensional para-Kenmotsu manifold with respect to the Levi-Civita connection is called -recurrent manifold if its -curvature tensor satisfies the condition

(3.4)

where A is some non-zero 1-form.

Now, let us establish a relation between -recurrent and -semisymmetric para-Kenmotsu manifolds.

For that, let us suppose that Now, we define a function by

(3.5)

Using the fact that from (3.5) we get

Since we have

(3.6)

Then, from (3.6), we get

(3.7)

and hence, we have

(3.8)

Therefore,

(3.9)

Since the left hand side of (3.9) is zero and we deduce that and it shows that the 1-form A is closed.

Then from (3.4), we get that

(3.10)

and hence, we get that

(3.11)

i.e., 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 -recurrent P-Kenmotsu manifold is -semisymmetric and hence we state that:

Theorem 3.1: A -recurrent para-Kenmotsu manifold is -semisymmetric.

Further we determine the curvature value of -semisymmetric P-Kenmotsu manifold.

From (3.2), we have

(3.12)

which implies

(3.13)

By putting in the above equation, we get

(3.14)

Now, by using (2.8) and (3.3), the above equation reduces to:

(3.15)

Again on using (3.3), we get that

Therefore, from (3.1) we have

(3.16)

On contracting the above equation, we get

(3.17)

Then, from equations (3.16) and (3.17), we have

(3.18)

This shows that the curvature of -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 -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 -recurrent P-Kenmotsu manifold is an SP-Kenmotsu manifold.

4. Ricci Semisymmetric Para-Kenmotsu Manifolds

Definition 4.1: An n-dimensional Riemannian manifold is said to be Ricci semisymmetric if its Ricci tensor of the Levi-Civita connection satisfies the condition

(4.1)

Theorem 4.1: An n-dimensional (n > 2) P-Kenmotsu manifold is Ricci semisymmetric if and only if it is an Einstein Manifold.

Proof: Let us suppose that a P-Kenmotsu manifold be Ricci semisymmetric. Then from (4.1), we have

(4.2)

By putting in (4.2), we get

(4.3)

Now by using the equations (2.8) (a) and (2.8) (c), the above equation reduces to

(4.4)

Again by putting in (4.4), we get

(4.5)

This proves that the manifold is an Einstein manifold.

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.

Acknowledgements

The authors acknowledge Prof. Kalpana of Banaras Hindu University, Varanasi for her valuable suggestions in preparation of the manuscript.

References