1. Introduction

The notion of an almost para contact Riemannian manifold was introduced by Sato ^[10], in 1976. After that, Adati and Matsumoto ^[1] further defined and studied -Sasakian and -Sasakian manifolds which are regarded as a special kind of an almost contact Riemannian manifolds. Before Sato, in 1972, Kenmotsu ^[9] defined a class of almost contact Riemannian manifold. In 1995, Sinha and Sai Prasad ^[13] have defined a class of almost para contact metric manifolds namely, para-Kenmotsu (briefly -Kenmotsu) and special para-Kenmotsu (briefly -Kenmotsu) manifolds.

Let (, g) be an n-dimensional, , differentiable manifold of class . Let be its Levi-Civita connection, be the Riemannian Christoffel curvature tensor and be the Weyl conformal curvature tensor is defined by

(1.1)

where is the Ricci operator, is the Ricci tensor and is the scalar curvature of ^[4]. The Ricci operator and the (0,2)-tensor are defined by

(1.2)

and

(1.3)

A manifold is conformally flat if and . If C = 0 then is called conformally symmetric and hence it is Weyl-semisymmetric ^[5].

For a (0, k)-tensor field T, , on (, g) we define the tensors , and Q(g, T) respectively as ^[8]:

(1.4)

(1.5)

(1.6)

where the endomorphism is defined by

(1.7)

If the tensors and are linearly dependent then the manifold is called Weyl-pseudosymmetric ^[8], and it is same as

(1.8)

which holds on the set = {x M: C at x}, where is some function of . If , then the manifold is called Weyl-semisymmetric ^[8].

Locally symmetric, semisymmetric and Pseudosymmetric Para-Sasakian manifolds are widely studied by many geometers ^{[2, 6, 7]}. By studying Weyl-semisymmetric para-Kenmotsu manifolds, Satyanarayana and Sai Prasad have shown that such a manifold is conformally flat and hence it is an SP-Kenmotsu manifold ^[12]. Later, they extended their work to find the characterisations of the Weyl-pseudosymmetric para-Kenmotsu manifolds which are regarded as the extended classes of Weyl-semisymmetric para-Kenmotsu manifolds.

Through this study, we could obtain the characterisations of the para-Kenmotsu manifolds with the conditions =0 and .

2. Para-Kenmotsu Manifold

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

(2.1)

(2.2)

Then is called an almost para contact manifold.

Let be the Riemannian metric satisfying such that, for all vector fields and on ,

(2.3)

(2.4)

(2.5)

Then the manifold ^[10] is said to admit an almost para contact Riemannian structure .

A manifold of dimension with Riemannian metric admitting a tensor field of type (1, 1), a vector field and a 1-form satisfying (2.1), (2.3) along with

(2.6)

(2.7)

(2.8)

(2.9)

is called a para-Kenmotsu manifold or briefly -Kenmotsu manifold ^[13], where is the covariant differentiation with respect to the metric .

Let be an -dimensional Riemannian manifold admitting a tensor field of type (1, 1), a vector field and a 1-form satisfying

(2.10)

(2.11)

Then is called special para-Kenmotsu manifold or in brief-Kenmotsu manifold ^[13].

It is known that, in a P-Kenmotsu manifold the following relations hold good ^[13]:

(2.12)

(2.13)

(2.14)

(2.15)

(2.16)

(2.17)

(2.18)

where is the Ricci operator.

An almost para-contact Riemannian manifold is said to be an -Einstein manifold ^[11] if the Ricci curvature tensor is of the form

(2.19)

where and are smooth functions on . In particular, if then it is said to be an Einstein manifold ^[11].

Moreover, it is also known that if a P-Kenmotsu manifold is projectively flat then it is an Einstein manifold and its scalar curvature has a negative constant value . Especially, if a P-Kenmotsu manifold is of constant curvature, then scalar curvature has a negative constant value ^[13] and in this case

(2.20)

(2.21)

(2.22)

3. Main Results

Through this study, we could obtain the characterisations of para-Kenmotsu manifolds with the conditions = 0 and .

The tensors and on (, g) are defined by:

(3.1)

and

(3.2)

Let () be a para-Kenmotsu manifold. Then from (3.1), (3.2) and (1.1), we have

(3.3)

and

(3.4)

By multiplying (3.3) and (3.4) with and on using the condition , we get

(3.5)

By putting in (3.5) and on using (2.12) and (2.13), we get

(3.6)

which on simplification gives

(3.7)

This shows that the manifold is an -Einstein manifold.Thus, we state the following theorem.

Theorem 3.1: A P-Kenmotsu manifold () with the condition = 0 is an -Einstein manifold.

Further, let us consider para-Kenmotsu manifold () with the condition .

Then, we have

(3.8)

By multiplying (3.8) with and on using (1.7), we get

(3.9)

By using the equations (2.12) and (2.13), the above equation reduces to

(3.10)

By interchanging and in (3.10) and on subtracting it from (3.10), we get

(3.11)

By putting in (3.11), we get

(3.12)

On contracting the above equation with respect to , we get

(3.13)

From (3.13), if , the manifold is Weyl-semisymmetric, and hence we have

(3.14)

which gives us

(3.15)

This shows that the manifold is an -Einstein manifold.

Now, by using the equations (3.14) and (3.15), the equation (3.11) takes the form , means that the manifold is conformally flat and hence it is an SP-Kenmotsu manifold ^[12].

If and in (3.13), we have . Thus, we state the following theorem.

Theorem 3.2: A Para-Kenmotsu manifold () with the condition is either conformally flat, in which is an SP-Kenmotsu manifold, or holds on .

4. Conclusion

In this paper, we have obtained the curvature properties of para-Kenmotsu manifold with the conditions = 0 and . Some of these results obtained are in similar to the results reported earlier in the case of para-Sasakian manifolds ^[3].

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, Dr. B. Satyanarayana of Nagarjuna University and Mr. T. Satyanarayana of Pragathi Engineering College for their valuable suggestions in preparing this manuscript. We would extend our heartful thanks to the referee for his valuable suggestions and comments.

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