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Research Article

Open Access Peer-reviewed

K. L. Sai Prasad^{ }, S. Sunitha Devi, G. V. S. R. Deekshitulu

Received September 11, 2018; Revised October 23, 2018; Accepted November 24, 2018

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.

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.

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.

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:

**Definitio****n 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.*

**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.

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

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

[1] | Sato, I, On a structure similar to the almost contact structure, Tensor (N.S.), 30, 219-224, 1976. | ||

In article | |||

[2] | Sato, I, On a structure similar to the almost contact structure II, Tensor (N.S.), 31, 199-205, 1977. | ||

In article | |||

[3] | Adati, T. and Matsumoto, K, On conformally recurrent and conformally symmetric P-Sasakian Manifolds, TRU Math., 13, 25-32, 1977. | ||

In article | |||

[4] | Adati, T. and Miyazawa, T, On P-Sasakian manifolds admitting some parallel and recurrent tensors, Tensor (N.S.), 33, 287-292, 1979. | ||

In article | |||

[5] | De, U. C. and Avijit Sarkar, On a type of P-Sasakian manifolds, Math. Reports, 11(2), 139-144, 2009. | ||

In article | |||

[6] | Matsumoto, K., Ianus, S. and Ion Mihai, On P-Saskian manifolds which admit certain tensor fields, Publ. Math. Debrecen, 33, 61-65, 1986. | ||

In article | |||

[7] | Kenmotsu, K, A class of almost contact Riemannian manifolds, Tohoku Math. Journal, 24, 93-103, 1972. | ||

In article | View Article | ||

[8] | Sinha, B. B. and Sai Prasad, K. L, A class of almost para contact metric Manifold, Bulletin of the Calcutta Math. Soc., 87, 307-312, 1995. | ||

In article | |||

[9] | Pokhariyal, G. P. and Mishra, R. S, The curvature tensors and their relativistic significance, Yokohoma Math. J., 18, 105-108, 1970. | ||

In article | |||

[10] | Pokharial, G. P, Study of a new curvature tensor in a Sasakian manifold, Tensor (N.S.), 36, 222-225, 1982. | ||

In article | |||

[11] | Sai Prasad, K. L. and Satyanarayana, T, Some curvature properties on a Special paracontact Kenmotsu manifold with respect to Semi-symmetric connection, Turkish Journal of Analysis and Number Theory, 3(4), 94-96, 2015. | ||

In article | View Article | ||

Published with license by Science and Education Publishing, Copyright © 2018 K. L. Sai Prasad, S. Sunitha Devi and G. V. S. R. Deekshitulu

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

K. L. Sai Prasad, S. Sunitha Devi, G. V. S. R. Deekshitulu. On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3). *Turkish Journal of Analysis and Number Theory*. Vol. 6, No. 6, 2018, pp 155-158. http://pubs.sciepub.com/tjant/6/6/2

Prasad, K. L. Sai, S. Sunitha Devi, and G. V. S. R. Deekshitulu. "On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3)." *Turkish Journal of Analysis and Number Theory* 6.6 (2018): 155-158.

Prasad, K. L. S. , Devi, S. S. , & Deekshitulu, G. V. S. R. (2018). On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3). *Turkish Journal of Analysis and Number Theory*, *6*(6), 155-158.

Prasad, K. L. Sai, S. Sunitha Devi, and G. V. S. R. Deekshitulu. "On a Class of P-Kenmotsu Manifolds Admitting Weyl-projective Curvature Tensor of Type (1, 3)." *Turkish Journal of Analysis and Number Theory* 6, no. 6 (2018): 155-158.

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[1] | Sato, I, On a structure similar to the almost contact structure, Tensor (N.S.), 30, 219-224, 1976. | ||

In article | |||

[2] | Sato, I, On a structure similar to the almost contact structure II, Tensor (N.S.), 31, 199-205, 1977. | ||

In article | |||

[3] | Adati, T. and Matsumoto, K, On conformally recurrent and conformally symmetric P-Sasakian Manifolds, TRU Math., 13, 25-32, 1977. | ||

In article | |||

[4] | Adati, T. and Miyazawa, T, On P-Sasakian manifolds admitting some parallel and recurrent tensors, Tensor (N.S.), 33, 287-292, 1979. | ||

In article | |||

[5] | De, U. C. and Avijit Sarkar, On a type of P-Sasakian manifolds, Math. Reports, 11(2), 139-144, 2009. | ||

In article | |||

[6] | Matsumoto, K., Ianus, S. and Ion Mihai, On P-Saskian manifolds which admit certain tensor fields, Publ. Math. Debrecen, 33, 61-65, 1986. | ||

In article | |||

[7] | Kenmotsu, K, A class of almost contact Riemannian manifolds, Tohoku Math. Journal, 24, 93-103, 1972. | ||

In article | View Article | ||

[8] | Sinha, B. B. and Sai Prasad, K. L, A class of almost para contact metric Manifold, Bulletin of the Calcutta Math. Soc., 87, 307-312, 1995. | ||

In article | |||

[9] | Pokhariyal, G. P. and Mishra, R. S, The curvature tensors and their relativistic significance, Yokohoma Math. J., 18, 105-108, 1970. | ||

In article | |||

[10] | Pokharial, G. P, Study of a new curvature tensor in a Sasakian manifold, Tensor (N.S.), 36, 222-225, 1982. | ||

In article | |||

[11] | Sai Prasad, K. L. and Satyanarayana, T, Some curvature properties on a Special paracontact Kenmotsu manifold with respect to Semi-symmetric connection, Turkish Journal of Analysis and Number Theory, 3(4), 94-96, 2015. | ||

In article | View Article | ||