Article
Kyungpook Mathematical Journal 2020; 60(4): 805-819
Published online December 31, 2020
Copyright © Kyungpook Mathematical Journal.
η-Einstein Solitons on (ε)-Kenmotsu Manifolds
Mohd Danish Siddiqi*, Sudhakar Kumar Chaubey
Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia
e-mail : msiddiqi@jazanu.edu.sa
Section of Mathematics, Department of Information Technology, Shinas College of Technology, Shinas, P. O. Box 77, Postal Code 324, Oman
e-mail : sudhakar.chaubey@shct.edu.om
Received: October 9, 2019; Revised: June 26, 2020; Accepted: June 29, 2020
Abstract
The objective of this study is to investigate η-Einstein solitons on (ε)-Kenmotsu manifolds when the Weyl-conformal curvature tensor satisfies some geometric properties such as being flat, semi-symmetric and Einstein semi-symmetric. Here, we discuss the properties of η-Einstein solitons on ⊎-symmetric (ε)-Kenmotsu manifolds.
Keywords: (ε)-Kenmotsu manifold, η-Einstein solitons, Weyl-conformal curvature, Einstein manifold
1. Introduction
In
The basic difference between the Riemannian and the semi-Riemannian (signature of the metric tensor
A tensor
for all
A Riemannian or a semi-Riemannian manifold is said to be
for all
Also, a Riemannian or a semi-Riemannian manifold is said to be
for all
In 1995, Alias, Romero and Sánchez [1] introduced the notion of
In general relativity, the perfect fluid spacetime is of special interest. Lorentzian manifolds with Ricci tensor of the form
where
Einstein's Field equations without cosmological constant are given by
for all vector fields
The energy momentum tensor
where µ is the energy density function,
The paper is organized as follows. In Section 2, some preliminary results are recalled. After preliminaries in Section 3, we prove that a ξ-conformally flat Lorentzian para-Sasakian type spacetime is a Robertson-Walker spacetime. Then we study ϕ-Weyl semisymmetric Lorentzian para-Sasakian type spacetime and prove that a ϕ-Weyl semisymmetric Lorentzian para-Sasakian type spacetime is a perfect fluid spacetime, provided
2. Preliminaries
A differentiable manifold
for all vector fields
where
If the relations
hold for a Lorentzian almost paracontact manifold
If a vector field
where β is a non-zero scalar function and
In the Lorentzian manifold
where α is a non-zero scalar function. It is to be noted that in Lorentzian para-Sasakian type spacetime ξ is a unit timelike vector.
Let us consider an
for all vector fields
Now we state the following results which will be needed in the later section.
3. ξ-conformally Flat Lorentzian Para-Sasakian Type Spacetime
In this section we characterize ξ-conformally flat Lorentzian para-Sasakian type spacetime. Let
Using 2.9 and 2.10 in the above equation, we obtain
Replacing
From (3.3), we get
That is, the spacetime is a perfect fluid spacetime. In view of the above discussions we state the following:
Since conformally flat implies ξ-conformally flat, therefore we can state:
Replacing
which implies ξ is an eigenvector of the Ricci tensor.
It is known [5] that if an
By virtue of (3.4) and (3.5), we get
Putting
which gives
Hence by Lemma 2.1 we conclude that ξ is torse-forming. Thus we can state the following:
In [10], the authors proved the following:
In view of the above Theorem we conclude the following:
Also in [10] the following result was stated
4. ϕ-Weyl Semisymmetric Lorentzian Para-Sasakian Type Spacetime
In 2012, Yildiz and De [17] introduced the notion of ϕ-Weyl semisymmetric
for arbitrary vector fields
We adopt the same definition in an
for any vector field
Putting
Taking the inner product on both sides by
Replacing
Replacing
provided
From (1.3) and (4.6), we obtain
The above equation is of the form of a perfect fluid spacetime, where
Therefore the state equation is
5. Ricci-semisymmetric Lorentzian Para-Sasakian Type Spacetime
In this section we deal with Lorentzian para-Sasakian type spacetime satisfying
From (5.1) it follows that
Replacing
Now putting
which implies that the manifold under consideration is an Einstein spacetime.
Now we consider perfect fluid spacetime obeying Einstein's equation whose velocity vector field is the characteristic vector field of the manifold. Then we get from Einstein's field equation
Using (5.4) in (5.5) and putting
Now taking a frame field and contracting
Equation 5.6 and 5.7 together yields that
Thus we have :
Acknowledgements.
The author is thankful to the referee and the Editor in Chief for their valuable suggestions towards the improvement of the paper.
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