Article
KYUNGPOOK Math. J. 2019; 59(3): 537-562
Published online September 23, 2019
Copyright © Kyungpook Mathematical Journal.
η-Ricci Solitons in δ-Lorentzian Trans Sasakian Manifolds with a Semi-symmetric Metric Connection
Mohd Danish Siddiqi
Department of Mathematics, Jazan University, Faculty of Science, Jazan, Kingdom of Saudi Arabia
e-mails : anallintegral@gmail.com, msiddiqi@jazanu.edu.sa
Received: February 14, 2018; Revised: August 29, 2018; Accepted: October 2, 2018
Abstract
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
The aim of the present paper is to study the
Keywords: η-Ricci Solitons, δ-Lorentzian trans-Sasakian manifold, semi-symmetric metric connection, curvature tensors, Einstein manifold.
Introduction
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
In 1924, the idea of a semi-symmetric linear connection on a differentiable manifold was introduced by A. Friedmann and J. A. Schouten [13]. In 1930, Bartolotti [5] gave a geometrical meaning of such a connection. In 1932, H. A. Hayden [16] defined and studied semi-symmetric metric connections. In 1970, K. Yano [42], started a systematic study of semi-symmetric metric connections in a Riemannian manifold and this was further studied by various authors such as Sharfuddin Ahmad and S. I. Hussain [31], M. M. Tripathi [34], I. E. Hirică and L. Nicolescu [17, 18], G. Pathak and U.C. De [27].
Let ∇ be a linear connection in an
The connection ∇ is said to be symmetric if its torsion tensor
A linear connection ∇ is said to be a semi-symmetric connection if its torsion tensor
The study of differentiable manifolds with Lorentizain metric is a natural and interesting topic in differential geometry. In 1996, Ikawa and Erdogan studied Lorentzian Sasakian manifold [20]. Also Lorentzian para contact manifolds were introduced by Matsumoto [24]. Trans Lorentzian para Sasakian manifolds have been used by Gill and Dube [15]. In [41], Yildiz et al. studied Lorentzian
The study of manifolds with indefinite metrics is of interest from the standpoint of physics and relativity. In 1969, Takahashi [36] has introduced the notion of al-most contact metric manifolds equipped with pseudo Riemannian metric. These indefinite almost contact metric manifolds and indefinite Sasakian manifolds are known as (
The semi Riemannian manifolds has the index 1 and the structure vector field
When
In 1982, R. S. Hamilton [19] said that the Rici solitons move under the Ricci flow simply by diffeomorphisms of the initial metric that is they are sationary points of the Ricci flow is given by
Definition 1.1
A
In 1925, Levy [22] obtained the necessary and sufficient conditions for the existence of such tensors. later, R. Sharma [30] initiated the study of Ricci solitons in contact Riemannian geometry. After that, Tripathi [35], Nagaraja et al. [25] and others like C. S. Bagewadi et al. [4] extensively studied Ricci soliton. In 2009, J. T. Cho and M. Kimura [9] introduced the notion of
Moreover, in this paper we introduced the relation between metric connection and semi-symmetric metric connection in an
Preliminaries
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Let
In [37], Tanno classified the connected almost contact metric manifold. For such a manifold the sectional curvature of the plane section containing
In Grey and Harvella [14] classification of almost Hermitian manifolds, there appears a class
With the above literature, we define the
Definition 2.1
A
If
Form
Further in an
An affine connection ∇̄ in
Let
Curvature Tensor on δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Let
Lemma 3.1
By the covariant differentiation of
Lemma 3.2
By replacing
Remark 3.1
Replace
Remark 3.2
Now, again replace
Remark 3.3
Replace
Now, contracting
This gives
Replace
Now, we have the following lemmas.
Lemma 3.3
By replacing
Lemma 3.4
From
Similarly taking
Quasi-projectively flat δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Let
Definition 4.1
A
Now, from
If
These results shows that the manifold under the consideration is an
Theorem 4.1
ϕ -Projectively flat δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
An
This result shows that the manifold under the consideration is an
Theorem 5.1
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfying R̄.S̄ = 0
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Now, suppose that
If
Theorem 6.1
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfying P̄.S̄ = 0
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Now, we consider
This result show that the manifold under the consideration is an
Theorem 7.1
Weyl Conformal Curvature Tensoron δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
The Weyl conformal curvature tensor
Now, taking inner product with
Theorem 8.1
δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Let us consider that the
Now, using
Let
Theorem 9.1
η -Ricci Solitons and Ricci Solitons in δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
Let
Using
Theorem 10.1
Corollary 10.1
Here is an example of
Example 10.1
Let ∇ be the Levi-Civita connection with respect to the metric
The Riemannian connection ∇ with respect to the metric
Now, we consider this example for semi-symmetric metric connection from
References
- Abstract
- Introduction
- Preliminaries
- Curvature Tensor on
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection - Quasi-projectively flat
δ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection ϕ -Projectively flatδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connectionδ -Lorentzian trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingR̄.S̄ = 0δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection satisfyingP̄.S̄ = 0- Weyl Conformal Curvature Tensoron
δ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection δ -Lorentzian Trans-Sasakian Manifold with Weyl Conformal Flat Conditions with a Semi-symmetric Metric Connectionη -Ricci Solitons and Ricci Solitons inδ -Lorentzian Trans-Sasakian Manifold with a Semi-symmetric Metric Connection- References
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