Article
Kyungpook Mathematical Journal 2022; 62(4): 737-749
Published online December 31, 2022
Copyright © Kyungpook Mathematical Journal.
Some Geometric Properties of η ∗ -Ricci Solitons on α -Lorentzian Sasakian Manifolds
Shashikant Pandey and Abhishek Singh, Rajendra Prasad
Department of Mathematics and Astronomy, University of Lucknow, Lucknow, 226007 Uttar Pradesh, India
e-mail : shashi.royal.lko@gmail.com and lkoabhi27@gmail.com
Department of Mathematics and Astronomy, University of Lucknow, Lucknow, 226007 Uttar Pradesh, India
e-mail : rp.manpur@rediffmail.com
Received: December 20, 2021; Revised: April 19, 2022; Accepted: May 3, 2022
Abstract
We investigate the geometric properties of
Keywords: η*-Ricci solitons, φ*-Symmetric, Ricci semisymmetric, φ*Ricci symmetric, η* Einstein manifolds
1. Introduction
The Ricci flow, which is used to compute the canonical metric based on the smooth manifold, was proposed by Hamilton [19] in 1982. The Ricci flow provides an evolution expression of metrics for a Riemannian manifold as follows:
The Einstein metric can be naturally generalized to Ricci solitons which are defined on the Riemannian manifold
If
Cho and Kimura [9] proposed the concept of
In the current scenario, η-Ricci solitons are studied by various researchers have considered such η-Ricci solitons, and have found interesting geometric properties in many contexts: on Lorentzian para-Sasakian manifolds [2], [28], gradient η-Ricci solitons [3], on ϵ-para Sasakian manifolds [21] and [4], quasi-Sasakian 3-manifolds [22], 3-dimensional Kenmotsu manifolds [24], Sasakian 3-manifolds [25], para-Sasakian manifolds [26] and para Kenmotsu manifolds [29] and studied Lorentzian Sasakian manifold [27] etc.
The structure of the paper is as follows. The neccessary basic theory about α-LS manifolds is given in Section 2. In Section 3, the geometric properties of
2. Preliminaries
A
for any vector fields
Also, α-LS manifolds satisfy [16],
where ∇ has the usual meaning.
Moreover, on α-LS manifolds the following relations hold (see [1]):
where
As per the definition of the Lie derivative, we have
Definition 2.1. ([18]) A
is called an
Definition 2.2. An α-LS manifold
is called Ricci semisymmetric.
Definition 2.3. ([13]) An α-LS manifold
is called
If
Definition 2.4. ([30]) An α-LS manifold
for all vector fields
3. Ricci and η ∗ -Ricci Solitons on α -LS Manifolds
Let
Because of (2.15), the (3.1) becomes
for any
for any
The data
From (3.2), we get
Taking
From (2.11) and (3.4), we obtain
4. Ricci Semisymmetric η ∗ -Ricci Solitons on α -LS Manifolds
In this section, we investigate Ricci semisymmetric α-LS manifolds on
which implies that
Taking
Using (2.8) in (4.2), we infer
Since
Using (3.2) and (3.4) in (4.4), we infer
Putting
which is equivalent to
Putting
Subtracting (4.7) and (4.8), we get
for any
Now from above, we are able to state our results.
Theorem 4.1. If
In case
Corollary 4.2. If α-LS manifolds
From (3.2), (3.5) and (4.7), we obtain
As a consequence, we can state following proposition.
Proposition 4.3. Let
5. φ ∗ -Symmetric η ∗ -Ricci Solitons on α -LS Manifolds
Consider
Using (2.1) and (5.1), we get
Taking inner product in (5.2) with
which implies
After simplification, we obtain
Putting
Using (2.5) and (3.3) in (5.6), we infer
Taking
Since
Now, using (3.3) in (5.9), we infer
Taking
Subtracting (5.10) from (5.11), we obtain
for any
Now, we are ready to state the following results.
Theorem 5.1. If
In case
Corollary 5.2. If α-LS manifolds
From (3.2), (3.4) and (5.10) we have
This leads to the following proposition.
Proposition 5.3. Let
6. φ ∗ -Ricci Symmetric η ∗ -Ricci Solitons on α -LS Manifolds
This section is devoted to the study of
Consider a
Using (2.1), we infer
Taking inner product with
Let
Putting
The second term of (6.5), takes the form
and we obtain
The equations (6.5) and (6.7) imply that
which gives
In view of (2.5) and (3.4), we have
Putting
Using (2.4), (2.7) and (3.3) in (6.9), we get
Since
Putting
Subtracting (6.11) from (6.12), we get
for any
Hence, we can state the following results.
Theorem 6.1. If
In case
Corollary 6.2. If α-LS manifolds
Using (3.2), (3.5) and (6.11) we have
This leads to the following proposition.
Proposition 6.3. Let
Example 6.4. Now, we assume the 3-dimensional manifold
where
The vector fields
are linearly independent at each point of
Let
Let
Let
for any vector field
Let
Then, using the linearity of
for any vector field
It is easy to observe
Thus for
Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric
Using Koszul's formula
One can easily obtain
Now, we see that the manifold is an α-LS manifold.
Also, the Riemannian curvature tensor
Then
Then, the Ricci tensor
From (3.2), we obtain
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