### 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 ^{3}

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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