#### Abstract

We categorize almost quasi-Yamabe solitons on -Sasakian manifolds and their -submanifolds whose potential vector field is torse-forming, admitting a generalized symmetric metric connection of type . Finally, a nontrivial example is provided to confirm some of our results.

#### 1. Introduction

Yamabe solitons (YS) are ideas that generate self-similar Yamabe flow (YF) solutions [1]:

Di Cerbo and Disconzi were the first to notice them in [2]. Chen and Deshmukh proposed the concept of quasi-Yamabe soliton (QYS) in [3], which we will explore in this study for a more broader situation when the constants are functions.

*Definition 1. *An almost quasi-Yamabe solitons (AQYS) on Remannian manifold is a set of data that fulfill the following equation:where is operator of the Lie derivative in view of , and are smooth functions on , is the 1-form of , and is the scalar curvature. If , , or , we call an AQYS shrinking, stable, or growing, respectively, using Yamabe solitons nomenclature.

If is a gradient type, (2) yieldsIt is nothing more than a generalized quasi-Yamabe gradient soliton (GQYGS) (see [4, 5]). Several authors have thoroughly investigated AQYS and QYGS in [6â€“13].

*Definition 2. *A vector field on meets the following conditions and is known as a torse-forming vector field [14]:where is a 1-form and is in .

We classify such vector field as(i)It is concircular if the 1-form vanishes identically [15](ii)For concurrent, and [16](iii)It is recurrent if (iv)Parallel if If the vector field fulfills (4) with , it called as torqued vector field [17].

The content of the paper is as follows. After the opening remark, Section 2 contains the fundamental result of an -Sasakian manifold. We discuss the concept of a generalized symmetric metric connection- (GSMC-) in Section 3. With regard to GSMC-, Section 4 is devoted to -submanifolds of an -Sasakian manifold. With respect to such a connection, we examine QYS in view of a torse-forming vector field on an -Sasakian manifold in Section 5. The study of QYS with a torse-forming vector field on -submanifolds of an -Sasakian manifolds is also covered in Section 6. Finally, in Section 7, we look at AQYGS with a torse-forming vector field by considering the tangential and normal components of such a vector field on -submanifolds.

#### 2. -Sasakian Manifolds

If a tensor field , a contravariant vector field , a 1-form , and the Lorentzian metric are admitted to a differentiable manifold ; it is termed as an -Sasakian manifold (see [18, 19]); then,where be the Levi-Civita connection along the metric . In an -Sasakian manifold, we yields

If we writethen is a symmetric tensor field. Now, is closed on (see [18, 20]); then,

In an -Sasakian manifold , the following relationships are maintained (see [20, 21]):for any vector fields , , and on , where and are the curvature tensor and Ricci tensor of , simultaneously.

Let be a submanifold of an -Sasakian manifold. The Gauss and Weingarten formulas are given bywhere and belong to and , respectively.

#### 3. Generalized Symmetric Metric Connection of Type

Let and be a linear and Levi-Civita connection on an -Sasakian manifold. Now, we will go through the results that will be used.

Lemma 1 (see [22]). *In an -Sasakian manifold , the GSMC of type is given byfor all and on .*

Lemma 2 (see [22]). *The following relations hold on an -Sasakian manifold in light of GSMC-:for any , .*

#### 4. -Submanifolds of an -Sasakian Manifold with GSMC-

Here, we have recall the well-known definition in the following manner.

*Definition 3 (see [23]). *A Riemannian manifold of an -Sasakian manifold is called a -submanifold if is tangent to and there exists on a differentiable distribution such that(i) is invariant under , i.e., (ii)The orthogonal complement distribution of the distribution on is totally real, i.e.,

*Definition 4 (see [23]). *If the distribution is horizontal (resp., vertical), then the pair is called -horizontal (resp., -vertical) if (resp., ). The -submanifold is also called -horizontal (resp., -vertical) if .

The orthogonal complement is given bywhere .

Let be a -submanifold of an -Sasakian manifold with a GSMC-. For any and , we can writeThe Gauss and Weingarten formulas with respect to are, respectively, given byfor any , where , . Here, , , and are called the induced connection on , the second fundamental form, and the Weingarten mapping with respect to , respectively. In view of (10), (12), and (17), we obtainUsing (15) and (16) in (19), we obtainfor any .

#### 5. Quasi-Yamabe Solitons (QYS) with Torse-Forming Vector Field

We classify QYS with torse-forming vector fields on an -Sasakian manifold admitting a GSMC- in this section. As a result, we can prove the theorem below.

Theorem 1. *An -Sasakian manifold , , with respect to GSMC- admitting QYS. If is a torse-forming vector field, then the data is growing, steadying, and contracting in accordance with , unless is constant.*

*Proof. *Let the data be a QYS on in terms of GSMC-. From (3), we haveFrom (3) and (12), we obtainfor all .

With the help of (23) and (24), we obtainOn contracting (25), we findwhere â€‰=â€‰ .

As a result, Theorem 1 is proven.

In this sequel, the corollaries are as follows.

Corollary 1. *If (2) defines a QYS on an -Sasakian manifold , , admitting a GSMC-, then there are the existing relationships representing in Table 1.*

In Table 1, , , , , , â€‰and .

Corollary 2. *Let data be a QYS on an -Sasakian manifold , , with respect to a GSMC-. If is torse-forming vector field, then is growing, steadying, and contracting according to , unless is constant.*

Corollary 3. *Let , , be an -Sasakian manifold endowed with a GSMC-. If a data be a QYS on and is a torse-forming vector field, then is growing, steadying, and contracting according to , unless is constant.*

Corollary 4. *If (2) defines a QYS on an -Sasakian manifold , , with respect to a GSMC-, then we obtain the relationship in Table 2.*

Corollary 5. *Let a data be a QYS on an -Sasakian manifold , , with respect to a GSMC-. Then, the following relationships are maintained in Table 3.*

In Table 3, , , , , , and .

#### 6. Quasi-Yamabe Solitons (QYS) with Potential Vector Field is Torse-Forming on -Submanifold

We investigate QYS in relating to a torse-forming vector field on -submanifolds of an -Sasakian manifold with regard to the induced connection of type in this section. The following is our theorem.

Theorem 2. *Let a -submanifold of an -Sasakian manifold be , , admitting a GSMC is -horizontal (resp. -vertical) and is parallel with respect to . If data is a QYS on and is a torse-forming vector field, then is growing, steadying, or contracting according to , unless is constant.*

*Proof. *If is -horizontal for all and is parallel in relation to , then there is, from (20),With the help of Lemma 1, the induced connection is also a GSMC-. This leads to the statement of Theorem 2.

In this sequel, we write the following corollaries.

Corollary 6. *Let a -submanifold of an -Sasakian manifold be , , admitting a GSMC is -horizontal (resp. -vertical) and is parallel in terms of . If (2) defines a QYS on and is a torse-forming vector field, then the results hold in Table 4.*

Corollary 7. *Let a -submanifold of an -Sasakian manifold be , , admitting a GSMC is -horizontal (resp. -vertical) and is parallel in term of of type . If data is a QYS on and is a torse-forming vector field, then, in Table 5, relationships must be true.*

Corollary 8. *Let a -submanifold of an -Sasakian manifold be , , admitting a GSMC of type is -horizontal (resp. -vertical) and is parallel with respect to . If is a QYS on and is a torse-forming vector field, then is growing, steadying, or contracting according to , unless is constant.*

Corollary 9. *Let a -submanifold of an -Sasakian manifold be , , in relation to a GSMC- which is -horizontal (resp. -vertical) and which is parallel in view og . If is a QYS on and is a torse-forming vector field, then is growing, steadying, or shrinking according as , unless is constant.*

Corollary 10. *If (2) defines a QYS on an -Sasakian manifold , , concerning a GSMC-, then, in Table 6, relationships are true.*

#### 7. Almost Quasi-Yamabe Solitons (AQYS) whose Potential Vector Field is Torse-Forming on -Submanifold

In this section, we classify AQYS whose potential field is torse-forming on -submanifold of an -Sasakian manifold with respect to a GSMC-. At this stage, we denote and as tangential and normal components of such vector field. To begin, we will prove the outcome.

Theorem 3. *An almost quasi-Yamabe soliton on -submanifold of an -Sasakian manifold in relation to a GSMC- satisfiesany type of vector field on .*

*Proof. *In light of (3), (12), (17), and (18), we haveWe obtain the following equation when we compare the tangential and normal components of (29):We may deduce from the concept of Lie derivative and (31) thatUsing (32) in (2), we yieldThis completed our assertion.

As a result, the following corollaries are stated.

Corollary 11. *If (2) defines AQYS on -submanifold of an -Sasakian manifold in relation to a GSMC- which is minimal, consequently, the following relationship holds:*

Corollary 12. *If (2) defines AQYS on -submanifold of an -Sasakian manifold and the distribution is -horizontal (resp. -vertical), , where is parallel with induced connection of type , then we havein all vector fields on .*

Corollary 13. *If data is a AQYS on -submanifold of an -Sasakian manifold and the distribution is -horizontal (resp. -vertical), , is parallel with induced connection of type is minimal, then the relation holds:*

#### 8. Example

A 4-dimensional differentiable manifold is taken into consideration, that is, , where is the standard coordinate in . At each point along , is a set of linearly independent vector fields and is described as

Also, the Lie bracketâ€™s nonvanishing components are as follows:

Let on be the Lorentzian metric as

Let be the 1-form corresponding to the Lorentzian metric :for any . If is defined as the -tensor field,

We can readily prove this using the linearity characteristics of and :for any . Thus, for , the frame leads to an -contact skeleton, which is known as the -contact manifold of dimension 4. Now, for , Koszulâ€™s formula gives the nonvanishing component:

Using the above equation, it can be easily verified that holds for each . Thus, an -contact manifold is a 4-dimensional -Sasakian manifold. From (12), we calculate as follows:

It is clear from (12) that holds for each . So, an -Sasakian manifold admitted a GSMC-.

The nonvanishing components of the curvature tensor using the aforementioned formulas are

The Ricci tensor of is defined as , where , and is given by

Also, the scalar curvature â€‰=â€‰.

Let any vector fields , and ; it is possible to writewhere , in order for

If we consider the 1-form by , for any and considering astherefore, the relation,holds. As per these consequences, from (24), we obtain

Also, we calculate

Also,