A Note on <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3240995pt" id="M1" height="12.223pt" version="1.1" viewBox="-0.0657574 -11.8989 32.0999 12.223" width="32.0999pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-13" d="M608 643C551 682 486 687 443 687C299 687 188 608 188 504C188 361 341 312 465 312C483 312 501 313 518 315C454 195 366 93 314 59C255 82 201 109 141 109C73 109 33 83 33 52C33 13 76 -9 159 -9C218 -9 267 -1 314 15C354 0 402 -15 478 -15C575 -15 671 36 759 141L740 155C657 63 571 16 475 16C448 16 411 26 372 39C458 74 565 199 627 333C806 381 931 499 931 587C931 634 908 669 848 669C767 669 671 582 620 506C591 462 569 411 534 347C503 343 490 342 471 342C330 342 229 399 229 500C229 603 346 657 451 657C486 657 540 657 596 624L608 643ZM890 588C890 519 822 457 747 414C712 395 678 378 643 369C683 484 760 632 841 632C873 632 890 615 890 588ZM266 35C235 24 200 21 161 21C103 21 74 32 74 54C74 75 122 79 141 79C172 79 209 69 266 36V35Z"/></g><g transform="matrix(.017,0,0,-0.017,16.233,0)"><path id="g198-17" d="M856 669C824 657 796 643 772 628C709 667 631 687 551 687C314 687 158 548 158 412C158 332 216 282 300 282C445 282 516 426 527 561H507C478 406 414 312 302 312C245 312 199 352 199 408C199 535 345 657 550 657C617 657 687 644 742 609C644 538 583 430 495 271C383 71 317 16 193 16C122 16 73 60 73 95H75C82 84 97 83 105 83C133 83 148 106 148 131C148 160 125 179 97 179C64 179 38 151 38 108C38 42 104 -15 211 -15C368 -15 496 71 610 302C672 426 704 522 770 588C803 559 825 518 825 465C825 382 787 316 731 316C710 316 697 328 697 345C697 358 700 380 733 399L723 417C688 398 665 367 665 338C665 303 688 272 737 272C821 272 886 359 886 447C886 516 850 572 797 612C816 627 836 639 863 650L856 669Z"/></g></svg>-</span>Sasakian Manifolds with Almost Quasi-Yamabe Solitons

Yadav, Sunil Kumar; Suthar, D. L.; Shimelis, Biniyam

doi:https://doi.org/10.1155/2021/5878006

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Research Article | Open Access

Volume 2021 | Article ID 5878006 | https://doi.org/10.1155/2021/5878006

A Note on -Sasakian Manifolds with Almost Quasi-Yamabe Solitons

Sunil Kumar Yadav,¹D. L. Suthar,²and Biniyam Shimelis²

Academic Editor: Ram Jiwari

Received27 Aug 2021

Revised16 Oct 2021

Accepted09 Nov 2021

Published10 Dec 2021

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 [6–13].

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 = .
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, , , , , , 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 bracket’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 , Koszul’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 = .

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,

With the help of (52) and (53), equation (51) reduces

Also,

We consider that , , and .

Thus, we get is a Yamabe soliton, that is, holds, unless

As a result, the existence of the YS on a 4-dimensional -Sasakian manifold admitting a GSMC- with potential vector field as torse-forming is justified. Then, Theorems 1 and 2 are verified.

Data Availability

No data were used to support this study.

Conflicts of Interest

There are no conflicts of interest regarding the publication of this article.

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Copyright © 2021 Sunil Kumar Yadav et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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