Isometries on almost Ricci–Yamabe solitons

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere Sn(r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^n(r)$$\end{document} are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with α≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ne 0$$\end{document} and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space En\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E^n$$\end{document} or Euclidean sphere Sn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^n.$$\end{document} Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.


Introduction
One of the most significant approaches to understanding the geometric structure in Riemannian geometry is to study the theory of geometric flows. The Ricci flow is a well-known geometric flow introduced by Hamilton [15], who used it to prove a three-dimensional sphere theorem [14]. The idea of the Ricci flow is contributed to the proof of Thurston's conjecture, including as a special case, the Poincaré conjecture. The Ricci soliton on a Riemannian manifold (M, g) are self-limiting solutions to Ricci flow and is defined by (1.1) where L V g denotes the Lie-derivative of g along potential vector field V, Ric is the Ricci curvature of M 2n+1 and λ, a real constant. When the vector field V is the gradient of a smooth function f on M 2n+1 , that is, V = ∇ f, then we say that Ricci soliton is gradient (for details see [9,20]). According to Petersen and Wylie [20], a gradient Ricci soliton is rigid if it is a flat N × R k , where N is Einstein and gave certain classification. The notion of almost Ricci soliton was introduced by Pigola et al. [21] by taking λ as a smooth function in the definition of Ricci soliton (1.1). The authors in [2] studied the rigidity of gradient almost Ricci solitons and showed that it is isometric to the Euclidean space R n or sphere S n . Barros et al. [3], Yang and Zhang [28], Cao et al. [8] obtain several rigidity results.
To tackle the Yamabe problem on manifolds of positive conformal Yamabe invariant, Hamilton introduced the geometric flow known as Yamabe flow. The Yamabe soliton is a self-similar solution to the Yamabe flow. On a Riemannian manifold (M, g), a Yamabe soliton is given by (1.2) where R is the scalar curvature of the manifold and λ, a real constant. Even though both the Ricci and Yamabe solitons are similar in dimension n = 2, the solitons behave differently for dimension n > 2 as the Yamabe soliton preserves the conformal class of the metric but the Ricci soliton does not in general. If λ is a smooth function in (1.2), then it is called almost Yamabe soliton. Alkhaldi et al. [1] gave a characterization of almost Yamabe soliton with conformal vector field. Barbosa and Ribeiro [4] gave some rigidity results for Yamabe almost soliton. Güler and Crasmareanu [13], in 2019, introduced the notion of the Ricci-Yamabe map which is a scalar combination of Ricci and Yamabe flow. In [13], the authors define the following: Definition 1.1 [13] The map RY (α,β,g) : I → T s 2 (M) given by: is called the (α, β)-Ricci-Yamabe map of the Riemannian flow (M, g). If then g(.) will be called an (α, β)-Ricci-Yamabe flow. The Ricci-Yamabe flow can be Riemannian or semi-Riemannian or singular Riemannian flow due to the involvement of scalars α and β. This kind of different choices can be useful in some physical models such as relativity theory. The Ricci-Yamabe soliton emerges as the limit of the solution of Ricci-Yamabe flow. Definition 1.2 A Riemannian manifold (M n , g), n > 2 is said to admit almost Ricci-Yamabe soliton (g, V, λ, α, β) if there exist smooth function λ such that where α, β ∈ R. Almost Ricci-Yamabe soliton is of particular interest as it generalizes a large group of well-known solitons such as: • Ricci almost soliton (α = 1, β = 0).
If V is a gradient of some smooth function f on M, then the above notion is called gradient almost Ricci-Yamabe soliton and then (1.3) reduces to where ∇ 2 f is the Hessian of f. The almost Ricci-Yamabe soliton (ARYS) is said to be expanding, shrinking or steady if λ < 0, λ > 0 or λ = 0 respectively. In particular, if λ is constant, then ARYS reduces to Ricci-Yamabe soliton. Many geometers such as [10,11,22] analyzed Ricci-Yamabe solitons. In [23,26], authors studied Ricci-Yamabe soliton in different spacetimes. Singh and Khatri [16,25] studied ARYS in almost contact manifolds. Siddiqi et al. [24] consider ARYS on static spacetimes.
Motivated by the above studies, we investigated the ARYS under certain conditions. The present paper is organized as follows: In Sect. 2, several rigidity results are obtained by following the methods of Barros and Ribeiro [5] for compact almost Ricci soliton. Also, we obtained the conditions under which compact gradient ARYS is isometric to the Euclidean sphere S n (r ). In Sect. 3, ARYS with conformal, solenoidal and torseforming vector fields are considered. We showed that a complete ARYS with α = 0 and potential vector field as conformal vector field is either isometric to Euclidean space E n or Euclidean sphere S n (r ). Also, complete gradient ARYS with conformal vector field is investigated. Lastly, ARYS with solenoidal and torse-forming vector fields are considered and obtained several rigidity results which are proved by constructing non-trivial examples.

Some rigidity results on ARYS
Before proceeding to the main results of this paper, we obtained several lemmas on ARYS and gradient ARYS which would be used later.

Lemma 2.1
For a gradient ARYS (M n , g, ∇ f, λ), the following formula holds: Proof Equation (1) is directly obtained by taking trace of the soliton equation. For Eq. (2), we consider Schur's Lemma (n > 2), we have Then, using Ricci identity in the above expression gives Thus, in regard of equation (1) yields
Petersen and Wylie [20] obtained the following Bochner formula for Killing and gradient field as:

Lemma 2.2 Given a vector field X on a Riemannian manifold
When X = ∇ f is a gradient field and Z is any vector field, we have Taking an inner product of Eq. (2) in Lemma 2.1 by arbitrary vector field Z gives In particular,

Lemma 2.3 For an ARYS
Making use of Schur's Lemma, Lemma 2.2, (2.4) and (2.5), we get the required results. This completes the proof.
Moreover, from (1.3) we have In consequence of this in Lemma 2.3, we get then X is Killing and M n is RYS.
Proof Since M n is compact, taking integration of Lemma 2.3 gives In view of our hypothesis and (2.7), we get ∇ X = 0 which implies L X g = 0, i.e., X is Killing vector field. In this case, ARYS will be simply RYS since M n will be Einstein manifold, which implies that λ is constant. This completes the proof.
then X is Killing.

Theorem 2.7 Let
Combining second argument of Lemma 2.1 and (2.8), then taking divergence of the obtained expression yields Now, using commuting covariant derivative and Ricci identity, we have Making use of the above expression in (2.9), we get Combining first argument of Lemma 2.1, (2.2) and (2.10), we obtain Making use of the fact that By hypothesis, since M n is compact, we get (2.14) Combining (2.2) in (2.14) proves the second part provided α + (n − 1)β = 0. This completes the proof.
Now, using the foregoing equation in (2.14) yields With regard to Theorem 2.7, Corollary 2.8 and Tashiro's result [27] which states that a compact Riemannian manifold (M n , g) is conformally equivalent to S n (r ) provided there exists a non-trivial function f : M n → R such that ∇ 2 f = f n g. We obtain the following result which is a generalization of Corollary 1 of [5] and Corollary 1.10 of [12].

β} is isometric to a Euclidean sphere S n (r ) if one of the following conditions hold:
(1) M n has constant scalar curvature.
(2) M n is a homogeneous manifold. Hodge-de Rham decomposition theorem states that we may decompose the vector field X over a compact oriented Riemannian manifold as a sum of the gradient of a function h and a divergence free vector field Y, i.e., where div Y = 0. Taking divergence of (2.16) gives div X = h. From the fundamental equation, we have 2div X + (2α + nβ)R = 2nλ. Therefore, combining both equations result in the following: On the other hand, if (M n , g, ∇ f, λ) is also a compact gradient ARYS, then from equation (1)

ARYS with certain conditions on the potential vector field
In this section, we consider ARYS whose potential vector field satisfies certain conditions such as conformal, solenoidal and torse-forming vector fields. First we recall the definition of conformal vector field. A smooth vector field X on a Riemannian manifold is said to be a conformal vector field if there exists a smooth function ψ on M that satisfies We say that X is non-trivial if X is not Killing, that is, ψ = 0. Conformal vector field under almost Ricci soliton and almost Ricci-Bourguignon solitons were considered by authors in [5,6] and obtained interesting results. Now, we state and prove the following lemma. (n ≥ 3) be ARYS with α = 0. If X is a conformal vector field with potential function ψ, then R and λ − ψ are constants.

Lemma 3.1 Let
Proof Since X is a conformal vector field, we have L X g = 2ψg. Making use of this in the soliton equation Making use of Schur's Lemma in (3.3) and inserting it in the covariant derivative of (3.2) results in (n − 2)α∇ R = 0. As α = 0, then R is constant, which implies then from (3.2) that λ − ψ is also constant. This completes the proof.
Theorem 3.2 Let (M n , g, X, λ) (n ≥ 3) be a compact ARYS with α = 0. If X is a non-trivial conformal vector field, then M n is isometric to Euclidean sphere S n (r ).
Proof In regard of Lemma 3.1, we know that R and λ − ψ are constants. Moreover, using Lemma 2.3 [29] we conclude that R = 0, otherwise ψ = 0, a contradiction as ψ = 0. Taking Lie derivative of (3.1) and using the fact that R and λ − ψ are constants give Now, applying Theorem 4.2 of [29] to conclude that M n is isometric to Euclidean sphere S n (r ). This completes the proof. Now, we look at gradient ARYS admitting conformal vector field on which we state and prove the following:

is a non-trivial conformal vector field with non-negative scalar curvature, then either
(1) M n is isometric to a Euclidean space E n . or (2) M n is isometric to a Euclidean sphere S n . Moreover, ψ is a first eigenfunction of Laplacian and λ = 2α+nβ 2n R − λ 1 n f + k, where k is a constant. Proof Since ∇ f is a non-trivial conformal vector field, we have L ∇ f g = 2ψg, ψ = 0. Now, in consequence of argument (1) of Lemma 2.1, we get ψ = f n = 0. Moreover, from Lemma 3.1, we know that R and λ − ψ are constants. Suppose R = 0, then this implies that M n is Ricci flat and by using Tashiro's theorem [27] in the fundamental equation, we conclude that M n is isometric to a Euclidean space E n . On the other hand, suppose R = 0. Then, making use of Lemma 2.1 in ψ = f n gives λ = ψ + ( 2α+nβ 2n )R. As a consequence, (3.1) becomes Ric = R n g for α = 0. Therefore, by involving a theorem by Nagano and Yano [18], we can conclude that M n is isometric to a Euclidean sphere S n . Furthermore, taking into account of the fact that Ric = R n g, we can use Lichnerowicz's theorem [17], the first eigenvalue of the Laplacian of M n is λ 1 = R n−1 . Now, we make use of well known formula by Obata and Yano [19], which gives In view of (3.4), one can easily obtain ψ = −λ 1 ψ, that is, ψ is a first eigenfunction of the Laplacian. Also, we get ( f +λ 1 f ) = 0. Then, by Hopf theorem, we obtain f +λ 1 f = c, where c is a constant. Combining the last expression with Lemma 2.1 give us the required expression for λ. This completes the proof.
In [6], the authors consider almost Ricci-Bourguignon soliton and almost η-Ricci-Bourguignon soliton with solenoidal and torse-forming vector field and obtained several rigidity results. Following similar methods, we examine ARYS (M n , g, ξ, λ) with solenoidal and torse-forming vector fields.