Almost Ricci–Yamabe soliton on contact metric manifolds

The theory of geometric flows plays a significant role in understanding the geometric structure in Riemannian geometry. Hamilton [1] introduced the concept of Ricci flow. A Ricci soliton is a self-similar solution to Ricci flow ∂_tg(t) = −2S(t), where S is the Ricci curvature. Ricci solitons are a generalization of Einstein manifolds. A Ricci soliton on a Riemannian manifold (M, g) is defined by

where L_Vg denotes the Lie derivative of g along a vector field V, λ a constant and arbitrary vector field X, Y on M. If λ < 0, λ > 0 or λ = 0 then the soliton is said to be shrinking, expanding or steady, respectively. A Ricci soliton is said to be a gradient Ricci soliton if V = ∇f, for a smooth function f. For a detailed study on Ricci soliton see Refs. [2, 3] and references therein.

Hamilton [1] introduced a geometric flow that is similar to Ricci flow and called it Yamabe flow. Yamabe solitons correspond to self-similar solutions of the Yamabe flow. A Yamabe soliton preserves the conformal class of the metric but the Ricci soliton does not in general [1]. In dimension n = 2, both the solitons are similar. On a Riemannian manifold (M, g) a Yamabe soliton is given by

for arbitrary vector fields X, Y on M, λ a scalar and r the scalar curvature of M. If λ is a smooth function, then equations (1) and (2) are called Ricci almost soliton given by Pigola et al. [4] and almost Yamabe soliton given by Barbosa and Ribeiro [5], respectively. For a detailed study on Yamabe soliton see Refs. [6–10] and references therein.

Recently, in 2019, Guler and Crasmareanu [11] introduced a new type of geometric flow, a scalar combination of Ricci flow and Yamabe flow under the name Ricci–Yamabe map. In Ref. [11], the authors define the following:

The Ricci–Yamabe flow can be Riemannian or semi-Riemannian or singular Riemannian flow due to the involvement of scalars α and β [11]. These kinds of choices can be useful when dealing with relativity. The Ricci–Yamabe soliton emerges as the self-similar solutions of the Ricci–Yamabe flow. The notion of Ricci–Yamabe soliton from the Ricci–Yamabe flow can be defined as follows:

The Ricci–Yamabe soliton is said to be expanding, shrinking or steady if λ < 0, λ > 0 or λ = 0 respectively. Therefore, equation (3) is the Ricci–Yamabe soliton of (α, β)-type which is a combination of Ricci soliton and Yamabe soliton. In particular, (1,0), (0,1), (1, −1) and (1, − 2ρ)-type is the Ricci–Yamabe soliton are Ricci soliton, Yamabe soliton, Einstein soliton and ρ-Einstein soliton, respectively. Therefore, the notion of the Ricci–Yamabe soliton generalizes a large class of soliton-like equations. Using the terminology of Ricci almost soliton, the notion of almost Ricci–Yamabe soliton can be defined as follows:

Recently, in [12], the author studied the Ricci–Yamabe soliton on almost Kenmotsu manifolds. He shows that a (k, μ)′-almost Kenmotsu manifolds admitting a Ricci–Yamabe soliton or gradient Ricci–Yamabe soliton is locally isometric to the Riemannian product $Hn+1(−4)×Rn$⁠. Siddiqi and Akyol [13], introduced the notion of η-Ricci–Yamabe soliton and establish the geometrical bearing on Riemannian submersions in terms of η-Ricci–Yamabe soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an η-Ricci–Yamabe soliton, η-Ricci soliton and η-Yamabe soliton. Ricci–Yamabe soliton in perfect fluid spacetime is analyzed by authors in Ref. [14]. Khatri and Singh [15] studied almost Ricci–Yamabe soliton in different classes of almost Kenmotsu manifolds. In Ref. [16], Ghosh shows that if the metric of a non-Sasakian (k, μ)-contact metric is a gradient Ricci almost soliton, then in Dimension 3, it is flat and in higher dimensions it is locally isometric to Eⁿ⁺¹ × Sⁿ(4). Thus a natural question arises. “What happens when the metric of a non-Sasakian (k, μ)-contact metric manifold is a gradient almost Ricci-Yamabe soliton.”

The result of which is shown in section 4. Motivated by the above studies, we study almost Ricci–Yamabe soliton on contact metric manifolds. This paper aims to investigate the properties of almost contact metric manifolds whose metric admits almost Ricci–Yamabe solitons. The classification of K-contact and (κ, μ)-contact admitting almost Ricci–Yamabe soliton is obtained. The significance of studying almost Ricci–Yamabe soliton is that it generalizes a number of previously obtained results by Ghosh [16], Sharma [17] and some well-known results in Ricci soliton, Yamabe soliton and ρ-Einstein soliton within the framework of contact geometry.

The result of which is shown in section 4. Motivated by the above studies, we study almost Ricci–Yamabe soliton on contact metric manifolds. The present paper is organized as follows: After preliminaries in section 2, in section 3 we study almost (α, β)-Ricci–Yamabe solitons with the potential vector field collinear with the Reeb vector field ξ and found interesting results. Next in section 4, the gradient almost Ricci–Yamabe soliton in K-contact metric manifold is analyzed. Moreover in Section 5, the gradient almost Ricci–Yamabe soliton in the framework of (k, μ)-contact metric manifold is investigated and obtained that it is locally isometric to Eⁿ⁺¹ × Sⁿ(4) for n > 1 and flat if n = 1. Finally, an example of a three-dimensional (k, μ)-contact metric manifold is constructed.

In this section, we give some of the basic results and formulas of (k, μ)-contact metric manifold and refer to Refs. [17–20] for more information and details.

A 2n+1-dimensional smooth manifold M is called a contact manifold if it admits a global differential 1-form η (called contact form) such that η ∧ (dη)ⁿ ≠ 0 everywhere on M. A contact manifold induced as almost contact metric structure (η, ξ, ϕ, g), that is, a vector field ξ called the characteristic vector field, a (1,1)-tensor field ϕ and Riemannian metric g such that

for a vector fields X, Y on M. If in addition, dη(X, Y) = g(X, ϕY) then M is called a contact metric manifold [21]. Moreover, if ∇ denotes the Riemannian connection of g, then the following relation holds:

From the definition, it pursues that ϕξ = 0 and η◦ϕ = 0. Then, the manifold M(ϕ, ξ, η, g) equipped with such a structure is called a contact metric manifold [21, 22].

Given a contact metric manifold M we define a symmetric (1,1)-tensor field h and self adjoint operator l by $h=12Lξϕ$ and l = R(., ξ)ξ, where L denotes Lie differentiation. Then, hϕ = −ϕh, Trh = Tr ϕh = 0, hξ = 0. Also from Ref. [21],

A normal contact metric manifold is called a Sasakian manifold. A contact metric manifold is Sasakian if and only if

for any vector fields X, Y on M. The vector field ξ is a Killing vector with respect to g if and only if h = 0. A contact metric manifold M(ϕ, ξ, η, g) for which ξ is killing (equivalently h = 0 or Trl = 2n) is said to be a K-contact metric manifold. On a K-contact manifold, the following formulas are known [21].

where ∇ is the operator of covariant differentiation of g, S is the Ricci tensor of type (0,2) such that S(X, Y) = g(QX, Y), where Q is Ricci operator and R is the Riemann curvature tensor of g. A Sasakian manifold is K-contact and the converse is not true except in Dimension 3.

As a generalization of the Sasakian case, Blair et al. [18] introduced (k, μ)-nullity distribution on a contact metric manifold and gave several reasons for studying it. A full classification of (k, μ)-spaces was given by Boeckx [19].

The (k, μ)-nullity distribution of a contact metric manifold M²ⁿ⁺¹(ϕ, ξ, η, g) is a distribution

for any X, Y, Z ∈ T_pM and real numbers k and μ. A contact metric manifold M²ⁿ⁺¹ with ξ ∈ N(k, μ) is called a (k, μ)-contact metric manifold. In particular, if μ = 0, then the notion of (k, μ)-nullity distribution reduces to the notion of k-nullity distribution, introduced by Tanno [23]. If k = 1, the structure is Sasakian, and if k < 1, the (k, μ)-nullity condition determines the curvature of the manifold completely.

In a (k, μ)-contact metric manifold the following relations hold [18, 20].

Here, r is the scalar curvature of the manifold.

Ghosh [16] obtained a result for contact metric manifold with potential vector field collinear with the Reeb vector field. Motivated by this study, we extended it to an almost (α, β)-Ricci–Yamabe soliton. We prove the following:

Proof. Suppose the potential vector field is collinear with the Reeb vector field, i.e. V = σξ, where σ is a non-zero function on M. Differentiating it along arbitrary vector field X gives

Using this in (5) and simplifying we obtain

Taking X = Y = ξ in (18) yields

Replacing Y by ξ in (18) gives

Suppose that the Reeb vector field ξ is an eigenvector of the Ricci operator at each point of M, then Qξ = (Trl)ξ. Using this in the forgoing equation along with (19) gives, Dσ = (ξσ)ξ. Differentiating it along with vector field X yields

Making use of Poincare lemma in (21), we obtain

Choosing X, Y ⊥ ξ and using the fact that dη ≠ 0 in (22), we see that ξσ = 0. Hence, Dσ = 0 i.e. σ is a constant. Then (18) becomes,

Contracting (23) and using the fact that Trhϕ = 0, we get

Differentiating (23) along arbitrary vector field X gives

Contracting (25) and using the fact that in contact metric manifold, div(hϕ)Y = g(Qξ, Y) − 2nη(Y), in the forgoing equation result in the following:

Taking Y ⊥ ξ and using (24) in (26) gives α = 0 or Yr = 0. Assuming α ≠ 0 and replacing Y by ϕ²Y shows Dr = (ξr)ξ. Differentiating along arbitrary vector field X gives, ∇_XDr = X(ξr)ξ − (ξr)(ϕX + ϕhX). Applying Poincare lemma, the forgoing equation yields

choosing X, Y ⊥ ξ, it follows that ξr = 0. Hence, Dr = 0 i.e. r is constant. Then (24) implies λ is constant and consequently from (19), Trl is constant. In view of (26) we get Trl = 2n, i.e. h = 0. Hence manifold is K-contact and then from (23), it is Einstein provided α ≠ 0. Suppose M is complete, then making use of results in Sharma [17] and Boyer and Galicki [24], we see that the manifold is compact Einstein Sasakian. This completes the proof. □

From (19) we get, 2αTrl = (2λ − βr). Using this in (20) gives

making use of result by Perrone [25] and (28), we can state the following:

In consequence of Theorem 1, considering a particular case when potential vector field V is the Reeb vector field ξ, we can easily prove the following:

In [17], Sharma proved that if a compact K-contact metric is a gradient Ricci soliton then it is Einstein Sasakian. Extending this for gradient Ricci almost soliton, Ghosh [16] proved that compact K-contact metric is Einstein Sasakian and isometric to a unit sphere S²ⁿ⁺¹. However, this result is also true if one relaxes the hypothesis compactness to completeness (see Ref. [26]). In this section, we consider the gradient almost Ricci–Yamabe soliton and extend these results and prove.

Proof. A gradient almost Ricci–Yamabe soliton is given by

Taking covarient differentiation of (29) along arbitrary vector field Y yields

Since R(X, Y)Df = ∇_X∇_YDf − ∇_Y∇_XDf − ∇_[X,Y]Df, then in consequence of (30) we get

Differentiating (11) along vector field Y and using (12) gives

Taking inner product of (31) with ξ and replacing Y by ξ and using the fact that g(R(X, Y)Df, ξ) = −g(R(X, Y)ξ, Df) along with (12) and (32), Eq. (31) reduces to X(f + 2λ − βr) = ξ(f + 2λ − βr)η(X), which can be written as d(f + 2λ − βr) = ξ(f + 2λ − βr)η. Then operating the last equation by d and using Poincare lemma, i.e. d² = 0 we get dξ(f + 2λ − βr) ∧ η + ξ(f + 2λ − βr)dη = 0. Taking the wedge product of forgoing equation with η and using the fact that η ∧ η = 0 yields ξ(f + 2λ − βr)dη ∧ η = 0. Therefore ξ(f + 2λ − βr) = 0 on M as dη is non-vanishing everywhere on M, consequently, D(f + 2λ − βr) = 0. Hence, f + 2λ − βr is constant on M.

Taking Lie differentiation of (29) along ξ and noting $LξQ=0$ (as ξ is Killing) we obtain

Lie differentiating Df along ξ and using (10) yields

Differentiating covariently (34) along vector field Y and using (10) we obtain

According to Yano [27], we have the commutative formula

Setting V = ξ and X = Df in (36) and noting $Lξ∇=0$ and using (33)-(35) yields

Replacing X by ϕX and Y by ϕY along with well-known formula

we get

Suppose α ≠ 0. Since f + 2λ − βr is constant Eq. (38) reduces to

for any vector field X on M. Taking an inner product of (31) along with f + 2λ − βr = constant yields

Let {e_i, ϕe_i, ξ; i = 1, 2, …n} be an orthonormal ϕ − basis of M such that Qe_i = σ_ie_i. Using this in (39) we get Qϕe_i = (4n − σ_i)ϕe_i. Then the scalar curvature is given by

Replacing X by ξ in (40) and using (32) yields QϕDf − 2nϕDf = 0. In consequence of this in (39), it reduces to ϕQDf = 2nϕDf. Operating last equation with ϕ and using (11) gives QDf = 2nDf. Then taking covariant derivative results in

Since r = 2n(2n + 1) is constant, then $divQ=12dr=0$⁠. Making use of this and contracting (41) we obtain ‖Q‖² = 2nr. In consequence of this with r = 2n(2n + 1), we can easily see that $‖Q−r2n+1I‖2=0$ i.e., length of the symmetric tensor $Q−r2n+1I$ vanishes, we must have QX = 2nX. Thus M is Einstein with Einstein constant 2n. Suppose M is complete, then by the result of Sharma [17] we can conclude that M is compact. Applying Boyer–Galicki [24] we conclude that it is Sasakian. Also, Eq. (29) can be rewritten as ∇_XDf = −ρX, where ρ = 4αn + βr − 2λ, then by Obata’s theorem [28] it is isometric to a unit sphere S²ⁿ⁺¹. This completes the proof. □

In [16], Ghosh proved that if the metric of a non-Sasakian (κ, μ)-contact metric manifold admits a gradient Ricci almost soliton, then it is locally isometric to Eⁿ⁺¹ × Sⁿ(4) for n > 1. Following his work, we explore the gradient almost Ricci–Yamabe soliton on non-Sasakian (κ, μ)-contact metric manifold and obtain the following:

Proof. Making use of R(X, Y)Df = ∇_X∇_YDf − ∇_Y∇_XDf − ∇_[X,Y]Df and (29), we get

Taking covariant derivative of (15) and using it in (42) yields

Taking the inner product of (43) with ξ gives

Taking the inner product of (14) with Df, we get

Combining (44) and (45) we get

Taking X = ϕX and Y = ϕY and using the fact that R(ϕX, ϕY)ξ = 0, Eq. (46) for α ≠ 0 reduces to

Replacing Y = ξ in (46) gives

In consequence of (15), replacing X by Df and simplifying we obtain

Making use of (49) in (48) gives

Taking an inner product of (50) with ξ we get, k(ξf) + 2(ξλ) = 0 and using this in forgoing equation

Differentiating (51) and simplifying, we obtain

Taking inner product of (52) with ξ gives, μh(2λ − βr − 4nαk) = 0, and using it in (52)

Operating h in the above equation and using (13), we get

We get the following cases:

• Case-I: For μ = 0. In consequence, equation (47) gives k = 0. Hence, R(X, Y)ξ = 0.

Now in Blair [29] proved that a (2n + 1)-dimensional contact metric manifold satisfying R(X, Y)ξ = 0 is locally isometric to Eⁿ⁺¹ × Sⁿ(4) for n > 1 and flat if n = 1.

Therefore, we conclude that the manifold under consideration is locally isometric to Eⁿ⁺¹ × Sⁿ(4) for n > 1 and flat if n = 1.

• Case-II: For ϕDf = 0. Operating ϕ on both sides gives Df = (ξf)ξ. Differentiating along arbitrary vector field X gives

Applying the Poincare lemma in the above equation yields

Taking X, Y ⊥ ξ and since dη is nowhere vanishing on M, it follows ξf = 0. Hence Df = 0 i.e. f is constant. Then from (29) we see that M is Einstein (i.e., 2αQY = (2λ − βr)Y). Taking a trace of the last equation yields 2αr = (2n + 1)(2λ − βr). Also, replacing Y by ξ in the second last equation and using the previous equation results in QY = 2nkY. Consequently, the scalar curvature is r = 2nk(2n + 1). Now proceeding similarly as in Theorem 4.1 of Ghosh [16], we also find that for n = 1, M is locally flat (as μ = 0 and k = 0 consequently R(X, Y)ξ = 0), using μ = 2(1 − n) in (47), we see that $k=n−1n>1$⁠, a contraction. Since M³ is flat and λ is constant in view of (29), we see that the vector field is homothetic.

• Case-III: For 2(n − 1) + μ − 2nμ = 0 implies $μ=2(1−n)1−2n$⁠.

Using this value of μ in the expression of k in (47), we get $k=1n−n$⁠.

Replacing X by Df in (15) then inserting it in (51) yields

Inserting $μ=2(1−n)1−2n$ and $k=1n−n$ in (57), we obtain Df = (ξf)ξ. Then proceeding similarly as in Case-II we obtain a similar conclusion. Since QX = 2nkX, taking covarient differentiation gives ∇Q = 0 and consequently (42) reduces to

Since R(X, Y)ξ = 0 and taking inner product of forgoing equation with ξ and replacing Y by ξ gives Xλ = (ξλ)η(X). Similarly as above we can easily see that λ is constant and consequently R(X, Y)Df = 0 i.e., Df is tangent to the flat factor Eⁿ⁺¹. This completes the proof. □

Finally, we construct an example for verifying the obtained result. In Ref. [30], De and Mandal constructed a 3-dimensional example of a generalized (κ, μ)-contact metric manifold with κ = 1 − (σ(z))² and μ = 2(1 + σ(z)). Let $k:I⊂R→R$ be a smooth function defined on an open interval I such that k(z) ≤ 1 for any z ∈ I. Set $σ(z)=1−k(z)≥0$ and let {e₁, e₂, e₃} be three linearly independent vector fields on $M=R2×I⊂R3$ given by

where f(z) and h(z) are arbitrary functions of z. Let g be the Riemannian metric defined by

Let η be the 1-form defined by η(X) = g(X, e₁) for all vector field X on M. Also, let ϕ be the (1,1)-tensor field define as

Clearly, M(ϕ, ξ = e₁, η, g) formed a contact metric manifold. The non-vanishing components of the curvature tensor are given by Ref. [30]:

The non-vanishing components of the Ricci tensor and scalar curvature are as follows:

Now let us take $k∈R$ as a constant. Then the manifold M becomes (κ, μ)-contact metric manifold. Moreover, we see that for k ≠ 1, M is a non-Sasakian manifold. Let the potential vector field V = e₁, then solving (5) gives ασ(σ + 1) = 0, which implies σ = 0, a contradiction. Thus, Corollary 3 is verified.

We are thankful to the reviewers for their valuable suggestions. The first author is thankful to the Department of Science and Technology, New Delhi, India for financial support in the form of Inspire Fellowship (DST/INSPIRE Fellowship/2018/IF180830).