YAMABE SOLITONS IN CONTACT GEOMETRY

. It is shown that the scalar curvature of a Yamabe soliton as a Sasakian manifold is constant and the soliton vector field is Killing. The same conclusion is shown to hold for a Yamabe soliton as a K -contact manifold M 2 n +1 if any one of the following conditions hold: (i) its scalar curvature is constant along the soliton vector field V , (ii) V is an eigenvector of the Ricci operator with eigenvalue 2 n , (iii) V is gradient.


Introduction
The evolution of a Riemannian metric g on a smooth manifold M to a metric g(t) in time t through the equation ∂ ∂t g(t) = −r(t)g(t), g(0) = g, where r(t) denotes the scalar curvature of g(t), is called the Yamabe flow, and was introduced by Hamilton [8].The Yamabe flow is a natural geometric deformation to metrics of constant scalar curvature, and corresponds to the fast diffusion case of the porous medium equation (the plasma equation) in mathematical physics (Burchard et al. [3]).Just as a Ricci soliton is a special solution of the Ricci flow, a Yamabe soliton is a special solution of the Yamabe flow that moves by a one parameter family of diffeomorphisms ϕ t generated by a time-dependent vector field W t on M , and homotheties, i.e., g(t) = σ(t)ϕ * t g, where σ is a positive real valued function of the parameter t.Substituting the foregoing equation in the Yamabe flow equation and setting σ(0) = 1, − σ(0) = c gives the equation where g is the initial metric g(0) of the Yamabe flow, V is a vector field on M such that W t = 1 2σ(t) V , £ the Lie-derivative operator, r the scalar curvature of g, and c is a real constant defined earlier.The Riemannian manifold (M, g) with a vector field V and a constant c, satisfying the equation (1.1) is called a Yamabe soliton.The Yamabe soliton is said to be shrinking, steady, or expanding when c > 0, c = 0, or c < 0 respectively.In particular, if V = Df (up to the addition of a Killing vector field) for a smooth function f , where D denotes the gradient operator of g, then a Yamabe soliton is called a gradient Yamabe soliton in which case (1.1) assumes the form ∇∇f = (c − r)g. (1. 2) The gradient Yamabe soliton is trivial when f is constant and r is constant.In [6], Daskalopoulos and Sesum showed that a compact gradient Yamabe soliton has constant scalar curvature (i.e., its metric is a Yamabe metric).Chen and Deshmukh [5] obtained some sufficient conditions on Yamabe solitons to be of constant scalar curvature.Ghosh [7] has shown that a Yamabe soliton on a Kenmotsu manifold has constant scalar curvature.In [15], Sharma studied Yamabe solitons in contact Riemannian geometry, and proved that a 3-dimensional Yamabe soliton whose metric is Sasakian has constant scalar curvature and the soliton vector field V is Killing.In this paper, we generalize this result in any dimension in the form of the following result.
Theorem 1.1.Let (M, g) be a Yamabe soliton with soliton vector field V .If g is a Sasakian metric, then it has constant scalar curvature and V is Killing.Furthermore, we show that the same conclusion (as in the above theorem) holds on a (2n + 1)-dimensional K-contact manifold M (a generalization of Sasakian manifold) under certain conditions specified in the following result.More precisely, we establish Theorem 1.2.Let (M, g) be a (2n + 1)-dimensional Yamabe soliton with soliton vector field V , and g be a K-contact metric.Then its scalar curvature is constant and V is Killing, if any one of the following conditions hold: (i) The scalar curvature is constant along V .(ii) V is an eigenvector of the Ricci operator with eigenvalue 2n.(iii) V is gradient.In case (iii), the Yamabe soliton becomes trivial (i.e., f is constant).

Remarks.
1.An odd dimensional analogue of Kaehler geometry is the Sasakian geometry.The Kaehler cone over a Sasakian Einstein manifold is a Calabi-Yau manifold which has application in superstring theory based on a 10-dimensional manifold that is the product of the 4-dimensional space-time and a 6-dimensional Ricci-flat Kaehler (Calabi-Yau) manifold (see Candelas et al. [4]).Sasakian geometry has been extensively studied since its recently perceived relevance in string theory.Sasakian Einstein metrics have received a lot of attention in physics, for example, p-brane solutions in superstring theory, Maldacena conjecture (AdS/CFS duality) [11].
2. The condition (ii) in Theorem 1.2 is motivated by the fact that, if we take V as the Reeb vector field ξ, then the Yamabe soliton equation (1.1) implies r = c because ξ is Killing for a K-contact metric.

A Brief Review Of Contact Geometry
A (2n + 1)-dimensional smooth manifold M is said to be contact if it has a global 1-form η such that η ∧ (dη) n ̸ = 0 on M .For a contact 1-form η there exists a unique vector field ξ (Reeb vector field) such that dη(ξ, .)= 0 and η(ξ) = 1.Henceforth X, Y, Z will denote arbitrary vector field on M .Polarizing dη on the contact subbundle η = 0, we obtain a Riemannian metric g and a (1,1)-tensor field φ such that g is called an associated metric of η and (φ, η, ξ, g) a contact metric structure.
The name contact seems to be due to Sophus Lie [10] and is natural in view of the simple example of Huygens' principle (cited in Blair [2]).Tangent wave fronts are mapped to tangent wave fronts through a contact transformation.According to Gibbs, the geometrical structure of thermodynamics is described by a contact manifold equipped with a contact form whose zeros define the laws of thermodynamics (Arnold [1]).
A contact metric structure is said to be K-contact if ξ is Killing with respect to g.
The following formulas are valid on a K-contact manifold.
Ric(X, ξ) = 2ng(X, ξ), i.e., Qξ = 2nξ, (2.3) where ∇, R, Ric and Q denote respectively, the Riemannian connection, curvature tensor, Ricci tensor and Ricci operator of g.A contact metric manifold (M, g) is said to be Sasakian if the almost Kaehler structure on the cone manifold (M ×R + , r 2 g + dr 2 ) over M is Kaehler.Sasakian manifolds are K-contact and the converse is true only in dimension 3.For a Sasakian manifold, the following formula holds: For details, we refer to the standard monograph of Blair [2].
It is evident from the defining equation (1.1) of a Yamabe soliton, that the associated vector field V is a conformal vector field with conformal scale function 2(c−r).A vector field V on an m-dimensional Riemannian manifold (M, g) is said to be a conformal vector field if £ V g = 2ρg (2.5) for a smooth function ρ on M .Denoting the gradient vector field of ρ by Dρ, the Laplacian −div.Dρ by ∆ρ, and Y ] Z, we have the following integrability conditions for the conformal vector field V (Yano [17]): (2.8)

Proofs of The Results
Proof of Theorem 1.1 First, we note that this theorem was already proved in dimension 3 by Sharma [15].So, we now prove it in higher dimensions.The crucial idea of the proof is based on the following results of Okumura [12] (Theorems 3.1 and 3.3): A conformal vector field V (defined by (2.5) on a Sasakian manifold of dimension 2n + 1 > 3 differs from −Dρ by a Killing vector field, i.e., V = W − Dρ where W is a Killing vector field on M , and the conformal scale function ρ satisfies the concircular equation: ∇∇ρ = −ρg.For a Yamabe soliton (1.1) we note that ρ = c − r, and therefore the above equation can be expressed as Proof of Theorem 1.2 Taking the Lie-derivative of the K-contact formula (2.3) along the Yamabe vector field V and using the integrability equation (2.7) with m = 2n + 1, we have At this point, we compute the term ∇ ξ Dρ as follows.As ρ = c − r, this term is −∇ ξ Dr.To compute its value, we take the Lie-derivative of dr(X) = g(Dr, X) along ξ and noting that the Lie-derivative operator commutes with the exterior derivative operator, we find that (d£ ξ r)(X) = g(∇ ξ Dr − ∇ Dr ξ, X).We note that ξr = 0 because, by definition, ξ is Killing for a K-contact manifold.As a consequence, the use of the formula (2.To prove part (ii), we assume the hypothesis: QV = 2nV and operate it by £ ξ , and noting that ξ is Killing and hence £ ξ Q = 0, we find that Q£ ξ V = 2n£ ξ V .On the other hand, equations (3.5) and (3.6) provide Combined use of these two results, and that £ V ξ = −£ ξ V shows that φDr = 0. Operating it by φ and using the last equation in (2.1) in conjunction with ξr = 0 (because ξ is Killing) we conclude that Dr = 0, i.e., r is constant.Hence V is homothetic.But a homothetic vector field on a K-contact manifold is Killing [14], and hence V is Killing.This proves part (ii).
In order to prove part (iii), we let V = Df up to the sum of a Killing vector field.So, we can present equation (1.2) in the form Applying the exterior derivation, using Poincare formula: d 2 = 0, and taking its wedge product with η provides (ξf )η ∧ dη = 0. Since η ∧ dη is nowhere zero on M , by definition of the contact structure, we conclude that ξf = 0. Hence Xf = Xr, i.e., f differs from r by a constant.Thus, we note, in passing, that equation (3.9) is same as (3.1).Virtually following the argument used in the proof of Theorem 1.1 from equation (3.1) onwards, we conclude that r is constant.As f differs from r by a constant, and r is constant, it follows that f is constant and therefore the Yamabe soliton is trivial, completing the proof.