Abstract

We study Ricci solitons in 𝛼-Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a 𝛼-Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if ℒ𝑉𝑔+2𝑆 is parallel where 𝑉 is a given vector field, then (𝑔,𝑉,𝜆) is Ricci soliton. Further, by virtue of this result, Ricci solitons for 𝑛-dimensional 𝛼-Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional 𝛼-Sasakian manifolds are discussed with an example.

1. Introduction

In 1982, Hamilton [1] introduced the concept of Ricci flow which smooths out the geometry of manifold that is if there are singular points these can be minimized under Ricci flow. Ricci solitons move under the Ricci flow simply by diffeomorphisms of the initial metric that is they are stationary points of the Ricci flow: 𝜕𝑔/𝜕𝑡=−2Ric(𝑔), (in this paper we use Ric=𝑆) in the space of metrics on 𝑀. Hence it is interesting to study Ricci solitons.

Definition 1.1. A Ricci soliton (𝑔,𝑉,𝜆) on a Riemannian manifold 𝑀 is defined by ℒ𝑉𝑔+2𝑆+2𝜆𝑔=0.(1.1) It is said to be shrinking, steady, or expanding according as 𝜆<0,𝜆=0 and 𝜆>0.

Note that here the metric 𝑔(𝑡) is the pull back of the initial metric 𝑔(0) by a 1-parameter family of diffeomorphisms generated by a vector field 𝑉 on a manifold 𝑀. Compact Ricci solitons are the fixed points of the Ricci flow: 𝜕𝑔/𝜕𝑡=−2Ric(𝑔) projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings and often arise as blow-up limits for the Ricci flow on compact manifolds.

In 1923, Eisenhart [2] proved that if a positive definite Riemannian manifold (𝑀,𝑔) admits a second order parallel symmetric covariant tensor other than a constant multiple of the metric tensor, then it is reducible. In 1925, Levy [3] obtained the necessary and sufficient conditions for the existence of such tensors. In 1989, 1990, and 1991, Sharma [4–6] has generalized Levy's result by showing that a second order parallel (not necessarily symmetric and nonsingular) tensor on an 𝑛-dimensional (𝑛>2) space of constant curvature is a constant multiple of the metric tensor. It is also proved that in a Sasakian manifold there is no nonzero parallel 2-form. In 2007, Das [7] in his paper proved that a second order symmetric parallel tensor on an 𝛼-K-contact (𝛼∈𝑅0) manifold is a constant multiple of the associated metric tensor and also proved that there is no nonzero skew symmetric second order parallel tensor on an 𝛼-Sasakian manifold. Note that 𝛼-Sasakian manifolds are generalisations of Sasakian manifolds. Hence one can find interest in generalisation, from Sasakian to 𝛼-Sasakian manifolds and study Ricci solitons in this manifold.

In 2008, Sharma [8] studied Ricci solitons in K-contact manifolds, where the structure field 𝜉 is killing and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. In 2010, Călin and Crasmareanu [9] extended the Eisenhart problem to Ricci solitons in 𝑓-Kenmotsu manifolds. They studied the case of 𝑓-Kenmotsu manifolds satisfying a special condition called regular and a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric. Using this result, they obtained the results on Ricci solitons. Recently, Bagewadi and Ingalahalli [10] studied Ricci solitons in Lorentzian 𝛼-Sasakian Manifolds.

In this paper, we obtain some results on Ricci solitons.

2. Preliminaries

Let 𝑀 be an almost contact metric manifold of dimension 𝑛, equipped with an almost contact metric structure (𝜙,𝜉,𝜂,𝑔) consisting of a (1,1) tensor field 𝜙, a vector field 𝜉, a 1-form 𝜂 and a Riemannian metric 𝑔, which satisfy 𝜙2=−𝐼+𝜂⊗𝜉,𝜂(𝜉)=1,𝜂∘𝜙=0,𝜙𝜉=0,(2.1)𝑔(𝜙𝑋,𝜙𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),𝜂(𝑋)=𝑔(𝑋,𝜉),(2.2) for all 𝑋,𝑌∈𝔛(𝑀). An almost contact metric manifold 𝑀(𝜙,𝜉,𝜂,𝑔) is said to be 𝛼-Sasakian manifold if the following conditions hold: ∇𝑋𝜙𝑌=𝛼(𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋),(2.3)∇𝑋∇𝜉=−𝛼𝜙𝑋,𝑋𝜂𝑌=𝛼𝑔(𝑋,𝜙𝑌).(2.4) Holds for some smooth function 𝛼 on 𝑀.

In an 𝛼-Sasakian manifold, the following relations hold: 𝑅(𝑋,𝑌)𝜉=𝛼2[]𝜂(𝑌)𝑋−𝜂(𝑋)𝑌+(𝑌𝛼)𝜙𝑋−(𝑋𝛼)𝜙𝑌,(2.5)𝑅(𝜉,𝑋)𝑌=𝛼2[]𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋+𝑔(𝑋,𝜙𝑌)(grad𝛼)+(𝑌𝛼)𝜙𝑋,(2.6)𝜂(𝑅(𝑋,𝑌)𝑍)=𝛼2[]𝑔(𝑌,𝑍)𝜂(𝑋)−𝑔(𝑋,𝑍)𝜂(𝑌)+(𝑋𝛼)𝑔(𝜙𝑌,𝑍)−(𝑌𝛼)𝑔(𝜙𝑋,𝑍),(2.7)𝑆(𝑋,𝜉)=𝛼2(𝑛−1)𝜂(𝑋)−((𝜙𝑋)𝛼),(2.8)𝑆(𝜉,𝜉)=𝛼2(𝑛−1),(2.9)𝑄𝜉=𝛼2(𝑛−1)𝜉+𝜙(grad𝛼),(2.10) for all 𝑋,𝑌,𝑍∈𝔛(𝑀), where 𝑅 is the Riemannian curvature tensor, 𝑆 is the Ricci tensor and 𝑄 is the Ricci operator.

3. Parallel Symmetric Second Order Tensors and Ricci Solitonsin 𝛼-Sasakian Manifolds

Fix ℎ a symmetric tensor field of (0,2)-type which we suppose to be parallel with respect to ∇ that is ∇ℎ=0. Applying the Ricci identity [4, 11] ∇2ℎ(𝑋,𝑌;𝑍,𝑊)−∇2ℎ(𝑋,𝑌;𝑊,𝑍)=0,(3.1) we obtain the relation ℎ(𝑅(𝑋,𝑌)𝑍,𝑊)+ℎ(𝑍,𝑅(𝑋,𝑌)𝑊)=0.(3.2) Replacing 𝑍=𝑊=𝜉 in (3.2) and by using (2.5) and by the symmetry of ℎ, we have 2[](𝑌𝛼)ℎ(𝜙𝑋,𝜉)−(𝑋𝛼)ℎ(𝜙𝑌,𝜉)+2𝛼2[]𝜂(𝑌)ℎ(𝑋,𝜉)−𝜂(𝑋)ℎ(𝑌,𝜉)=0.(3.3) Put 𝑋=𝜉 in (3.3) and by virtue of (2.1), we have 2𝛼2[]𝜂(𝑌)ℎ(𝜉,𝜉)−ℎ(𝑌,𝜉)−2(𝜉𝛼)ℎ(𝜙𝑌,𝜉)=0.(3.4) Replacing 𝑌=𝜙𝑌 in (3.4), we have []2(𝜉𝛼)ℎ(𝑌,𝜉)−𝜂(𝑌)ℎ(𝜉,𝜉)−2𝛼2ℎ(𝜙𝑌,𝜉)=0.(3.5) Solving (3.4) and (3.5), we have 𝛼4+(𝜉𝛼)2[]𝜂(𝑌)ℎ(𝜉,𝜉)−ℎ(𝑌,𝜉)=0.(3.6) Since 𝛼4+(𝜉𝛼)2≠0, it results ℎ(𝑌,𝜉)=𝜂(𝑌)ℎ(𝜉,𝜉).(3.7) Differentiating (3.7) covariantly with respect to 𝑋, we have î€·âˆ‡ğ‘‹â„Žî€¸î€·âˆ‡(𝑌,𝜉)+â„Žğ‘‹î€¸î€·ğ‘Œ,𝜉+â„Žğ‘Œ,∇𝑋𝜉=∇𝑋𝜂(∇𝑌)+ğœ‚ğ‘‹ğ‘Œâˆ‡î€¸î€»â„Ž(𝜉,𝜉)+𝜂(𝑌)î€ºî€·ğ‘‹â„Žî€¸(∇𝑌,𝜉)+2â„Žğ‘‹.𝜉,𝜉(3.8) By using the parallel condition ∇ℎ=0, 𝜂(∇𝑋𝜉)=0 and (3.7) in (3.8), we have â„Žî€·ğ‘Œ,∇𝑋𝜉=∇𝑋𝜂(𝑌)ℎ(𝜉,𝜉).(3.9) By using (2.4) in (3.9), we get âˆ’ğ›¼â„Ž(𝑌,𝜙𝑋)=𝛼𝑔(𝑋,𝜙𝑌)ℎ(𝜉,𝜉).(3.10) Replacing 𝑋=𝜙𝑋 in (3.10), we get 𝛼[]ℎ(𝑌,𝑋)−𝑔(𝑌,𝑋)ℎ(𝜉,𝜉)=0.(3.11) Since 𝛼 is a nonzero smooth function in 𝛼-Sasakian manifold and this implies that ℎ(𝑋,𝑌)=𝑔(𝑋,𝑌)ℎ(𝜉,𝜉),(3.12) the above equation implies that ℎ(𝜉,𝜉) is a constant, via (3.7). Now by considering the above condition we state the following theorem.

Theorem 3.1. A symmetric parallel second order covariant tensor in an 𝛼-Sasakian manifold is a constant multiple of the metric tensor.

Corollary 3.2. A locally Ricci symmetric (∇𝑆=0)𝛼-Sasakian manifold is an Einstein manifold.

Remark 3.3. The following statements for 𝛼-Sasakian manifold are equivalent: (1)Einstein, (2)locally Ricci symmetric, (3)Ricci semi-symmetric that is 𝑅⋅𝑆=0.
The implication (1)→(2)→(3) is trivial. Now, we prove the implication (3)→(1) and 𝑅⋅𝑆=0 means exactly (3.2) with replaced ℎ by 𝑆 that is, (𝑅(𝑋,𝑌)⋅𝑆)(𝑈,𝑉)=−𝑆(𝑅(𝑋,𝑌)𝑈,𝑉)−𝑆(𝑈,𝑅(𝑋,𝑌)𝑉).(3.13) Considering 𝑅⋅𝑆=0 and putting 𝑋=𝜉 in (3.13), we have 𝑆(𝑅(𝜉,𝑌)𝑈,𝑉)+𝑆(𝑈,𝑅(𝜉,𝑌)𝑉)=0.(3.14) By using (2.6) in (3.14), we obtain 𝑔(𝜙𝑈,𝑌)𝑆(grad𝛼,𝑉)+(𝑈𝛼)𝑆(𝜙𝑌,𝑉)+𝛼2[]+𝑔𝑔(𝑌,𝑈)𝑆(𝜉,𝑉)−𝜂(𝑈)𝑆(𝑌,𝑉)(𝜙𝑉,𝑌)𝑆(𝑈,grad𝛼)+(𝑉𝛼)𝑆(𝑈,𝜙𝑌)+𝛼2[𝑔](𝑌,𝑉)𝑆(𝑈,𝜉)−𝜂(𝑉)𝑆(𝑈,𝑌)=0.(3.15) Putting 𝑈=𝜉 in (3.15) and by using (2.1), (2.8), and (2.9) on simplification, we obtain (𝜉𝛼)𝑆(𝜙𝑌,𝑉)−𝛼2𝜂(𝑌)((𝜙𝑉)𝛼)−𝛼2𝑆(𝑌,𝑉)+𝑔(𝑌,𝜙𝑉)𝑆(𝜉,grad𝛼)+𝛼4(𝑛−1)𝑔(𝑌,𝑉)+𝛼2𝜂(𝑉)((𝜙𝑌)𝛼)=0.(3.16) Interchanging 𝑌 and 𝑉 in (3.16), we have (𝜉𝛼)𝑆(𝜙𝑉,𝑌)−𝛼2𝜂(𝑉)((𝜙𝑌)𝛼)−𝛼2𝑆(𝑉,𝑌)+𝑔(𝑉,𝜙𝑌)𝑆(𝜉,grad𝛼)+𝛼4(𝑛−1)𝑔(𝑉,𝑌)+𝛼2𝜂(𝑌)((𝜙𝑉)𝛼)=0.(3.17) Adding (3.16) and (3.17), we obtain 𝑆(𝑌,𝑉)=(𝑛−1)𝛼2𝑔(𝑌,𝑉).(3.18) We conclude the following.

Proposition 3.4. A Ricci semi-symmetric 𝛼-Sasakian manifold is an Einstein manifold.

Corollary 3.5. Suppose that on a 𝛼-Sasakian manifold the (0,2)-type field ℒ𝑉𝑔+2𝑆 is parallel where 𝑉 is a given vector field. Then (𝑔,𝑉,𝜆) yield a Ricci soliton. In particular, if the given 𝛼-Sasakian manifold is Ricci semi-symmetric with ℒ𝑉𝑔 parallel, one has the same conclusion.

Proof. Follows from Theorem 3.1 and Corollary 3.2.
A Ricci soliton in 𝛼-Sasakian manifold defined by (1.1). Thus ℒ𝑉𝑔+2𝑆 is parallel. In Theorem 3.1 we proved that if an 𝛼-Sasakian manifold admits a symmetric parallel (0,2) tensor, then the tensor is a constant multiple of the metric tensor. Hence ℒ𝑉𝑔+2𝑆 is a constant multiple of the metric tensor 𝑔 that is (ℒ𝑉𝑔+2𝑆)(𝑋,𝑌)=𝑔(𝑋,𝑌)ℎ(𝜉,𝜉), where ℎ(𝜉,𝜉) is a nonzero constant. We close this section with applications of our Theorem 3.1 to Ricci solitons.

Corollary 3.6. If a metric 𝑔 in an 𝛼-Sasakian manifold is a Ricci soliton with 𝑉=𝜉 then it is Einstein.

Proof. Putting 𝑉=𝜉 in (1.1), then we have ℒ𝜉𝑔+2𝑆+2𝜆𝑔(𝑋,𝑌)=0,(3.19) where ℒ𝜉𝑔∇(𝑋,𝑌)=𝑔𝑋𝜉,𝑌+𝑔𝑋,∇𝑌𝜉=0.(3.20) Substituting (3.20) in (3.19), then we get the result.

Hence we state the following result.

Corollary 3.7. A Ricci soliton (𝑔,𝜉,𝜆) in an 𝑛-dimensional 𝛼-Sasakian manifold cannot be steady but is shrinking.

Proof. From Linear Algebra either the vector field 𝑉∈Span𝜉 or 𝑉⟂𝜉. However the second case seems to be complex to analyse in practice. For this reason we investigate for the case 𝑉=𝜉.
A simple computation of ℒ𝜉𝑔+2𝑆 gives ℒ𝜉𝑔(𝑋,𝑌)=0.(3.21) From (1.1), we have ℎ(𝑋,𝑌)=−2𝜆𝑔(𝑋,𝑌) and then putting 𝑋=𝑌=𝜉, we have ℎ(𝜉,𝜉)=−2𝜆,(3.22) where ℎℒ(𝜉,𝜉)=𝜉𝑔(𝜉,𝜉)+2𝑆(𝜉,𝜉),(3.23) by using (2.9) and (3.21) in the above equation, we have ℎ(𝜉,𝜉)=2𝛼2(𝑛−1).(3.24) Equating (3.22) and (3.24), we have 𝜆=−(𝑛−1)𝛼2.(3.25) Since 𝛼 is some nonzero function, we have 𝜆≠0, that is Ricci soliton in an 𝑛-dimensional 𝛼-Sasakian manifold cannot be steady but is shrinking because 𝜆<0.

Corollary 3.8 3.8. If an 𝑛-dimensional 𝛼-Sasakian manifold is 𝜂-Einstein then the Ricci solitons in 𝛼-Sasakian manifold that is (𝑔,𝜉,𝜆), where 𝜆=−(𝑛−1)𝛼2 with varying scalar curvature cannot be steady but it is shrinking.

Proof. The proof consists of three parts. (i)We prove 𝛼-Sasakian manifold is 𝜂-Einstein. (ii)We prove the Ricci soliton in 𝛼-Sasakian manifold is consisting of varying scalar curvature. (iii)We find that the Ricci soliton in 𝛼-Sasakian manifold is shrinking.
First we prove that the 𝛼-Sasakian manifold is 𝜂-Einstein: the metric 𝑔 is called 𝜂-Einstein if there exists two real functions ğ‘Ž and 𝑏 such that the Ricci tensor of 𝑔 is given by the general equation 𝑆(𝑋,𝑌)=ğ‘Žğ‘”(𝑋,𝑌)+𝑏𝜂(𝑋)𝜂(𝑌).(3.26) Now by simple calculations we find the values of ğ‘Ž and 𝑏. Let 𝑒𝑖,  {𝑖=1,2,…𝑛} be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=𝑌=𝑒𝑖 in (3.26) and taking summation over 𝑖, we get 𝑟=ğ‘›ğ‘Ž+𝑏.(3.27) Again putting 𝑋=𝑌=𝜉 in (3.26) then by using (2.9), we have ğ‘Ž+𝑏=(𝑛−1)𝛼2.(3.28) Then from (3.27) and (3.28), we have î‚¸ğ‘Ÿğ‘Ž=(𝑛−1)−𝛼2,𝑏=𝑛𝛼2−𝑟.(𝑛−1)(3.29) Substituting the values of ğ‘Ž and 𝑏 in (3.26), we have 𝑟𝑆(𝑋,𝑌)=(𝑛−1)−𝛼2𝑔(𝑋,𝑌)+𝑛𝛼2−𝑟(𝑛−1)𝜂(𝑋)𝜂(𝑌),(3.30) the above equation is an 𝜂-Einstein 𝛼-Sasakian manifold.
Now, we have to show that the scalar curvature 𝑟 is not a constant and it is varying. For an 𝑛-dimensional 𝛼-Sasakian manifolds the symmetric parallel covariant tensor ℎ(𝑋,𝑌) of type (0,2) is given by ℎℒ(𝑋,𝑌)=𝜉𝑔(𝑋,𝑌)+2𝑆(𝑋,𝑌).(3.31) By using (3.21) and (3.30) in (3.31), we have ℎ(𝑋,𝑌)=2𝑟(𝑛−1)−2𝛼2𝑔(𝑋,𝑌)+2𝑛𝛼2−2𝑟(𝑛−1)𝜂(𝑋)𝜂(𝑌).(3.32) Differentiating (3.32) covariantly with respect to 𝑍, we have î€·âˆ‡ğ‘â„Žî€¸îƒ¬2∇(𝑋,𝑌)=𝑍𝑟2∇(𝑛−1)−4𝛼(𝑍𝛼)𝑔(𝑋,𝑌)+4𝑛𝛼(𝑍𝛼)−𝑍𝑟+(𝑛−1)𝜂(𝑋)𝜂(𝑌)2𝑛𝛼2−2𝑟(𝑔𝑛−1)𝑋,∇𝑍𝜉𝜂(𝑌)+𝑔𝑌,∇𝑍𝜉.𝜂(𝑋)(3.33) By substituting 𝑍=𝜉 and 𝑋=𝑌∈(Span𝜉)⟂ in (3.33) and by using ∇ℎ=0, we have ∇𝜉𝑟=2(𝑛−1)𝛼(𝜉𝛼)⟹∇𝜉𝑟=(𝑛−1)∇𝜉𝛼2.(3.34) On integrating (3.34), we have 𝑟=(𝑛−1)𝛼2+𝑐,(3.35) where 𝑐 is some integral constant. Thus from (3.35), we have 𝑟 is a varying scalar curvature.
Finally, we have to check the nature of the soliton that is Ricci soliton in 𝛼-Sasakian manifold:
From (1.1), we have ℎ(𝑋,𝑌)=−2𝜆𝑔(𝑋,𝑌) then putting 𝑋=𝑌=𝜉, we have ℎ(𝜉,𝜉)=−2𝜆.(3.36) If we put 𝑋=𝑌=𝜉 in (3.32), that is ℎ(𝜉,𝜉)=2𝑟(𝑛−1)−2𝛼2𝑔(𝜉,𝜉)+2𝑛𝛼2−2𝑟(𝑛−1)𝜂(𝜉)𝜂(𝜉),(3.37) Above equation reduced as, ℎ(𝜉,𝜉)=2(𝑛−1)𝛼2.(3.38) Equating (3.36) and (3.38), we have 𝜆=−(𝑛−1)𝛼2.(3.39) Since, 𝜆≠0 because 𝛼 is some smooth function and 𝜆<0, that is the Ricci soliton in an 𝛼-Sasakian manifold is shrinking.

4. Ricci Solitons in 3-Dimensional 𝛼-Sasakian Manifold

In this section we restrict our study to 3-dimensional 𝛼-Sasakian manifold, that is Ricci solitons in 3-dimensional 𝛼-Sasakian manifold.

Corollary 4.1. If a Ricci soliton (𝑔,𝜉,𝜆) where 𝜆=−2𝛼2 of 3-dimensional 𝛼-Sasakian manifold with varying scalar curvature cannot be steady but it is shrinking.

Proof. The proof consists of three parts. (i)We prove that the Riemannian curvature tensor of 3-dimensional 𝛼-Sasakian manifold is 𝜂-Einstein. (ii)We prove that the Ricci soliton in 3-dimensional 𝛼-Sasakian manifold is consisting of varying scalar curvature. (iii)We find that the Ricci soliton in a 3-dimensional 𝛼-Sasakian manifold is shrinking.
First we consider: the Riemannian curvature tensor of 3-dimensional 𝛼-Sasakian manifold and it is given by 𝑟𝑅(𝑋,𝑌)𝑍=𝑔(𝑌,𝑍)𝑄𝑋−𝑔(𝑋,𝑍)𝑄𝑌+𝑆(𝑌,𝑍)𝑋−𝑆(𝑋,𝑍)𝑌−2[𝑔].(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌(4.1) Put 𝑍=𝜉 in (4.1) and by using (2.5) and (2.8), we have [](𝑌𝛼)𝜙𝑋−(𝑋𝛼)𝜙𝑌+𝛼2[]𝜂(𝑌)𝑋−𝜂(𝑋)𝑌=𝜂(𝑌)𝑄𝑋−𝜂(𝑋)𝑄𝑌+2𝛼2[]𝑟𝜂(𝑌)𝑋−𝜂(𝑋)𝑌−((𝜙𝑌)𝛼)𝑋+((𝜙𝑋)𝛼)−2[].𝜂(𝑌)𝑋−𝜂(𝑋)𝑌(4.2) Again put 𝑌=𝜉 in (4.2) and by using (2.1) and (2.10), on simplification we get 𝑟𝑄𝑋=2−𝛼2𝑋+3𝛼2−𝑟2𝜂(𝑋)𝜉+(𝜉𝛼)𝜙𝑋+𝜂(𝑋)(𝜙(grad𝛼))+((𝜙𝑋)𝛼)𝜉.(4.3) By taking an inner product 𝑌 in (4.3), we have 𝑟𝑆(𝑋,𝑌)=2−𝛼2𝑔(𝑋,𝑌)+3𝛼2−𝑟2𝜂(𝑋)𝜂(𝑌)+(𝜉𝛼)𝑔(𝜙𝑋,𝑌)−𝜂(𝑋)((𝜙𝑌)𝛼)+𝜂(𝑌)((𝜙𝑋)𝛼).(4.4) Interchanging 𝑋 and 𝑌 in (4.4), we have 𝑟𝑆(𝑌,𝑋)=2−𝛼2(𝑌,𝑋)+3𝛼2−𝑟2𝜂(𝑋)𝜂(𝑌)+(𝜉𝛼)𝑔(𝜙𝑌,𝑋)−𝜂(𝑌)((𝜙𝑋)𝛼)+𝜂(𝑋)((𝜙𝑌)𝛼).(4.5) Adding (4.4) and (4.5), we have 𝑟𝑆(𝑋,𝑌)=2−𝛼2𝑔(𝑋,𝑌)+3𝛼2−𝑟2𝜂(𝑋)𝜂(𝑌).(4.6) Equation (4.6) shows that a 3-dimensional 𝛼-Sasakian manifold is 𝜂-Einstein.
Now, we have to show that the scalar curvature 𝑟 is not a constant that is 𝑟 is varying. Now, ℎℒ(𝑋,𝑌)=𝜉𝑔(𝑋,𝑌)+2𝑆(𝑋,𝑌).(4.7) By using (3.21) and (4.6) in (4.7), we have ℎ(𝑋,𝑌)=𝑟−2𝛼2𝑔(𝑋,𝑌)+6𝛼2−𝑟𝜂(𝑋)𝜂(𝑌).(4.8) Differentiating the above equation covariantly with respect to 𝑍, we have î€·âˆ‡ğ‘â„Žî€¸î€ºâˆ‡(𝑋,𝑌)=𝑍𝑔𝑟−4𝛼(𝑍𝛼)(𝑋,𝑌)+12𝛼(𝑍𝛼)−∇𝑍𝑟𝜂+(𝑋)𝜂(𝑌)6𝛼2𝑔−𝑟𝑋,∇𝑍𝜉𝜂(𝑌)+𝑔𝑌,∇𝑍𝜉.𝜂(𝑋)(4.9) Substituting 𝑍=𝜉,  𝑋=𝑌∈(Span𝜉)⟂ in (4.9) and by virtue of ∇ℎ=0, we have ∇𝜉𝑟=4𝛼(𝜉𝛼)⟹∇𝜉𝑟=∇𝜉2𝛼2.(4.10) On integrating (4.10), we have 𝑟=2𝛼2+𝑐,(4.11) where 𝑐 is some integral constant. Thus from (4.11), we have 𝑟 a varying scalar curvature.
Finally, we have to check the nature of the soliton that is Ricci soliton (𝑔,𝜉,𝜆) in 3-dimensional 𝛼-Sasakian manifold.
From (1.1), we have ℎ(𝑋,𝑌)=−2𝜆𝑔(𝑋,𝑌) and then putting 𝑋=𝑌=𝜉, we have ℎ(𝜉,𝜉)=−2𝜆.(4.12) If 𝑋=𝑌=𝜉 in (4.8), that is ℎ(𝜉,𝜉)=𝑟−2𝛼2𝑔(𝜉,𝜉)+6𝛼2−𝑟𝜂(𝜉)𝜂(𝜉).(4.13) Above equation reduced as ℎ(𝜉,𝜉)=4𝛼2.(4.14) Equating (4.12) and (4.14), we have 𝜆=−2𝛼2.(4.15) Since from (4.15), we have 𝜆≠0. Therefore Ricci soliton (𝑔,𝜉,𝜆) in 3-dimensional 𝛼-Sasakian manifold is shrinking.

Example 4.2. Let 𝑀={(𝑥,𝑦,𝑧)∈𝑅3}. Let (𝐸1,𝐸2,𝐸3) be linearly independent vector fields given by 𝐸1=𝑒𝑥𝜕𝜕𝑦,𝐸2=𝑒𝑥𝜕𝜕𝜕𝑥+2𝑦𝜕𝑧,𝐸3=𝜕.𝜕𝑧(4.16) Let 𝑔 be the Riemannian metric defined by 𝑔(𝐸1,𝐸2)=𝑔(𝐸2,𝐸3)=𝑔(𝐸1,𝐸3)=0, 𝑔(𝐸1,𝐸1)=𝑔(𝐸2,𝐸2)=𝑔(𝐸3,𝐸3)=1, where 𝑔 is given by 1𝑔=𝑒2𝑥1−4𝑒2𝑥𝑦2𝑑𝑥⊗𝑑𝑥+𝑑𝑦⊗𝑑𝑦+𝑒2𝑥.𝑑𝑧⊗𝑑𝑧(4.17) Let 𝜂 be the 1-form defined by 𝜂(𝑈)=𝑔(𝑈,𝐸3) for any 𝑈∈𝔛(𝑀). Let 𝜙 be the (1,1) tensor field defined by 𝜙𝐸1=𝐸2,𝜙𝐸2=−𝐸1,𝜙𝐸3=0. Then using the linearity of 𝜙 and 𝑔 yields that 𝜂(𝐸3)=1,𝜙2𝑈=−𝑈+𝜂(𝑈)𝐸3 and 𝑔(𝜙𝑈,𝜙𝑊)=𝑔(𝑈,𝑊)−𝜂(𝑈)𝜂(𝑊) for any vector fields 𝑈,𝑊∈𝔛(𝑀). Thus for 𝐸3=𝜉, (𝜙,𝜉,𝜂,𝑔) defines a Sasakian structure on 𝑀. By definition of Lie bracket, we have 𝐸1,𝐸2=−𝑒𝑥𝐸1+2𝑒2𝑥𝐸3,𝐸1,𝐸3=𝐸2,𝐸3=0.(4.18) Let ∇ be the Levi-Civita connection with respect to above metric 𝑔 Koszula formula is given by ∇2𝑔𝑋[][][]𝑌,𝑍=𝑋(𝑔(𝑌,𝑍))+𝑌(𝑔(𝑍,𝑋))−𝑍(𝑔(𝑋,𝑌))−𝑔(𝑋,𝑌,𝑍)−𝑔(𝑌,𝑋,𝑍)+𝑔(𝑍,𝑋,𝑌).(4.19) Then ∇𝐸1𝐸1=𝑒𝑥𝐸2,∇𝐸2𝐸2=0,∇𝐸3𝐸3∇=0,𝐸1𝐸2=−𝑒𝑥𝐸1+𝑒2𝑥𝐸3,∇𝐸2𝐸1=−𝑒2𝑥𝐸3,∇𝐸2𝐸3=𝑒2𝑥𝐸1,∇𝐸1𝐸3=−𝑒2𝑥𝐸2,∇𝐸3𝐸1=−𝑒2𝑥𝐸2,∇𝐸3𝐸2=𝑒2𝑥𝐸1.(4.20) Clearly (𝜙,𝜉,𝜂,𝑔) structure is an 𝛼-Sasakian structure and satisfy, ∇𝑋𝜙𝑌=𝛼(𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋),∇𝑋𝜉=−𝛼𝜙𝑋,(4.21) where 𝛼=𝑒2𝑥≠0. Hence (𝜙,𝜉,𝜂,𝑔) structure defines 𝛼-Sasakian structure. Thus 𝑀 equipped with 𝛼-Sasakian structure is a 𝛼-Sasakian manifold. The tangent vectors 𝑋 and 𝑌 to 𝑀 are expressed as linear combination of 𝐸1,𝐸2,𝐸3, that is ∑𝑋=3𝑖=1ğ‘Žğ‘–ğ¸ğ‘– and ∑𝑌=3𝑖=1𝑏𝑖𝐸𝑖, where ğ‘Žğ‘– and 𝑏𝑖(𝑖=1,2,3) are scalars.
Using 𝛼=𝑒2𝑥 in (4.11), we have 𝑟=2𝑒4𝑥+𝑐≠0,(4.22) and it shows that the scalar curvature is not constant.
Using 𝛼=𝑒2𝑥 in (4.15), we have 𝜆=−2𝑒4𝑥≠0.(4.23) In this example 𝛼=𝑒2𝑥≠0, this implies that 𝜆<0, that is the Ricci soliton in 3-dimensional 𝛼-Sasakian manifold is shrinking.

5. Conclusion

In this paper we have shown that the Ricci soliton in an 𝛼-Sasakian manifold cannot be steady but it is shrinking accordingly because 𝜆 is negative.

Acknowledgment

The authors express their thanks to DST (Department of Science and Technology), Government of India, for providing financial assistance under the major research project (no. SR/S4/MS: 482/07).