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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 421384, 13 pages
http://dx.doi.org/10.5402/2012/421384
Research Article

Ricci Solitons in 𝛼-Sasakian Manifolds

Department of Mathematics, Kuvempu University, Shankaraghatta 577 451, India

Received 20 April 2012; Accepted 28 May 2012

Academic Editors: F. P.Β Schuller and I.Β Strachan

Copyright Β© 2012 Gurupadavva Ingalahalli and C. S. Bagewadi. 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.

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,πœ†) 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: πœ•π‘”/πœ•π‘‘=βˆ’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,πœ†) on a Riemannian manifold 𝑀 is defined by ℒ𝑉𝑔+2𝑆+2πœ†π‘”=0.(1.1) It is said to be shrinking, steady, or expanding according as πœ†<0,πœ†=0 and πœ†>0.

Note that here the metric 𝑔(𝑑) 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: πœ•π‘”/πœ•π‘‘=βˆ’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 (π›Όβˆˆπ‘…0) 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 πœ‰ is killing and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. In 2010, CΔƒlin 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 (πœ™,πœ‰,πœ‚,𝑔) consisting of a (1,1) tensor field πœ™, a vector field πœ‰, a 1-form πœ‚ and a Riemannian metric 𝑔, which satisfy πœ™2=βˆ’πΌ+πœ‚βŠ—πœ‰,πœ‚(πœ‰)=1,πœ‚βˆ˜πœ™=0,πœ™πœ‰=0,(2.1)𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ),πœ‚(𝑋)=𝑔(𝑋,πœ‰),(2.2) for all 𝑋,π‘Œβˆˆπ”›(𝑀). An almost contact metric manifold 𝑀(πœ™,πœ‰,πœ‚,𝑔) is said to be 𝛼-Sasakian manifold if the following conditions hold: ξ€·βˆ‡π‘‹πœ™ξ€Έπ‘Œ=𝛼(𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋),(2.3)βˆ‡π‘‹ξ€·βˆ‡πœ‰=βˆ’π›Όπœ™π‘‹,π‘‹πœ‚ξ€Έπ‘Œ=𝛼𝑔(𝑋,πœ™π‘Œ).(2.4) Holds for some smooth function 𝛼 on 𝑀.

In an 𝛼-Sasakian manifold, the following relations hold: 𝑅(𝑋,π‘Œ)πœ‰=𝛼2[]πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ+(π‘Œπ›Ό)πœ™π‘‹βˆ’(𝑋𝛼)πœ™π‘Œ,(2.5)𝑅(πœ‰,𝑋)π‘Œ=𝛼2[]𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋+𝑔(𝑋,πœ™π‘Œ)(grad𝛼)+(π‘Œπ›Ό)πœ™π‘‹,(2.6)πœ‚(𝑅(𝑋,π‘Œ)𝑍)=𝛼2[]𝑔(π‘Œ,𝑍)πœ‚(𝑋)βˆ’π‘”(𝑋,𝑍)πœ‚(π‘Œ)+(𝑋𝛼)𝑔(πœ™π‘Œ,𝑍)βˆ’(π‘Œπ›Ό)𝑔(πœ™π‘‹,𝑍),(2.7)𝑆(𝑋,πœ‰)=𝛼2(π‘›βˆ’1)πœ‚(𝑋)βˆ’((πœ™π‘‹)𝛼),(2.8)𝑆(πœ‰,πœ‰)=𝛼2(π‘›βˆ’1),(2.9)π‘„πœ‰=𝛼2(π‘›βˆ’1)πœ‰+πœ™(grad𝛼),(2.10) for all 𝑋,π‘Œ,π‘βˆˆπ”›(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 β„Ž a symmetric tensor field of (0,2)-type which we suppose to be parallel with respect to βˆ‡ that is βˆ‡β„Ž=0. Applying the Ricci identity [4, 11] βˆ‡2β„Ž(𝑋,π‘Œ;𝑍,π‘Š)βˆ’βˆ‡2β„Ž(𝑋,π‘Œ;π‘Š,𝑍)=0,(3.1) we obtain the relation β„Ž(𝑅(𝑋,π‘Œ)𝑍,π‘Š)+β„Ž(𝑍,𝑅(𝑋,π‘Œ)π‘Š)=0.(3.2) Replacing 𝑍=π‘Š=πœ‰ in (3.2) and by using (2.5) and by the symmetry of β„Ž, we have 2[](π‘Œπ›Ό)β„Ž(πœ™π‘‹,πœ‰)βˆ’(𝑋𝛼)β„Ž(πœ™π‘Œ,πœ‰)+2𝛼2[]πœ‚(π‘Œ)β„Ž(𝑋,πœ‰)βˆ’πœ‚(𝑋)β„Ž(π‘Œ,πœ‰)=0.(3.3) Put 𝑋=πœ‰ in (3.3) and by virtue of (2.1), we have 2𝛼2[]πœ‚(π‘Œ)β„Ž(πœ‰,πœ‰)βˆ’β„Ž(π‘Œ,πœ‰)βˆ’2(πœ‰π›Ό)β„Ž(πœ™π‘Œ,πœ‰)=0.(3.4) Replacing π‘Œ=πœ™π‘Œ in (3.4), we have []2(πœ‰π›Ό)β„Ž(π‘Œ,πœ‰)βˆ’πœ‚(π‘Œ)β„Ž(πœ‰,πœ‰)βˆ’2𝛼2β„Ž(πœ™π‘Œ,πœ‰)=0.(3.5) Solving (3.4) and (3.5), we have 𝛼4+(πœ‰π›Ό)2ξ€Έ[]πœ‚(π‘Œ)β„Ž(πœ‰,πœ‰)βˆ’β„Ž(π‘Œ,πœ‰)=0.(3.6) Since 𝛼4+(πœ‰π›Ό)2β‰ 0, it results β„Ž(π‘Œ,πœ‰)=πœ‚(π‘Œ)β„Ž(πœ‰,πœ‰).(3.7) Differentiating (3.7) covariantly with respect to 𝑋, we have ξ€·βˆ‡π‘‹β„Žξ€Έξ€·βˆ‡(π‘Œ,πœ‰)+β„Žπ‘‹ξ€Έξ€·π‘Œ,πœ‰+β„Žπ‘Œ,βˆ‡π‘‹πœ‰ξ€Έ=βˆ‡ξ€Ίξ€·π‘‹πœ‚ξ€Έ(ξ€·βˆ‡π‘Œ)+πœ‚π‘‹π‘Œβˆ‡ξ€Έξ€»β„Ž(πœ‰,πœ‰)+πœ‚(π‘Œ)ξ€Ίξ€·π‘‹β„Žξ€Έ(ξ€·βˆ‡π‘Œ,πœ‰)+2β„Žπ‘‹.πœ‰,πœ‰ξ€Έξ€»(3.8) By using the parallel condition βˆ‡β„Ž=0, πœ‚(βˆ‡π‘‹πœ‰)=0 and (3.7) in (3.8), we have β„Žξ€·π‘Œ,βˆ‡π‘‹πœ‰ξ€Έ=ξ€·βˆ‡π‘‹πœ‚ξ€Έ(π‘Œ)β„Ž(πœ‰,πœ‰).(3.9) By using (2.4) in (3.9), we get βˆ’π›Όβ„Ž(π‘Œ,πœ™π‘‹)=𝛼𝑔(𝑋,πœ™π‘Œ)β„Ž(πœ‰,πœ‰).(3.10) Replacing 𝑋=πœ™π‘‹ in (3.10), we get 𝛼[]β„Ž(π‘Œ,𝑋)βˆ’π‘”(π‘Œ,𝑋)β„Ž(πœ‰,πœ‰)=0.(3.11) Since 𝛼 is a nonzero smooth function in 𝛼-Sasakian manifold and this implies that β„Ž(𝑋,π‘Œ)=𝑔(𝑋,π‘Œ)β„Ž(πœ‰,πœ‰),(3.12) the above equation implies that β„Ž(πœ‰,πœ‰) 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 (βˆ‡π‘†=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)β†’(2)β†’(3) is trivial. Now, we prove the implication (3)β†’(1) and 𝑅⋅𝑆=0 means exactly (3.2) with replaced β„Ž by 𝑆 that is, (𝑅(𝑋,π‘Œ)⋅𝑆)(π‘ˆ,𝑉)=βˆ’π‘†(𝑅(𝑋,π‘Œ)π‘ˆ,𝑉)βˆ’π‘†(π‘ˆ,𝑅(𝑋,π‘Œ)𝑉).(3.13) Considering 𝑅⋅𝑆=0 and putting 𝑋=πœ‰ in (3.13), we have 𝑆(𝑅(πœ‰,π‘Œ)π‘ˆ,𝑉)+𝑆(π‘ˆ,𝑅(πœ‰,π‘Œ)𝑉)=0.(3.14) By using (2.6) in (3.14), we obtain 𝑔(πœ™π‘ˆ,π‘Œ)𝑆(grad𝛼,𝑉)+(π‘ˆπ›Ό)𝑆(πœ™π‘Œ,𝑉)+𝛼2[]+𝑔𝑔(π‘Œ,π‘ˆ)𝑆(πœ‰,𝑉)βˆ’πœ‚(π‘ˆ)𝑆(π‘Œ,𝑉)(πœ™π‘‰,π‘Œ)𝑆(π‘ˆ,grad𝛼)+(𝑉𝛼)𝑆(π‘ˆ,πœ™π‘Œ)+𝛼2[𝑔](π‘Œ,𝑉)𝑆(π‘ˆ,πœ‰)βˆ’πœ‚(𝑉)𝑆(π‘ˆ,π‘Œ)=0.(3.15) Putting π‘ˆ=πœ‰ in (3.15) and by using (2.1), (2.8), and (2.9) on simplification, we obtain (πœ‰π›Ό)𝑆(πœ™π‘Œ,𝑉)βˆ’π›Ό2πœ‚(π‘Œ)((πœ™π‘‰)𝛼)βˆ’π›Ό2𝑆(π‘Œ,𝑉)+𝑔(π‘Œ,πœ™π‘‰)𝑆(πœ‰,grad𝛼)+𝛼4(π‘›βˆ’1)𝑔(π‘Œ,𝑉)+𝛼2πœ‚(𝑉)((πœ™π‘Œ)𝛼)=0.(3.16) Interchanging π‘Œ and 𝑉 in (3.16), we have (πœ‰π›Ό)𝑆(πœ™π‘‰,π‘Œ)βˆ’π›Ό2πœ‚(𝑉)((πœ™π‘Œ)𝛼)βˆ’π›Ό2𝑆(𝑉,π‘Œ)+𝑔(𝑉,πœ™π‘Œ)𝑆(πœ‰,grad𝛼)+𝛼4(π‘›βˆ’1)𝑔(𝑉,π‘Œ)+𝛼2πœ‚(π‘Œ)((πœ™π‘‰)𝛼)=0.(3.17) Adding (3.16) and (3.17), we obtain 𝑆(π‘Œ,𝑉)=(π‘›βˆ’1)𝛼2𝑔(π‘Œ,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,πœ†) 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,π‘Œ)=𝑔(𝑋,π‘Œ)β„Ž(πœ‰,πœ‰), where β„Ž(πœ‰,πœ‰) 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=πœ‰ then it is Einstein.

Proof. Putting 𝑉=πœ‰ in (1.1), then we have ξ€·β„’πœ‰ξ€Έπ‘”+2𝑆+2πœ†π‘”(𝑋,π‘Œ)=0,(3.19) where ξ€·β„’πœ‰π‘”ξ€Έξ€·βˆ‡(𝑋,π‘Œ)=π‘”π‘‹ξ€Έξ€·πœ‰,π‘Œ+𝑔𝑋,βˆ‡π‘Œπœ‰ξ€Έ=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,πœ‰,πœ†) in an 𝑛-dimensional 𝛼-Sasakian manifold cannot be steady but is shrinking.

Proof. From Linear Algebra either the vector field π‘‰βˆˆSpanπœ‰ or π‘‰βŸ‚πœ‰. However the second case seems to be complex to analyse in practice. For this reason we investigate for the case 𝑉=πœ‰.
A simple computation of β„’πœ‰π‘”+2𝑆 gives ξ€·β„’πœ‰π‘”ξ€Έ(𝑋,π‘Œ)=0.(3.21) From (1.1), we have β„Ž(𝑋,π‘Œ)=βˆ’2πœ†π‘”(𝑋,π‘Œ) and then putting 𝑋=π‘Œ=πœ‰, we have β„Ž(πœ‰,πœ‰)=βˆ’2πœ†,(3.22) where β„Žξ€·β„’(πœ‰,πœ‰)=πœ‰π‘”ξ€Έ(πœ‰,πœ‰)+2𝑆(πœ‰,πœ‰),(3.23) by using (2.9) and (3.21) in the above equation, we have β„Ž(πœ‰,πœ‰)=2𝛼2(π‘›βˆ’1).(3.24) Equating (3.22) and (3.24), we have πœ†=βˆ’(π‘›βˆ’1)𝛼2.(3.25) Since 𝛼 is some nonzero function, we have πœ†β‰ 0, that is Ricci soliton in an 𝑛-dimensional 𝛼-Sasakian manifold cannot be steady but is shrinking because πœ†<0.

Corollary 3.8 3.8. If an 𝑛-dimensional 𝛼-Sasakian manifold is πœ‚-Einstein then the Ricci solitons in 𝛼-Sasakian manifold that is (𝑔,πœ‰,πœ†), where πœ†=βˆ’(π‘›βˆ’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 πœ‚-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 πœ‚-Einstein: the metric 𝑔 is called πœ‚-Einstein if there exists two real functions π‘Ž and 𝑏 such that the Ricci tensor of 𝑔 is given by the general equation 𝑆(𝑋,π‘Œ)=π‘Žπ‘”(𝑋,π‘Œ)+π‘πœ‚(𝑋)πœ‚(π‘Œ).(3.26) Now by simple calculations we find the values of π‘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=π‘Œ=𝑒𝑖 in (3.26) and taking summation over 𝑖, we get π‘Ÿ=π‘›π‘Ž+𝑏.(3.27) Again putting 𝑋=π‘Œ=πœ‰ in (3.26) then by using (2.9), we have π‘Ž+𝑏=(π‘›βˆ’1)𝛼2.(3.28) Then from (3.27) and (3.28), we have ξ‚Έπ‘Ÿπ‘Ž=(π‘›βˆ’1)βˆ’π›Ό2ξ‚Ήξ‚Έ,𝑏=𝑛𝛼2βˆ’π‘Ÿξ‚Ή.(π‘›βˆ’1)(3.29) Substituting the values of π‘Ž and 𝑏 in (3.26), we have ξ‚Έπ‘Ÿπ‘†(𝑋,π‘Œ)=(π‘›βˆ’1)βˆ’π›Ό2𝑔(𝑋,π‘Œ)+𝑛𝛼2βˆ’π‘Ÿξ‚Ή(π‘›βˆ’1)πœ‚(𝑋)πœ‚(π‘Œ),(3.30) the above equation is an πœ‚-Einstein 𝛼-Sasakian manifold.
Now, we have to show that the scalar curvature π‘Ÿ is not a constant and it is varying. For an 𝑛-dimensional 𝛼-Sasakian manifolds the symmetric parallel covariant tensor β„Ž(𝑋,π‘Œ) of type (0,2) is given by β„Žξ€·β„’(𝑋,π‘Œ)=πœ‰π‘”ξ€Έ(𝑋,π‘Œ)+2𝑆(𝑋,π‘Œ).(3.31) By using (3.21) and (3.30) in (3.31), we have ξ‚Έβ„Ž(𝑋,π‘Œ)=2π‘Ÿ(π‘›βˆ’1)βˆ’2𝛼2𝑔(𝑋,π‘Œ)+2𝑛𝛼2βˆ’2π‘Ÿξ‚Ή(π‘›βˆ’1)πœ‚(𝑋)πœ‚(π‘Œ).(3.32) Differentiating (3.32) covariantly with respect to 𝑍, we have ξ€·βˆ‡π‘β„Žξ€Έξƒ¬2ξ€·βˆ‡(𝑋,π‘Œ)=π‘π‘Ÿξ€Έξƒ­ξƒ¬2ξ€·βˆ‡(π‘›βˆ’1)βˆ’4𝛼(𝑍𝛼)𝑔(𝑋,π‘Œ)+4𝑛𝛼(𝑍𝛼)βˆ’π‘π‘Ÿξ€Έξƒ­+ξ‚Έ(π‘›βˆ’1)πœ‚(𝑋)πœ‚(π‘Œ)2𝑛𝛼2βˆ’2π‘Ÿ(ξ‚Ήξ€Ίπ‘”ξ€·π‘›βˆ’1)𝑋,βˆ‡π‘πœ‰ξ€Έξ€·πœ‚(π‘Œ)+π‘”π‘Œ,βˆ‡π‘πœ‰ξ€Έξ€».πœ‚(𝑋)(3.33) By substituting 𝑍=πœ‰ and 𝑋=π‘Œβˆˆ(Spanπœ‰)βŸ‚ in (3.33) and by using βˆ‡β„Ž=0, we have βˆ‡πœ‰π‘Ÿ=2(π‘›βˆ’1)𝛼(πœ‰π›Ό)βŸΉβˆ‡πœ‰π‘Ÿ=(π‘›βˆ’1)βˆ‡πœ‰π›Ό2.(3.34) On integrating (3.34), we have π‘Ÿ=(π‘›βˆ’1)𝛼2+𝑐,(3.35) where 𝑐 is some integral constant. Thus from (3.35), we have π‘Ÿ 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 β„Ž(𝑋,π‘Œ)=βˆ’2πœ†π‘”(𝑋,π‘Œ) then putting 𝑋=π‘Œ=πœ‰, we have β„Ž(πœ‰,πœ‰)=βˆ’2πœ†.(3.36) If we put 𝑋=π‘Œ=πœ‰ in (3.32), that is ξ‚Έβ„Ž(πœ‰,πœ‰)=2π‘Ÿ(π‘›βˆ’1)βˆ’2𝛼2𝑔(πœ‰,πœ‰)+2𝑛𝛼2βˆ’2π‘Ÿξ‚Ή(π‘›βˆ’1)πœ‚(πœ‰)πœ‚(πœ‰),(3.37) Above equation reduced as, β„Ž(πœ‰,πœ‰)=2(π‘›βˆ’1)𝛼2.(3.38) Equating (3.36) and (3.38), we have πœ†=βˆ’(π‘›βˆ’1)𝛼2.(3.39) Since, πœ†β‰ 0 because 𝛼 is some smooth function and πœ†<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,πœ‰,πœ†) where πœ†=βˆ’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 πœ‚-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 π‘Ÿπ‘…(𝑋,π‘Œ)𝑍=𝑔(π‘Œ,𝑍)π‘„π‘‹βˆ’π‘”(𝑋,𝑍)π‘„π‘Œ+𝑆(π‘Œ,𝑍)π‘‹βˆ’π‘†(𝑋,𝑍)π‘Œβˆ’2[𝑔].(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(4.1) Put 𝑍=πœ‰ in (4.1) and by using (2.5) and (2.8), we have [](π‘Œπ›Ό)πœ™π‘‹βˆ’(𝑋𝛼)πœ™π‘Œ+𝛼2[]πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ=πœ‚(π‘Œ)π‘„π‘‹βˆ’πœ‚(𝑋)π‘„π‘Œ+2𝛼2[]π‘Ÿπœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œβˆ’((πœ™π‘Œ)𝛼)𝑋+((πœ™π‘‹)𝛼)βˆ’2[].πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ(4.2) Again put π‘Œ=πœ‰ in (4.2) and by using (2.1) and (2.10), on simplification we get ξ‚ƒπ‘Ÿπ‘„π‘‹=2βˆ’π›Ό2𝑋+3𝛼2βˆ’π‘Ÿ2ξ‚„πœ‚(𝑋)πœ‰+(πœ‰π›Ό)πœ™π‘‹+πœ‚(𝑋)(πœ™(grad𝛼))+((πœ™π‘‹)𝛼)πœ‰.(4.3) By taking an inner product π‘Œ in (4.3), we have ξ‚ƒπ‘Ÿπ‘†(𝑋,π‘Œ)=2βˆ’π›Ό2𝑔(𝑋,π‘Œ)+3𝛼2βˆ’π‘Ÿ2ξ‚„πœ‚(𝑋)πœ‚(π‘Œ)+(πœ‰π›Ό)𝑔(πœ™π‘‹,π‘Œ)βˆ’πœ‚(𝑋)((πœ™π‘Œ)𝛼)+πœ‚(π‘Œ)((πœ™π‘‹)𝛼).(4.4) Interchanging 𝑋 and π‘Œ in (4.4), we have ξ‚ƒπ‘Ÿπ‘†(π‘Œ,𝑋)=2βˆ’π›Ό2(π‘Œ,𝑋)+3𝛼2βˆ’π‘Ÿ2ξ‚„πœ‚(𝑋)πœ‚(π‘Œ)+(πœ‰π›Ό)𝑔(πœ™π‘Œ,𝑋)βˆ’πœ‚(π‘Œ)((πœ™π‘‹)𝛼)+πœ‚(𝑋)((πœ™π‘Œ)𝛼).(4.5) Adding (4.4) and (4.5), we have ξ‚ƒπ‘Ÿπ‘†(𝑋,π‘Œ)=2βˆ’π›Ό2𝑔(𝑋,π‘Œ)+3𝛼2βˆ’π‘Ÿ2ξ‚„πœ‚(𝑋)πœ‚(π‘Œ).(4.6) Equation (4.6) shows that a 3-dimensional 𝛼-Sasakian manifold is πœ‚-Einstein.
Now, we have to show that the scalar curvature π‘Ÿ is not a constant that is π‘Ÿ is varying. Now, β„Žξ€·β„’(𝑋,π‘Œ)=πœ‰π‘”ξ€Έ(𝑋,π‘Œ)+2𝑆(𝑋,π‘Œ).(4.7) By using (3.21) and (4.6) in (4.7), we have ξ€Ίβ„Ž(𝑋,π‘Œ)=π‘Ÿβˆ’2𝛼2𝑔(𝑋,π‘Œ)+6𝛼2ξ€»βˆ’π‘Ÿπœ‚(𝑋)πœ‚(π‘Œ).(4.8) Differentiating the above equation covariantly with respect to 𝑍, we have ξ€·βˆ‡π‘β„Žξ€Έξ€Ίβˆ‡(𝑋,π‘Œ)=π‘ξ€»π‘”ξ€Ίπ‘Ÿβˆ’4𝛼(𝑍𝛼)(𝑋,π‘Œ)+12𝛼(𝑍𝛼)βˆ’βˆ‡π‘π‘Ÿξ€»πœ‚+ξ€Ί(𝑋)πœ‚(π‘Œ)6𝛼2π‘”ξ€·βˆ’π‘Ÿξ€»ξ€Ίπ‘‹,βˆ‡π‘πœ‰ξ€Έξ€·πœ‚(π‘Œ)+π‘”π‘Œ,βˆ‡π‘πœ‰ξ€Έξ€».πœ‚(𝑋)(4.9) Substituting 𝑍=πœ‰,  𝑋=π‘Œβˆˆ(Spanπœ‰)βŸ‚ in (4.9) and by virtue of βˆ‡β„Ž=0, we have βˆ‡πœ‰π‘Ÿ=4𝛼(πœ‰π›Ό)βŸΉβˆ‡πœ‰π‘Ÿ=βˆ‡πœ‰ξ€·2𝛼2ξ€Έ.(4.10) On integrating (4.10), we have π‘Ÿ=2𝛼2+𝑐,(4.11) where 𝑐 is some integral constant. Thus from (4.11), we have π‘Ÿ a varying scalar curvature.
Finally, we have to check the nature of the soliton that is Ricci soliton (𝑔,πœ‰,πœ†) in 3-dimensional 𝛼-Sasakian manifold.
From (1.1), we have β„Ž(𝑋,π‘Œ)=βˆ’2πœ†π‘”(𝑋,π‘Œ) and then putting 𝑋=π‘Œ=πœ‰, we have β„Ž(πœ‰,πœ‰)=βˆ’2πœ†.(4.12) If 𝑋=π‘Œ=πœ‰ in (4.8), that is ξ€Ίβ„Ž(πœ‰,πœ‰)=π‘Ÿβˆ’2𝛼2𝑔(πœ‰,πœ‰)+6𝛼2ξ€»βˆ’π‘Ÿπœ‚(πœ‰)πœ‚(πœ‰).(4.13) Above equation reduced as β„Ž(πœ‰,πœ‰)=4𝛼2.(4.14) Equating (4.12) and (4.14), we have πœ†=βˆ’2𝛼2.(4.15) Since from (4.15), we have πœ†β‰ 0. Therefore Ricci soliton (𝑔,πœ‰,πœ†) in 3-dimensional 𝛼-Sasakian manifold is shrinking.

Example 4.2. Let 𝑀={(π‘₯,𝑦,𝑧)βˆˆπ‘…3}. Let (𝐸1,𝐸2,𝐸3) be linearly independent vector fields given by 𝐸1=𝑒π‘₯πœ•πœ•π‘¦,𝐸2=𝑒π‘₯ξ‚ƒπœ•πœ•πœ•π‘₯+2π‘¦ξ‚„πœ•π‘§,𝐸3=πœ•.πœ•π‘§(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=𝑒2π‘₯ξ€Ίξ€·1βˆ’4𝑒2π‘₯𝑦2𝑑π‘₯βŠ—π‘‘π‘₯+π‘‘π‘¦βŠ—π‘‘π‘¦+𝑒2π‘₯ξ€».π‘‘π‘§βŠ—π‘‘π‘§(4.17) Let πœ‚ be the 1-form defined by πœ‚(π‘ˆ)=𝑔(π‘ˆ,𝐸3) for any π‘ˆβˆˆπ”›(𝑀). Let πœ™ be the (1,1) tensor field defined by πœ™πΈ1=𝐸2,πœ™πΈ2=βˆ’πΈ1,πœ™πΈ3=0. Then using the linearity of πœ™ and 𝑔 yields that πœ‚(𝐸3)=1,πœ™2π‘ˆ=βˆ’π‘ˆ+πœ‚(π‘ˆ)𝐸3 and 𝑔(πœ™π‘ˆ,πœ™π‘Š)=𝑔(π‘ˆ,π‘Š)βˆ’πœ‚(π‘ˆ)πœ‚(π‘Š) for any vector fields π‘ˆ,π‘Šβˆˆπ”›(𝑀). Thus for 𝐸3=πœ‰, (πœ™,πœ‰,πœ‚,𝑔) defines a Sasakian structure on 𝑀. By definition of Lie bracket, we have 𝐸1,𝐸2ξ€»=βˆ’π‘’π‘₯𝐸1+2𝑒2π‘₯𝐸3,𝐸1,𝐸3ξ€»=𝐸2,𝐸3ξ€»=0.(4.18) Let βˆ‡ be the Levi-Civita connection with respect to above metric 𝑔 Koszula formula is given by ξ€·βˆ‡2𝑔𝑋[][][]π‘Œ,𝑍=𝑋(𝑔(π‘Œ,𝑍))+π‘Œ(𝑔(𝑍,𝑋))βˆ’π‘(𝑔(𝑋,π‘Œ))βˆ’π‘”(𝑋,π‘Œ,𝑍)βˆ’π‘”(π‘Œ,𝑋,𝑍)+𝑔(𝑍,𝑋,π‘Œ).(4.19) Then βˆ‡πΈ1𝐸1=𝑒π‘₯𝐸2,βˆ‡πΈ2𝐸2=0,βˆ‡πΈ3𝐸3βˆ‡=0,𝐸1𝐸2=βˆ’π‘’π‘₯𝐸1+𝑒2π‘₯𝐸3,βˆ‡πΈ2𝐸1=βˆ’π‘’2π‘₯𝐸3,βˆ‡πΈ2𝐸3=𝑒2π‘₯𝐸1,βˆ‡πΈ1𝐸3=βˆ’π‘’2π‘₯𝐸2,βˆ‡πΈ3𝐸1=βˆ’π‘’2π‘₯𝐸2,βˆ‡πΈ3𝐸2=𝑒2π‘₯𝐸1.(4.20) Clearly (πœ™,πœ‰,πœ‚,𝑔) structure is an 𝛼-Sasakian structure and satisfy, ξ€·βˆ‡π‘‹πœ™ξ€Έπ‘Œ=𝛼(𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋),βˆ‡π‘‹πœ‰=βˆ’π›Όπœ™π‘‹,(4.21) where 𝛼=𝑒2π‘₯β‰ 0. Hence (πœ™,πœ‰,πœ‚,𝑔) structure defines 𝛼-Sasakian structure. Thus 𝑀 equipped with 𝛼-Sasakian structure is a 𝛼-Sasakian manifold. The tangent vectors 𝑋 and π‘Œ to 𝑀 are expressed as linear combination of 𝐸1,𝐸2,𝐸3, that is βˆ‘π‘‹=3𝑖=1π‘Žπ‘–πΈπ‘– and βˆ‘π‘Œ=3𝑖=1𝑏𝑖𝐸𝑖, where π‘Žπ‘– and 𝑏𝑖(𝑖=1,2,3) are scalars.
Using 𝛼=𝑒2π‘₯ in (4.11), we have π‘Ÿ=2𝑒4π‘₯+𝑐≠0,(4.22) and it shows that the scalar curvature is not constant.
Using 𝛼=𝑒2π‘₯ in (4.15), we have πœ†=βˆ’2𝑒4π‘₯β‰ 0.(4.23) In this example 𝛼=𝑒2π‘₯β‰ 0, this implies that πœ†<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 πœ† 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).

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