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Volume 2013 (2013), Article ID 573925, 4 pages
Certain Results on Ricci Solitons in -Sasakian Manifolds
Department of Mathematics, Kuvempu University, Shankaraghatta, Shimoga, Karnataka 577 451, India
Received 1 April 2013; Revised 26 June 2013; Accepted 26 June 2013
Academic Editor: Giovanni Calvaruso
Copyright © 2013 S. R. Ashoka et al. 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.
We study Ricci solitons in -Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field. Further, we prove that if is conformal Killilng vector field, then the Ricci soliton in 3-dimensional -Sasakian manifolds is shrinking or expanding but cannot be steady.
A Ricci soliton is a generalization of an Einstein metric and is defined on a Riemannian manifold by where is a complete vector field on and is a constant. The Ricci soliton is said to be shrinking, steady, or expanding according as is negative, zero, and positive, respectively. Long-existing solutions, that is, solutions which exist on an infinite time interval, are the self-similar solutions, which in Ricci flow are called Ricci soliton.
Compact Ricci solitons are the fixed points of the Ricci flow 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. If the vector field is the gradient of a potential function , then is called a gradient Ricci soliton and (1) assumes the form A Ricci soliton on a compact manifold is a gradient Ricci soliton. A Ricci soliton on a compact manifold has constant curvature in dimension  and also in dimension . In , Perelman proved that a Ricci soliton on a compact -manifold is a gradient Ricci soliton. In , Sharma studied Ricci solitons in -contact manifolds, where the structure field is Killing, and he proved that a complete -contact gradient soliton is compact Einstein and Sasakian. In , Tripathi studied Ricci solitons in -contact metric and manifolds. In , Ghosh and Sharma studied -contact metrics as Ricci solitons. In , Nagaraja and Premalatha studied Ricci solitons in -Kenmotsu manifolds and 3-dimensional trans-Sasakian manifolds. Recently, Bagewadi and Ingalahalli  studied Ricci solitons in Lorentzian -Sasakian manifolds. Motivated by the previous studies on Ricci solitons, in this paper, we study Ricci solitons in an -Sasakian manifolds, where is some constant.
Let be an almost contact metric manifold of dimension equipped with an almost contact metric structure consisting of a tensor field , a vector field , a 1-form , and a Riemannian metric , which satisfy for all . An almost contact metric manifold is said to be -Sasakian manifold if the following conditions hold: for some nonzero constant on .
In an -Sasakian manifold, we have the following relations: for all , where is the Riemannian curvature tensor, is the Ricci tensor, and is the Ricci operator.
3. Ricci Solitons in -Sasakian Manifold
In this section, we prove some theorems on Ricci solitons in -Sasakian manifold.
Proposition 1. A complete Einstein -Sasakian manifold is compact.
Proof. Let be a complete Einstein -Sasakian manifold, then the general form is given by Operating in (12) and using (11) show . Hence we get . So the Ricci curvatures are equal to which is a positive constant. By Myers's theorem , we conclude that is compact.
Theorem 2. If the metric of an -Sasakian manifold is a gradient Ricci soliton, then the Ricci soliton is a shrinking soliton and is compact Einstein.
Proof. Equation (2) can be written as
where denotes the gradient operator of and denotes an arbitrary vector field on . Using this we derive
Taking its inner product with , substituting , and using (7) and (11), we have
Substituting (15) in (13), we get
Interchanging and in (16), we have
Adding (16) and (17), we have
Putting in (18), we have
The use of the previous two equations provides
Consequently, (13) assumes the form
Using this, we compute , and taking inner product with (bearing in mind that ), we obtain . Therefore, from (19), we have ; that is, is constant or . Hence (15) can be written as . Its exterior derivative implies . Hence . Thus is constant.
Consequently, (13) reduces to ; that is, an -Sasakian manifold is an Einstein. Also, as is negative for or ; that is, Ricci soliton in -Sasakian manifolds is shrinking.
From above-mentioned theorem, we state the following corollary.
Corollary 3. If a metric of a compact -Sasakian manifold is a Ricci soliton, then is a shrinking soliton and the manifold is Einstein.
Theorem 4. If a metric in an -Sasakian manifold is a Ricci soliton with , then it is Einstein.
Proposition 5. If an -Sasakian manifold is a Ricci soliton with point-wise collinear with , then is a constant multiple of and the manifold is Einstein.
Proof. From (1), we have where Substituting (25) in (24), then we obtain Putting in (26), we get Putting in (27), we have Again putting in (27), we obtain Equation (30) implies that Applying on both sides, Equation (31) implies that , but is nowhere vanishing. Therefore, which implies ; that is, is constant. As is Killing, we conclude that the manifold is Einstein which completes the proof.
Definition 6. A vector field is said to be conformal Killing vector field if it satisfies for some scalar function .
Theorem 7. Let be a Ricci soliton in an -Sasakian manifolds . Then is Ricci-semisymmetric if and only if is conformal Killing.
Proof. Suppose that is a conformal Killing vector field, and from (1), we have
The previous equation implies that
This shows that the Ricci soliton is Einstein as follows:
Let be an -Sasakian manifolds; then we have  (1)Einstein, (2)locally Ricci symmetric, (3)Ricci semisymmetric; that is, .
The implication is trivial. Now, we prove the implication .
Now, Considering and putting in (36), we have By using (7) in (37), we obtain Putting in (38) and by using (3), (9), and (10) on simplification, we obtain Substituting (39) in (1), we have where ; that is, is conformal Killing.
Now we study Ricci solitons in 3-dimensional -Sasakian manifolds.
Theorem 8. In a 3-dimensional -Sasakian manifolds, a Ricci soliton , where is conformal Killing, is (i)shrinking for ,(ii)expanding for and .
Proof. Suppose that is a 3-dimensional -Sasakian manifolds and is a Ricci soliton in . If is a conformal Killing vector field, then
for some scalar function .
In a 3-dimensional -Sasakian manifolds and from (1), we have In a 3-dimensional -Sasakian manifolds, the curvature tensor is given by Using (42), (43), and (44) in (45), we get Putting in (46), we get In an -Sasakian manifolds is given by From (47) and (48), we have The previous equation implies that From (44) and (50), we have (i)If implies , that is, . Hence Ricci soliton is shrinking. (ii)Let , suppose implies , that is, . Hence Ricci soliton is expanding. (iii)Let implies . If then implies . Hence Ricci soliton is expanding.
If in (41), then ; that is, conformal vector field does not exist, and then the Ricci soliton is generalization of Einstein metric, then is a Killing vector field. On base of this condition, we state the following.
Remark 9. A Ricci soliton in -Sasakian manifolds is shrinking, if is a Killing vector field.
The authors are grateful to the referee for their valuable suggestions to improve this paper.
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