Ricci Solitons in -Sasakian Manifolds
We study Ricci solitons in -Sasakian manifolds. It is shown that a symmetric parallel second order-covariant tensor in a -Sasakian manifold is a constant multiple of the metric tensor. Using this, it is shown that if is parallel where is a given vector field, then is Ricci soliton. Further, by virtue of this result, Ricci solitons for -dimensional -Sasakian manifolds are obtained. Next, Ricci solitons for 3-dimensional -Sasakian manifolds are discussed with an example.
In 1982, Hamilton  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: , (in this paper we use ) in the space of metrics on . Hence it is interesting to study Ricci solitons.
Definition 1.1. A Ricci soliton on a Riemannian manifold is defined by It is said to be shrinking, steady, or expanding according as and .
Note that here the metric is the pull back of the initial metric by a 1-parameter family of diffeomorphisms generated by a vector field on a manifold . 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.
In 1923, Eisenhart  proved that if a positive definite Riemannian manifold admits a second order parallel symmetric covariant tensor other than a constant multiple of the metric tensor, then it is reducible. In 1925, Levy  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 -dimensional 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  in his paper proved that a second order symmetric parallel tensor on an -K-contact 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 -Sasakian manifold. Note that -Sasakian manifolds are generalisations of Sasakian manifolds. Hence one can find interest in generalisation, from Sasakian to -Sasakian manifolds and study Ricci solitons in this manifold.
In 2008, Sharma  studied Ricci solitons in K-contact manifolds, where the structure field is killing and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. In 2010, Călin and Crasmareanu  extended the Eisenhart problem to Ricci solitons in -Kenmotsu manifolds. They studied the case of -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  studied Ricci solitons in Lorentzian -Sasakian Manifolds.
In this paper, we obtain some results on Ricci solitons.
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: Holds for some smooth function on .
In an -Sasakian manifold, the following relations hold: for all , where is the Riemannian curvature tensor, is the Ricci tensor and is the Ricci operator.
3. Parallel Symmetric Second Order Tensors and Ricci Solitonsin -Sasakian Manifolds
Fix a symmetric tensor field of -type which we suppose to be parallel with respect to that is . Applying the Ricci identity [4, 11] we obtain the relation Replacing in (3.2) and by using (2.5) and by the symmetry of , we have Put in (3.3) and by virtue of (2.1), we have Replacing in (3.4), we have Solving (3.4) and (3.5), we have Since , it results Differentiating (3.7) covariantly with respect to , we have By using the parallel condition , and (3.7) in (3.8), we have By using (2.4) in (3.9), we get Replacing in (3.10), we get Since is a nonzero smooth function in -Sasakian manifold and this implies that the above equation implies that 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 -Sasakian manifold is a constant multiple of the metric tensor.
Corollary 3.2. A locally Ricci symmetric -Sasakian manifold is an Einstein manifold.
Remark 3.3. The following statements for -Sasakian manifold are equivalent: (1)Einstein, (2)locally Ricci symmetric, (3)Ricci semi-symmetric that is .
The implication (1)→(2)→(3) is trivial. Now, we prove the implication (3)→(1) and means exactly (3.2) with replaced by that is, Considering and putting in (3.13), we have By using (2.6) in (3.14), we obtain Putting in (3.15) and by using (2.1), (2.8), and (2.9) on simplification, we obtain Interchanging and in (3.16), we have Adding (3.16) and (3.17), we obtain We conclude the following.
Proposition 3.4. A Ricci semi-symmetric -Sasakian manifold is an Einstein manifold.
Corollary 3.5. Suppose that on a -Sasakian manifold the -type field is parallel where is a given vector field. Then yield a Ricci soliton. In particular, if the given -Sasakian manifold is Ricci semi-symmetric with parallel, one has the same conclusion.
Proof. Follows from Theorem 3.1 and Corollary 3.2.
A Ricci soliton in -Sasakian manifold defined by (1.1). Thus is parallel. In Theorem 3.1 we proved that if an -Sasakian manifold admits a symmetric parallel tensor, then the tensor is a constant multiple of the metric tensor. Hence is a constant multiple of the metric tensor that is , where is a nonzero constant. We close this section with applications of our Theorem 3.1 to Ricci solitons.
Corollary 3.6. If a metric in an -Sasakian manifold is a Ricci soliton with then it is Einstein.
Hence we state the following result.
Corollary 3.7. A Ricci soliton in an -dimensional -Sasakian manifold cannot be steady but is shrinking.
Proof. From Linear Algebra either the vector field or . However the second case seems to be complex to analyse in practice. For this reason we investigate for the case .
A simple computation of gives From (1.1), we have and then putting , we have where by using (2.9) and (3.21) in the above equation, we have Equating (3.22) and (3.24), we have Since is some nonzero function, we have , that is Ricci soliton in an -dimensional -Sasakian manifold cannot be steady but is shrinking because .
Corollary 3.8 3.8. If an -dimensional -Sasakian manifold is -Einstein then the Ricci solitons in -Sasakian manifold that is , where with varying scalar curvature cannot be steady but it is shrinking.
Proof. The proof consists of three parts. (i)We prove -Sasakian manifold is -Einstein. (ii)We prove the Ricci soliton in -Sasakian manifold is consisting of varying scalar curvature. (iii)We find that the Ricci soliton in -Sasakian manifold is shrinking.
First we prove that the -Sasakian manifold is -Einstein: the metric is called -Einstein if there exists two real functions and such that the Ricci tensor of is given by the general equation Now by simple calculations we find the values of and . Let , be an orthonormal basis of the tangent space at any point of the manifold. Then putting in (3.26) and taking summation over , we get Again putting in (3.26) then by using (2.9), we have Then from (3.27) and (3.28), we have Substituting the values of and in (3.26), we have the above equation is an -Einstein -Sasakian manifold.
Now, we have to show that the scalar curvature is not a constant and it is varying. For an -dimensional -Sasakian manifolds the symmetric parallel covariant tensor of type is given by By using (3.21) and (3.30) in (3.31), we have Differentiating (3.32) covariantly with respect to , we have By substituting and in (3.33) and by using , we have On integrating (3.34), we have where is some integral constant. Thus from (3.35), we have is a varying scalar curvature.
Finally, we have to check the nature of the soliton that is Ricci soliton in -Sasakian manifold:
From (1.1), we have then putting , we have If we put in (3.32), that is Above equation reduced as, Equating (3.36) and (3.38), we have Since, because is some smooth function and , that is the Ricci soliton in an -Sasakian manifold is shrinking.
4. Ricci Solitons in 3-Dimensional -Sasakian Manifold
In this section we restrict our study to -dimensional -Sasakian manifold, that is Ricci solitons in -dimensional -Sasakian manifold.
Corollary 4.1. If a Ricci soliton where of 3-dimensional -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 -Sasakian manifold is -Einstein. (ii)We prove that the Ricci soliton in 3-dimensional -Sasakian manifold is consisting of varying scalar curvature. (iii)We find that the Ricci soliton in a 3-dimensional -Sasakian manifold is shrinking.
First we consider: the Riemannian curvature tensor of 3-dimensional -Sasakian manifold and it is given by Put in (4.1) and by using (2.5) and (2.8), we have Again put in (4.2) and by using (2.1) and (2.10), on simplification we get By taking an inner product in (4.3), we have Interchanging and in (4.4), we have Adding (4.4) and (4.5), we have Equation (4.6) shows that a 3-dimensional -Sasakian manifold is -Einstein.
Now, we have to show that the scalar curvature is not a constant that is is varying. Now, By using (3.21) and (4.6) in (4.7), we have Differentiating the above equation covariantly with respect to , we have Substituting , in (4.9) and by virtue of , we have On integrating (4.10), we have where is some integral constant. Thus from (4.11), we have a varying scalar curvature.
Finally, we have to check the nature of the soliton that is Ricci soliton in 3-dimensional -Sasakian manifold.
From (1.1), we have and then putting , we have If in (4.8), that is Above equation reduced as Equating (4.12) and (4.14), we have Since from (4.15), we have . Therefore Ricci soliton in 3-dimensional -Sasakian manifold is shrinking.
Example 4.2. Let . Let be linearly independent vector fields given by
Let be the Riemannian metric defined by , , where is given by
Let be the 1-form defined by for any . Let be the tensor field defined by . Then using the linearity of and yields that and for any vector fields . Thus for , defines a Sasakian structure on . By definition of Lie bracket, we have
Let be the Levi-Civita connection with respect to above metric Koszula formula is given by
Clearly structure is an -Sasakian structure and satisfy,
where . Hence structure defines -Sasakian structure. Thus equipped with -Sasakian structure is a -Sasakian manifold. The tangent vectors and to are expressed as linear combination of , that is and , where and are scalars.
Using in (4.11), we have and it shows that the scalar curvature is not constant.
Using in (4.15), we have In this example , this implies that , that is the Ricci soliton in 3-dimensional -Sasakian manifold is shrinking.
In this paper we have shown that the Ricci soliton in an -Sasakian manifold cannot be steady but it is shrinking accordingly because is negative.
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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