Research Article | Open Access
Certain Results on Ricci Solitons in Trans-Sasakian Manifolds
We study and obtain results on Ricci solitons in trans-Sasakian manifolds satisfying , , , and , where , , and are quasiconformal, projective, and conharmonic curvature tensors.
During 1982, Hamilton  made the fundamental observation that Ricci flow is an excellent tool for simplifying the structure of a manifold. It is a process which deforms the metric of a Riemannian manifold analogous to the diffusion of heat there by smoothing out the regularity in the metric. It is given by .
(i) If the manifold is an Euclidean space or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. Conversely, any metric unchanged by Ricci flow is Ricci-flat.
(ii) Let and the standard metric on the unit -sphere in the Euclidean space. If for some ( is the radius), then is a solution. So the Ricci flow with collapses to a point at time .
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 space of metrics on . 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 .
Let be a smooth function of time. Since is a diffeomorphism and is a Riemannian metric on (codomain), then by definition of pull back is a metric on (domain).
Set ; then we have  (This follows from the definition of the Lie derivative.)
Suppose we have a metric , a vector field , and (all independent of time) such that If we choose , , and , then it gives a family of diffeomorphisms with identity; then using (2) in (1), defined above is a Ricci flow with , that is, Hence, is a solution of the Ricci flow and is known as a Ricci soliton.
Definition 1 (see ). A Ricci soliton on a Riemannian manifold is defined by It is said to be shrinking, steady, or expanding according to , , and .
(1) Hamilton Cigar Soliton . Let , and defined by forms a family of a one-parameter group of diffeomorphisms. The vector field generated by is . The metric is obtained as , , , . Using (4), we have . Hence, this Ricci soliton is steady and is called cigar soliton as it is asymptotic to a flat cylinder at infinity.
(2) Cylinder Shrinking Soliton . Consider the product of the sphere with a like , where . Its Ricci tensor is given by . If we set , , , , then , . Hence is a shrinking soliton.
In , Sharma obtained some interesting results on Ricci solitons in K-contact manifolds. Cǎlin 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 showed that 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 concerned in -Kenmotsu manifolds and 3-dimensional -Kenmotsu manifolds.
In [6, 7], Bagewadi and Ingalahalli studied Ricci solitons in -Sasakian and Lorentzian -Sasakian manifolds and proved that a symmetric parallel tensor field of second order is a constant multiple of the Riemannian metric; based on these they obtained the results on Ricci Solitons concerned in these manifolds. In , Tripathi obtained some results on Ricci solitons in contact metric manifolds. In , Nagaraja and Premalatha used semisymmetric conditions in Kenmotsu manifolds and obtained reuslts on Ricci solitons in Kenmotsu manifolds.
Motivated by all this work in this paper we are studying the Ricci solitons in trans-Sasakian manifolds.
Trans-Sasakian manifolds arose in a natural way from the classification of almost contact metric structures, and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds. In , Gray-Hervella classification of almost Hermite manifolds appears as a class of Hermitian manifolds which are closely related to locally conformally Kähler manifolds. An almost contact metric structure on a manifold is called a trans-Sasakian structure  if the product manifold belongs to the class . The class  coincides with the class of trans-Sasakian structures of type . The local nature of the two subclasses and of trans-Sasakian structures is characterized completely. It is known that  trans-Sasakian structures of type , , and , are cosymplectic, -Sasakian, and -Kenmotsu, respectively, where . In , Marrero has shown that a trans-Sasakian manifold of dimension is either cosymplectic or -Sasakian or -Kenmotsu manifold. Later many authors worked in this topic, trans-Sasakian manifolds, like Blair and Oubiña , Janssens and Vanhecke , Bagewadi and Venkatesha , U. De and K. De , De and Tripathi , Shaikh et al. , Nagaraja et al. . Now we move on to the preliminaries.
An -dimensional differential manifold is said to be an almost contact metric manifold  if it admits a tensor field , a vector field , a -form , and a Riemannian metric , satisfying for all vector fields , on .
An almost contact metric manifold is said to be trans-Sasakian manifold if belongs to the class of the Hermitian manifolds , where is the almost complex structure of defined by for all vector fields on and smooth function on , and is the product metric on . This may be expressed by the following condition : where and are scalar functions on , and such a structure is said to be the trans-Sasakian structure of type . From (7), we have
Note 1. (1) For some smooth functions and , if and , , and , then it reduces to -Sasakian and -Kenmotsu manifold, respectively.
(2) If and are scalars and and , and , then it reduces to Sasakian and Kenmotsu manifold, respectively.
(3) If and , then it reduces to cosymplectic manifold, respectively.
In a trans-Sasakian manifold, we have  where , are functions and is the Riemannian curvature tensor. From (8) we have From (4) and (14), we get The above equation yields where is the Ricci operator and is the scalar curvature on .
Remark 2. Since our study deals with Ricci solitons of trans-Sasakian manifolds and by Definition 1, it is known that is a scalar quantity. Further, from (18) it is seen that is related to . Hence in the calculations of our results which will be proved in the following it is necessary to assume and as simply scalar quantities.
2.1. Example for 3-Dimensional Trans-Sasakian Manifold
We consider the -dimensional manifold , where are the standard co-ordinates in . Let be linearly independent global frame field on given by Let be the Riemannian metric defined by , , where is given by The is given by , , , , . The linearity property of and yields that , , , for any vector fields on . By the definition of Lie bracket, we have
Let be the Levi-Civita connection with respect to the above metric given by the Koszula formula Then,
2.2. Example for 3-Dimensional -Sasakian Manifolds
Let . Let be linearly independent vector fields given by Let be the Riemannian metric defined by , , where is given by The is given by The linearity property of and yields that , , , for any vector fields . By the definition of Lie bracket, we have Let be the Levi-Civita connection; with respect to the above metric given by Koszula formula (22) and by virtue of it we have The tangent vectors and to are expressed as linear combinations of , that is, and , where and () are scalars. This becomes -Sasakian manifold with .
2.3. Example for 3-Dimensional -Kenmotsu Manifold
Let . Let be linearly independent vector fields given by Let be the Riemannian metric defined by , , where is given by The is given by The linearity property of and yields that , , , for any vector fields on . By the definition of Lie bracket, we have Let be the Levi-Civita connection; with respect to above metric given by Koszula formula (22) and by virtue of it, we have The tangent vectors and to are expressed as linear combinations of ; that is, and , () are scalars. Clearly is a -Kenmotsu manifold with .
3. Ricci Soliton in Trans-Sasakian Manifolds Satisfying
The quasiconformal curvature tensor is defined by where are constants. Taking in (34) and using (10), (15), (16), and (17), we get Similarly using (12), (15), (16), and (17) in (34), we get We assume that the condition ; then we have for all vector fields , , , and on . Using (10) in (37), we have By taking an inner product with , we get In view of (34), (35), and (36) and Remark 2 in (39), then we have Taking in (40) and summing over , we get Putting in (41) and summing over , we get from (18) Since by , a trans-Sasakian manifold of dimension is either cosymplectic or -Sasakian or -Kenmotsu manifold. So, based on this we state the following.
Theorem 3. A Ricci soliton in a quasiconformally semisymmetric -Sasakian manifold is shrinking.
Theorem 4. A Ricci soliton in a quasiconformally semisymmetric -Kenmotsu manifold is expanding.
Proof. In (42) put ; then we have
If , then
The above equation implies that . Hence Ricci soliton is expanding.
Theorem 5. A Ricci soliton in a quasiconformally semisymmetric cosympletic manifold is steady.
Proof. In (42) put and ; then we have The above equation implies that . Hence Ricci soliton is steady.
4. Ricci Soliton in Trans-Sasakian Manifolds Satisfying
The projective curvature tensor is defined by Putting in (47), we get We assume that ; then we have Using (48) in (49), we obtain By taking an inner product with and by using Remark 2 in (50), then, we obtain By using (34), (35), and (36) in (51) and on simplification, we obtain where .
Taking in (52) and summing over , and on simplification, we obtain Taking in (53) and summing over , and on simplification, we obtain If and on simplification, we get Again by , a trans-Sasakian manifold of dimension is either cosymplectic or -Sasakian or -Kenmotsu manifold. So, based on this we state the following.
Theorem 6. A Ricci soliton in an -Sasakian manifold satisfying is shrinking, provided .
Proof. In (55) put ; then we have that is, and . This completes the proof of the theorem.
Theorem 7. A Ricci soliton in a -Kenmotsu manifold satisfying is expanding, provided .
Proof. In (55) put ; then we have On simplifying the above quadratic equation then we get that is either on simplification we get In both cases . This completes the proof of the theorem.
Theorem 8. A Ricci soliton in a cosympletic manifolds satisfying is steady, provided .
Proof. In (55) put and ; then we have since . This completes the proof of the theorem.