Abstract
We study 3-dimensional compact and simply connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. The first two results deal with finding necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold and in the third result deals with finding necessary and sufficient condition on a compact and simply connected trans-Sasakian manifold to be homothetic to a Sasakian manifold.
1. Introduction
It is well known that for an almost contact metric manifold (cf. [1]), the product has an almost complex structure , which with product metric makes an almost Hermitian manifold. The properties of the almost Hermitian manifold control the properties of the almost contact metric manifold and provide several structures on such as a Sasakian structure and a quasi-Sasakian structure (cf. [1–3]). There are known sixteen different types of structures on (cf. [4]), and using the structure in the class on , a structure was introduced on , which is called trans-Sasakian structure (cf. [5]), that generalizes Sasakian structure, Kenmotsu structure, and cosymplectic structure on a contact metric manifold (cf. [2, 3]), where being the real functions defined on .
Recall that a trans-Sasakian manifold is called a trans-Sasakain manifold of type , and trans-Sasakian manifolds of type , , and are called a cosymplectic, a -Sasakian, and a -Kenmotsu manifolds, respectively. It is on account of a result proved in [6] that a trans-Sasakian manifold of dimension five or greater than five reduces to a cosymplectic manifold, a -Sasakian manifold, or a -Kenmotsu manifold, so there is an emphasis on studying three-dimensional trans-Sasakian manifolds.
Among other questions, finding conditions under which a compact 3-dimensional trans-Sasakian manifold is homothetic to a Sasakian manifold is of prime importance. The geometry of -dimensional trans-Sasakian manifold is also important owing to Thurston’s conjecture (cf. [7]), and fetching conditions on a 3-dimensional trans-Sasakian manifold in matching it among Thurston’s eight geometries becomes more interesting. It is worth noting that in Thurston’s eight geometries, the first place is occupied by the spherical geometry .
In ([8–13]), the authors have studied compact 3-dimensional trans-Sasakian manifolds with some suitable restrictions on functions appearing in the definition of a trans-Sasakian manifold for getting conditions under which a trans-Sasakian manifold is homothetic to a Sasakian manifold. In particular, it is known that a 3-dimensional compact simply connected trans-Sasakian manifold satisfying Poisson equations and , respectively, is necessarily homothetic to a Sasakian manifold (cf. [10]).
An interesting work on 3-dimensional trans-Sasakian manifolds is found in [14, 15], where the authors have considered other aspects in Thurston’s eight geometries. In [10], it is asked whether the function on a 3-dimensional compact trans-Sasakian manifold satisfying necessitates the trans-Sasakian manifold to be homothetic to a Sasakian manifold. In [15], it is shown that this question has negative answer.
Einstein Sasakian manifolds are very important due to their geometric importance (cf. [16]). In this paper, in our first two results, we find necessary and sufficient conditions on a compact simply connected 3-dimensional trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold, and in the third, we find a necessary and sufficient condition on a compact simply connected 3-dimensional trans-Sasakian to be homothetic to a Sasakian manifold.
In the first result, we consider a compact and simply connected trans-Sasakian manifold of positive constant scalar curvature , the function satisfying Fischer-Marsden equation shows that the functions are related to by the inequality , and the Ricci operator satisfying Codazzi-type equation with respect to vector field necessarily implies that is homothetic to an Einstein Sasakian manifold. In the second result, we show that a compact simply connected trans-Sasakian manifold with function constant along the integral curves of , scalar curvature satisfying the inequality , and the Ricci operator satisfying Codazzi-type equation with respect to vector field necessarily imply that is homothetic to an Einstein Sasakian manifold. Finally, in the last result, we show that on a compact and simply connected trans-Sasakian manifold, the function satisfies the differential inequality , and vector fields are orthogonal, which necessarily imply that is homothetic to a Sasakian manifold, where the covariant derivative for a smooth vector field on .
2. Preliminaries
Let be an almost contact metric manifold , where being a -tensor field, a unit vector field, and smooth -form dual to with respect to the Riemannian metric satisfying where is the space of smooth sections of the tangent bundle (cf. [1]). If there exist functions on an almost contact metric manifold satisfying then is said to be a trans-Sasakian manifold, where , , and is the Levi-Civita connection with respect to the metric (cf. [8–15]). Using equations (1) and (2), it follows that
Using the Ricci tensor of a Riemannian manifold , the Ricci operator is defined by and . We have the following for a 3-dimensional trans-Sasakian manifold :
Note that equation (3) implies and using this equation together with equation (4), we have
Thus, on compact 3-dimensional trans-Sasakian manifold , using equation (6) and the above equation, we have
Now, we state the following result of Okumura.
Theorem 1. [17] Let be a Riemannian manifold. If admits a Killing vector field of constant length satisfying for nonzero constant and any vector fields and , then is homothetic to a Sasakian manifold.
For a smooth function on the Riemannian manifold , then the operator defined by is called the Hessian operator of , and it is a symmetric operator. Moreover, the Hessian of is defined by
The Laplace operator on is defined by , and we also have
Fischer-Marsden differential equation on a Riemannian manifold is (cf. [18])
3. Trans-Sasakian Manifolds Homothetic to Einstein Sasakian Manifolds
In this section, we find necessary and sufficient conditions for a compact and simply connected 3-dimensional trans-Sasakian manifold to be homothetic to an Einstein Sasakian manifold.
Theorem 2. A compact and simply connected 3-dimensional trans-Sasakian manifold with positive constant scalar curvature and the function a solution of Fischer-Marsden equation satisfying is homothetic to an Einstein Sasakian manifold of positive scalar curvature, if and only if, the Ricci operator satisfies
Proof. Suppose is a compact simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, equation (13) gives and taking trace in above equation and using equation (12), we have
Note that by equation (3), we have , and therefore, . Using this equation and equation (17) in equation (16), we get
Now, using equation (5), we have . Thus, the above equation becomes
Using equation (6), we have , and inserting it in the above equation, we conclude
Integrating the above equation, we get
Using the inequality in the statement, we conclude
Since is simply connected, it is connected, and therefore equation (22) implies either (i) or (ii) . Suppose (ii) holds, then as is a constant, we get , which in view of equation (4) implies ; that is, . Thus, we have
Using equation (6), we have , and inserting it in above equation, we get
Integrating the above equation, we get
Now, using (ii) in above integral, we have and since the scalar curvature , through above integral, we conclude that . Thus, using equations (2), (3), (4), and (5), take the forms
Taking the covariant derivative in the second equation of equation (28), we get and using equation (27) in above equation, we arrive at
Now, using the Codazzi equation type condition on in the hypothesis, we get
Using the second equation in equation (27), we compute the Lie derivative of with respect to to conclude that is, is a Killing vector field and that the flow of consists of isometries of the Riemannian manifold . Thus, we have and using equation (27), we conclude
Combining the above equation with equation (31), we have
Taking the inner product with in above equation, we conclude
We claim that being simply connected, ; for if , then by equation (27), we see that is parallel and that is closed, which implies is exact; that is, for a smooth function on . This implies , and being compact, there is a point such that , and we get , a contradiction to the fact that is a unit vector field. Hence, , and equation (36) implies ; that is, is a nonzero constant.
Now, equation (28) gives , and taking the covariant derivative in this equation yields
Using the condition in the hypothesis and equation (34) with , in above equation, we get
Operating on above equation while using equation (1) and , we conclude
This proves that is an Einstein manifold. Finally, using equation (27), with a nonzero constant, we compute
Hence, by Theorem 1, we conclude that is homothetic to a compact simply connected Einstein Sasakian manifold of positive scalar curvature. The converse is trivial.
Theorem 3. A compact and simply connected 3-dimensional trans-Sasakian manifold satisfying and the scalar curvature satisfying is homothetic to an Einstein Sasakian manifold, if and only if, the Ricci operator satisfies
Proof. Suppose is a compact and simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, using equation (4) and the condition , we get . However, if , then equation (3) implies that is closed, and as seen in the proof of Theorem 2, we get a contradiction owing to simply connectedness of . Thus, , and on connected , equation implies that . Therefore, equations (27) and (28) hold. Now, using equation (28), we have , which gives
Taking covariant derivative in above equation, we have and using equation (27), we get
Using condition in the hypothesis, we have
Also, equation (27) implies that is a Killing vector field, and therefore, using its outcome equation (34) as well as equation (28), we conclude That is, on using equation (1), we have
Thus, we have and integrating the above equation, we get
Using the condition in the hypothesis, we have , and as , we conclude . Consequently, equation (49) implies that , and being compact, we conclude that is a constant. Thus, being nonzero constant, equation (48) gives
This proves that is an Einstein manifold, and using equation (27), we see that the unit Killing vector field satisfies
Hence, in view of Theorem 1, we conclude that is homothetic to an Einstein Sasakian manifold. The converse is trivial.
Theorem 4. A compact and simply connected 3-dimensional trans-Sasakian manifold satisfying is homothetic to a Sasakian manifold, if and only if, the vector fields and are orthogonal.
Proof. Suppose is a compact and simply connected 3-dimensional trans-Sasakian manifold satisfying the hypothesis. Then, using equation (5), we have and taking covariant derivative in above equation with respect to , while using equation (3), we get
Taking trace in above equation and noting the following outcome of equation (2), for a local orthonormal frame on , we get where we have used , , , and well known formula
Now, using equation (4) and in equation (57), we have that is,
Note that on using equation (6), we have and
Inserting these equations in equation (60), we arrive at and integrating the above equation while keeping in mind equation (8), we get
Using the condition in the hypothesis, we conclude and ; that is, . We get , which in view of equation (6) implies
Integrating the above equation yields . Thus, equations (27) and (28) are now available to us. Taking covariant derivative in second equation of equation (28) with respect to , we get where we have used the second equation in equation (27). Now, using equation (27) and noting that in above equation, we conclude
Taking the inner product with in above equation, we get
Now, using equation (28), we have and inserting it in equation (67), we get
Taking in above equation and using the condition in the hypothesis, we conclude
Note that being compact and simply connected, is not allowed to be zero. Hence, the above equation implies that is nonzero constant. Thus, we have by virtue of equation (27) that
This proves that is homothetic to a Sasakian manifold. The converse is trivial.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is supported by the Deanship of Scientific Research, Imam Mohammed Ibn Saud Islamic University, Saudi Arabia, Grant No. 20-13-12-001.