Abstract
We study -homothetic deformations of -contact manifolds. We prove that -homothetically deformed -contact manifold is a generalized Sasakian space form if it is conharmonically flat. Further, we find expressions for scalar curvature of -homothetically deformed -contact manifolds.
1. Introduction
In 1968 Tanno [1] introduced the notion of -homothetic deformations. Carriazo and Martín-Molina [2] studied -homothetic deformation of generalized space forms and gave several examples for manifolds of dimension 3. De and Ghosh [3] studied -homothetic deformation of almost normal contact metric manifolds and prove that is invariant under such transformation. Bagewadi and Venkatesha [4] studied concircularly semisymmetric trans-Sasakian manifolds and De et al. [5] studied conharmonically semisymmetric, conharmonically flat, -conharmonically flat, and conharmonically recurrent generalized Sasakian space forms. Several authors [6–11] studied -contact manifolds and proved conditions for these manifolds to be of -conformally flat, -conformally flat, quasi-conharmonically flat, and -conharmonically flat. Motivated by the above studies, in this paper we study -homothetic deformations of -contact manifolds by considering conharmonic and projective curvature tensor. The paper is organized as follows. After Preliminaries, we give a brief account of information of -homothetic deformation of -contact manifolds in Section 3. In Section 4, we study conharmonically flat, semisymmetric, -conharmonically flat, quasi-conharmonically flat, and -conharmonically flat -contact manifolds with respect to -homothetic deformation. In the last section, we consider Weyl projective curvature in -contact manifolds with respect to -homothetic deformation.
2. Preliminaries
Let be a -dimensional almost contact metric manifold [12], consisting of a tensor field , a vector field , a 1-form , and Riemannian metric . Then for all , . If is a Killing vector field, then is called a -contact Riemannian manifold [13]. A -contact Riemannian manifold is called Sasakian [12], if the relation holds, where denotes the operator of covariant differentiation with respect to .
If is a -contact Riemannian manifold, then besides (1), (2), (3), and (4) the following relations hold [14]: for any vector fields and , where and denote, respectively, the curvature tensor of type and the Ricci tensor of type .
Definition 1. A contact metric manifold is said to be -Einstein if , where and are smooth functions on .
3. -Homothetic Deformation of -Contact Manifolds
Let be a -dimensional almost contact metric manifold. A -homothetic deformation is defined by with being a positive constant [1].
It is clear that the is also an almost contact metric manifold.
If is a -contact manifold with Riemannian connection , the connection of the -deformed -contact manifold can be calculated from and . Using Koszul's formula and (5), (6), and (11), of is given by Using (12), we obtain The curvature tensor of is given by Using (9), (10), and (14), we have From (14), we get where and are the Ricci tensors of and , respectively.
It follows from (16) that Again contracting (16) over , , we get where and are the scalar curvatures of and , respectively.
4. Conharmonic Curvature Tensor in -Homothetically Deformed -Contact Manifolds
The conharmonic tensor of a -homothetically deformed -contact manifold is defined by [15] for , , , where , , and are the Riemannian curvature tensor, Ricci tensor, and Ricci operator of .
Definition 2. An almost contact metric manifold is said to be(1)conharmonically flat if (2)conharmonically semisymmetric if (3)-conharmonically flat if (4)quasi-conharmonically flat if (5)-conharmonically flat if for all vector fields , , and .
Assume that is conharmonically flat -contact manifold with respect to -homothetic deformation. So, we have .
Then from (20), we have Setting , contracting (26) with , and using (7), (9), (14), and (16), we obtain Taking in (27) and using (1), (7), and (16), it follows that Thus, is -Einstein.
Using (28) in (26), we obtain From (29), we get Hence, it reduces to a generalized Sasakian space form with , , and . Thus, (30) leads to the following.
Theorem 3. A conharmonically flat -contact manifold admitting -homothetic deformation reduces to a generalized Sasakian space form with associated functions , , and .
Let us now consider a conharmonically semisymmetric -contact manifold admitting -homothetic deformation. Then the condition holds on for all vector fields , .
From (8), (14), (17), and (20), we obtain Setting , in (32), we get Again taking in (32) and using (17), we obtain Now, (21) yields Therefore, From this it follows that where Taking in (37) and making use of (32) and (33), we obtain If is a local orthonormal basis of vector fields in , then, from (39), we get From (20), it follows that Using (41) in (40), we obtain where In view of (34), (42) yields where Thus, is -Einstein.
If is a local orthonormal basis of vector fields in , then, from (44), we get where So, we can state the following.
Theorem 4. In a -dimensional conharmonically semisymmetric -contact manifold admitting -homothetic deformation, scalar curvature is given by (46).
Analogous to the definition of -conharmonically flat -contact manifolds [8], we define -conharmonically flat -contact manifolds with respect to -homothetic deformation. Let us assume that is a -conharmonically flat -contact manifold with respect to -homothetic deformation. It can be easily seen that where , , , .
Using (20), (48) yields for all , , , .
If is a local orthonormal basis of vector fields in , then is also a local orthonormal basis. So, using (1), (6), (14), (16), and (18), it can be easily verified that For a local orthonormal basis of vector fields in , putting in (49) and summing up with respect to , we have for all , . The previous equation, in view of (50), becomes for all , .
Using (2) and (18), (52) reduces to Setting in (53), summing up with respect to , and using (19), we obtain Replacing by and by in (53) and using (54), we obtain for all .
Now using the previous expression in (49), we obtain for all , , , .
The converse is obvious. Thus we have the following.
Theorem 5. A -dimensional -contact manifold is -conharmonically flat with respect to -homothetic deformation if and only if satisfies (55).
From (20), we obtain for all , , , .
Suppose that is quasi-conharmonically flat -contact manifold with respect to -homothetic deformation; that is, Then (56) reduces to For a local orthonormal basis of vector fields in , putting and in (58) and summing up with respect to , we obtain Using (2), (10), (14), and (17) in (59), we obtain Taking and using (1) and (17), we obtain and using (61) in (19), we obtain Hence, we can state the following.
Theorem 6. Let be a -contact manifold. Suppose that is obtained from by -homothetic deformation. If is quasi-conharmonically flat, then the scalar curvatures and of and are, respectively, given by (61) and (62).
Suppose that is -conharmonically flat. Then from (20), we have Contracting the above equation with respect to , we obtain for all , , , .
For a local orthonormal basis of vector fields in , using (64), we obtain Therefore, Using (17) in (66), we obtain Taking in (64) and using (10), (14), and (17), we obtain From this we can conclude that is -Einstein. Thus, we have the following.
Theorem 7. A -Homothetically deformed -conharmonically flat -contact manifold is -Einstein and its scalar curvature vanishes.
5. -Homothetic Deformation of -Weyl Projectively Flat -Contact Manifolds
Suppose that, in a -dimensional -contact manifold with -homothetic deformation, the Ricci tensor vanishes; that is, Then from (16), we have The Weyl projective curvature tensor of is given by [16] If , then (71) reduces to Using (70) in (72), we obtain The Weyl projective curvature tensor of with respect to -homothetic deformation is given by Now using (14) and (69) in (74), we get From (73), the (75) reduces to Thus, we can state the following.
Theorem 8. Let be obtained from a -contact manifold by -homothetic deformation. If the Ricci tensor of vanishes, then it is -Weyl projectively flat.