On three-dimensional quasi-Sasakian manifolds admitting semi-symmetric metric connection

The object of the present paper is to study three-dimensional quasi-Sasakian manifold equipped with semi-symmetric metric connection. The geometrical properties of conformal curvature tensor and the conservative quasi-conformal curvature tensor are discussed with such connection. Among other we have deal the conservative properties of quasi-conformal curvature with respect to semi-symmetric metric connection.


INTRODUCTION
introduced the notion of semi-symmetric linear connection on a differential manifold.Hayden (1932) introduced the idea of metric connection with torsion on Riemannian manifold.A systematic study of semi-symmetric metric connection on a Riemannian manifold has been given by Yano (1970) and later studied by Amur and Maralabhavi (1970), Bagewadi (1982), Bagewadi et al. (2007), and Prakasha et al. (2008).In the study Bagewadi and Venkatesh (2007), Friedmann and Schouten (1924) and Sharafuddin and Hussian, (1976), the others have obtained results on the conservativeness of projective, pseudo projective, conformal.Concircular, quasi-conformal curvature tensor on K -contact, Kenmotsu and Trans-Sasakian manifolds.Olszak (1986) considered the three-dimensional cases of normal almost contact metric manifold and Yadav et al. (2011) obtained some results on K-Contact and trans-Sasakian manifolds.Bagewadi et al. (2007), have been studied conservative projective curvature tensor on a trans-Sasakian manifold with respect to semi-symmetric metric connection.Prakasha et al. (2008) have studied the conservativeness of conformal and quasi-conformal curvature tensor on a trans-Sasakian manifold under the condition that admitting the semi-symmetric connection.
In this study we stabilized some basic results and studied conformal curvature tensor the conservativeness of conformal quasi conformal curvature tensor under the condition ) , ( ) ( X gradf g X df = on a three-dimensional qusi-Sasakian manifold admitting a semi -symmetric connection.

PRELIMINARIES
Let M be a where φ is a tensor field of type ) 1 , 1 ( vector field, η is an 1 -form and g is the Riemannian metric on M such that (Amur and Maralabhavi, 1970;Blair, 2002).
Let Φ be the fundamental − , which was first introduced by Blair (2002).The normal condition gives that the induced almost contact structure of R M × is integrable or equivalently, the torsion tensor field η ξ φ φ vanishes identically on M the rank of quasi Sasakian structure is always odd (Blair, 1967).It is equal to 1 if the structure is cosympletic and it is equal to 1 2 + n if the structure is Sasakian.

QUASI-SASAKIAN STRUCTURE OF DIMENSION THREE
An almost contact metric manifold of dimension three is called quasi-Sasakian manifold if and only if (De and Absos, 1997) for a function β defined on the manifold, ∇ being the operator of covariant differentiation with respect to Livi-civita connection of the manifold.Also we note that there is a function β on the manifold satisfying X Therefore, taking the skew properties, we can easily verify that 0 = ξβ .Clearly such a quasi-Sasakian manifold is cosympletic if and only if 0 = β as the consequence of Equation (4), we have ( ) In three-dimensional Riemannian manifold the Weyl-conformal curvature tensor vanishes, that is and r is the scalar curvature of the manifold.
Let 3 M be a three-dimensional quasi-Sasakian manifold the Ricci tensor S of 3 M the is given by (Bagewadi and Venkatesh,  2007) Now as the consequence of Equation ( 11), we get the Ricci the gradient of the function f is related to the exterior Moreover, as the consequence of Equations ( 10) and ( 13) we get from Equation ( 10), we get orthogonal to ξ , we get from above π is 1 -form on with ρ as the associated vector field that is with ξ is the associate vector field ,that is have been obtained by Yano (1970).
further, relation between the curvature tensor R and R ~ of type ) 3 , 1 ( of the connection ∇ and ∇ ~ respectively is given by where K is a tensor of type From Equation ( 19) it follows that where S ~is the Ricci tensor of the connection ∇ ~, Differentiating Equation ( 21) conveniently with respect to X , we obtain Now let ( ) i e be the orthonormal basis of the tangent space at each point of manifold for 22) and taking summation over i , we get BASIC RESULTS

Theorem 1
On a three-dimensional quasi-Sasakian manifold where the gradient of a function f is related to the exterior derivative df by the formula

Proof
Differentiating conveniently Equation (11) along the vector field X , we get ) Putting ξ = X Equation ( 24) and using Equations ( 1) and (3), we get Again from Equation ( 24), we get From Equations ( 24) and ( 25), we get the required result.

Theorem 2
A three-dimensional quasi-Sasakian manifold with semisymmetric metric connection under the condition that , we have

Proof
From Equation ( 20), we have Using Equations ( 4) and ( 8) in ( 27), we get the first result in the first result and using Equation (3) we get the second results that is Next by contracting in result (i) and using Equations ( 1), ( 3) and (4), we get the third result.
Again put in result (i), we get the (iv) result.

Theorem 3
A three-dimensional quasi-Sasakian manifold with semisymmetric metric connection under the condition that we have following results.

Proof
From Equation (20), we have Using the result of theorem 2,(i-iii-iv) in Equation ( 26), we get Using Equation ( 30) we get [ ] and also Using Equation ( 4) and the result of theorem (2b) (i-ii), we get Subtracting Equations ( 34) and ( 32) we get the required result.

Theorem 4
A three-dimensional quasi-Sasakian manifold admitting a semi-symmetric metric connection whose quasiconformal curvature tensor with respect to this connection is conservative then the scalar curvature of the manifold is given by

Conclusion
The layout of the work is to characterize the geometrical properties of conformal curvature tensor and the conservative quasi-conformal curvature tensor are on three-dimensional quasi-Sasakian manifold with respect to semi-symmetric metric connection.