On an ( ε , δ )-trans-Sasakian structure

In this paper we investigate (ε,δ )-trans-Sasakian manifolds which generalize the notion of (ε)-Sasakian and (ε)-Kenmotsu manifolds. We prove the existence of such a structure by an example and we consider φ -recurrent, pseudoprojectively flat and pseudo-projective semi-symmetric (ε,δ )-trans-Sasakian manifolds.


INTRODUCTION
The study of manifolds with indefinite metrics is of interest from the standpoint of physics and relativity.Manifolds with indefinite metrics have been studied by several authors.In 1993, Bejancu and Duggal [1] introduced the concept of (ε)-Sasakian manifolds and Xufeng and Xiaoli [6] established that these manifolds are real hypersurfaces of indefinite Kahlerian manifolds.Kumar et al. [4] studied the curvature conditions of these manifolds.Tripathi et al. [5] introduced and studied (ε)-almost para contact manifolds.Recently De and Sarkar [3] introduced (ε)-Kenmotsu manifolds and studied conformally flat, Weyl semisymmetric, φ -recurrent (ε)-Kenmotsu manifolds.The existence of a new structure on indefinite metrics influences the curvature.Motivated by the above studies, in this paper we introduce the concept of (ε, δ )-trans-Sasakian manifolds which generalizes the notion of (ε)-Sasakian as well as (ε)-Kenmotsu manifolds.
The paper is organized as follows.Section 2 covers some preliminary facts on (ε, δ )-trans-Sasakian structures.In Section 3, we give an example of such a structure and present some basic results.Also an explicit formula for the curvature tensor and Ricci tensors are obtained.Section 4 is devoted to φ -recurrent (ε, δ )-trans-Sasakian manifolds.In Section 5, we consider pseudo-projectively flat (ε, δ )-trans-Sasakian manifolds and prove that these manifolds are Einstein.In the last section, we consider (ε, δ )-trans-Sasakian manifolds with the condition R.P = 0 and prove that these manifolds are pseudo-projectively flat.
From (3.4) it follows that Example.Let (x, y, z) be Cartesian coordinates in R 3 and let Then e 1 , e 2 , e 3 are linearly independent at each point of M. We define Then the (ϕ, ξ , η, g) structure is an (ε)-almost contact metric structure in R 3 .
Further from Koszul's formula we have Using the above relations, for any vector field X on M we have Using (2.7), the above relation yields (3.6).
We note that for constants α and β , Consequently, Using (2.7) in the above equation, we have Again using (2.7) and in view of we obtain (3.9).Replacing Z by φ Z in (3.9) and making use of (2.1), we obtain Contracting the above with ξ and using (2.4) and (2.5), we obtain (3.10).

φ -RECURRENT (ε, δ )-TRANS-SASAKIAN MANIFOLDS
Let M be an (ε, δ )-trans-Sasakian manifold.Then M is said to be a φ -recurrent manifold if there exists a nonzero 1-form for arbitrary vector fields X,Y, Z,W on M. If the 1-form vanishes identically, then the manifold will be called a φ -symmetric manifold.Suppose that the (ε, δ )-trans-Sasakian manifold under consideration is φ -recurrent.Then from (4.1) and (2.1), we obtain By the above relation, Bianchi's identity yields Taking Y = Z = e i in the above equation, where (e i ) is an orthogonal basis of the tangent space at each point of the manifold and using (3.10) in (4.3), we obtain For X = ξ , this equation yields If α and β are constants, then from (2.6) and (3.8) in (4.6), we obtain If we consider X,Y orthogonal to ξ , then in view of (3.8), we have η( From (4.7), we obtain Suppose that the manifold is φ -recurrent.Then using (4.2), we obtain from the above equation i.e.
From the pseudo-projective curvature tensor as given in (5.1), a symmetric tensor of type (0, 2) can be defined as follows: RicP(X,Y ) = P(X, e i , e i ,Y ), (6.12) where P(X,Y, Z,U) = g( P(X,Y )Z,U) and (e i ) is an orthonormal basis of the tangent space at any point and summing over 1 ≤ i ≤ n in (6.12).From (5.1) and (6.12), we have For constants α and β , taking X = ξ in (6.14) and using (3.7), we have Thus we have Theorem 6.3.Let M be an (ε, δ )-trans-Sasakian manifold and let R(X,Y ).RicP = 0 hold in M. Then either α = ±β or M is an Einstein manifold in which case the curvature is given by r = n(n − 1)ε(εα 2 − δ β 2 ).

CONCLUSIONS
The (ε, δ )-trans-Sasakian manifolds generalize the (ε)-Sasakian and the (ε)-Kenmotsu manifolds.The indefinite metrics which arose during the study of physics and relativity from the geometric point of view influences the curvature.In this paper we proved under certain conditions that the (ε, δ )-trans-Sasakian manifolds reduce to the manifolds of constant curvature.Further we showed that (ε, δ )-trans-Sasakian manifolds with a pseudo-projective curvature tensor are Einstein.The study of these structures with semisymmetry conditions would give interesting results.