ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2014, Vol. 35, No. 2, pp. 56–64. c  Pleiades Publishing, Ltd., 2014. Some Curvature Properties of Trans-Sasakian Manifolds Ali Akbar * and Avijit Sarkar ** (Submitted by M. A. Malakhaltsev) Department of Mathematics, University of Kalyani, Kalyani-741235, West Bengal, India Received August 23, 2013 Abstract—The object of the present paper is to study quasi-conformally flat trans-Sasakian manifolds. We consider trans-Sasakian manifolds with η-parallel and cyclic parallel Ricci tensors. φ-Ricci symmetric quasi-conformally flat trans-Sasakian manifolds have been studied. We also investigates quasi-conformally flat trans-Sasakian manifolds which are Einstein Semi-symmetric. DOI: 10.1134/S1995080214020024 Keywords and phrases: trans-Sasakian manifolds, locally φ-symmetric, η-parallel Ricci tensor, quasi-conformal curvature tensor and Einstein Semisymmetric. 1. INTRODUCTION In 1985 J.A. Oubina [7] introduced a new class of almost contact metric manifolds, called trans- Sasakian manifolds, which includes Sasakian, Kenmotso and Cosymplectic structures. The authors in the paper [1, 3] and [12] studied such manifolds and obtained some interesting results. In the paper [5] the author studied conformally flat φ-recurrent trans-Sasakian manifolds. It is known that [6] trans- Sasakian structure of type (0, 0), (0,β) and (α, 0) are Cosymplectic, β-Kenmotsu and α-Sasakian respectively, where α, β ∈ R. In [8] J.C. Marrero proved that a trans-Sasakian manifold of dimension n ≥ 5 is either a Cosymplectic manifold or an α-Sasakian or a β-Kenmotsu manifold. In the present pa- per I have studied quasi-conformally flat trans-Sasakian manifolds of dimension n ≥ 5. Such manifolds are either a cosymplectic or a β-Kenmotsu or an α-Sasakian. The present paper is organized as follows: After introduction in Section 1, some preliminaries are given in Section 2. In Section 3 I study quasi- conformally flat trans-Sasakian manifolds. Section 4 is devoted to study trans-Sasakian manifolds with η-parallel and cyclic parallel Ricci tensors. φ-Ricci symmetric quasi-conformally flat trans-Sasakian manifolds are also considered in this section. In section 5 I study quasi-conformally flat trans-Sasakian manifolds which are Einstein Semi-symmetric. 2. PRELIMINARIES Let M be a (2n + 1)-dimensional connected differentiable manifold endowed with an almost contact metric structure (φ,ξ,η,g), where φ is a tensor field of type (1, 1),ξ is a vector field, η is an 1-form and g is a Riemannian metric on M such that [2] φ 2 X = −X + η(X)ξ, η(ξ )=1. (2.1) g(φX, φY )= g(X, Y ) − η(X)η(Y ). X, Y ∈ T (M ) (2.2) Then also φξ =0, η(φX)=0, η(X)= g(X, ξ ). (2.3) g(φX, X)=0. (2.4) * E-mail: aliakbar.akbar@rediffmail.com ** E-mail: avjaj@yahoo.co.in 56