extracta mathematicae Vol. 23, N´ um. 3, 265 – 277 (2008) On Three-Dimensional Trans-Sasakian Manifolds U. C. De, Avijit Sarkar Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India, e-mail: uc de@yahoo.com Department of Mathematics, University of Burdwan, Burdwan 713104, West Bengal, India, e-mail: avjaj@yahoo.co.in Presented by Oscar Garc´ ıa Prada Received February 23, 2008 Abstract : The object of the present paper is to study 3-dimensional trans-Sasakian manifolds which are locally φ-symmetric and have η-parallel Ricci tensor. Also 3-dimensional trans- Sasakian manifolds of constant curvature have been considered. An example of a three- dimensional locally φ-symmetric trans-Sasakian manifold is given. Key words : trans-Sasakian manifold, scalar curvature, locally φ-symmetric, η-parallel Ricci tensor, constant curvature. AMS Subject Class. (2000): 53C25. 1. Introduction Trans-Sasakian manifolds arose in a natural way from the classification of almost contact metric structures by D. Chinea and C. Gonzales [3], and they appear as a natural generalization of both Sasakian and Kenmotsu manifolds. Again in the Gray-Hervella classification of almost Hermite manifolds [7], there appears a class W 4 of Hermitian manifolds which are closely related to locally conformally K¨ahler manifolds. An almost contact metric structure on a mani- fold M is called a trans-Sasakian structure [13] if the product manifold M × R belongs to the class W 4 . The class C 6 ⊕ C 5 ([10], [11]) coincides with the class of trans-Sasakian structures of type (α, β). In [11], the local nature of the two subclasses C 5 and C 6 of trans-Sasakian structures is characterized completely. In [4], some curvature identities and sectional curvatures for C 5 , C 6 and trans- Sasakian manifolds are obtained. It is known that ([8]) trans-Sasakian struc- tures of type (0,0), (0,β ) and (α, 0) are cosymplectic, β-Kenmotsu and α- Sasakian respectively. In [15], it is proved that trans-Sasakian structures are generalized quasi-Sasakian structures [12]. Thus, trans-Sasakian structures also provide a large class of generalized quasi-Sasakian structures. The local structure of trans-Sasakian manifolds of dimension n ≥ 5 has been completely characterized by J. C. Marrero [10]. He proved that a trans- 265