The object ofthe present paper is to study the geometry oftrans-Sasakian manifold when it is projectively semi-symmetric, Weyl semi-symmetric and concircularly semi-symmetric. | Turk J Math 31 (2007) , 111 – 121. ¨ ITAK ˙ c TUB Some Curvature Tensors on a Trans-Sasakian Manifold C. S. Bagewadi and Venkatesha Abstract The object of the present paper is to study the geometry of trans-Sasakian manifold when it is projectively semi-symmetric, Weyl semi-symmetric and concircularly semi-symmetric. Key words and phrases: Trans-Sasakian, projectively flat, concircularly flat. 1. Introduction In 1985, . Oubina [9] introduced a new class of almost contact manifold namely trans-Sasakian manifold. Many geometers like [1, 2, 6], [5], [9], have studied this manifold and obtained many interesting results. The notion of semi-symmetric manifold is defined by R(X, Y )·R = 0 and studied by many authors [10, 11, 12]. The conditions R(X,Y)·P=0, R(X,Y)·C=0 and R(X,Y)·C = 0 are called projectively semi-symmetric, Weyl semisymmetric and concircularly semi-symmetric respectively, where R(X,Y) is considered as derivation of tensor algebra at each point of the manifold. In this paper we consider the trans-Sasakian manifold under the condition φ (grad α) = (2m-1) grad β satisfying the properties R(X,Y)·P = 0, R(X,Y)·C = 0 and R(X,Y)·C = 0 and show that such a manifold is either Einstein or η-Einstein. AMS Mathematics Subject Classification: Primary 55S10, 55S05 111 BAGEWADI, VENKATESHA 2. Preliminaries Let M be a (2m + 1) dimensional almost contact metric manifold with an almost contact metric structure (φ, ξ, η, g), where φ is a (1,1) tensor field, ξ is a vector field, η is a 1-form and g is the associated Riemannian metric such that [3], φ2 = −I + η ⊗ ξ, η(ξ) g(φX, φY ) = 1, φξ = 0, ηoφ = 0, = g(X, Y ) − η(X)η(Y ), g(X, φY ) = −g(φX, Y ) and g(X, ξ) = η(X)∀X, Y ∈ T M. () () An almost Contact metric structure (φ, ξ, η, g) on M is called a trans-Sasakian structure [9],if (M × R, J, G) belongs to the class W4 [7], where J is the almost complex d d ) = (φX − fξ, η(X) dt ) for all vector fields X on structure on M ×R defined by J(X, f dt M and smooth .
