PLOVDIVSKI UNIVERSITET “PAISI HILENDARSKI”, B_LGARI NAUQNI TRUDOVE, TOM 34, KN. 3, 2004 – MATEMATIKA PLOVDIV UNIVERSITY “PAISSII HILENDARSKI”, BULGARIA SCIENTIFIC WORKS, VOL. 34, BOOK 3, 2004 – MATHEMATICS CURVATURE PROPERTIES OF SOME THREE-DIMENSIONAL ALMOST CONTACT B-METRIC MANIFOLDS Mancho Manev, Galia Nakova Abstract. The curvature tensor on an arbitrary 3-dimensional Lorentz manifold is expressed by the Ricci tensor and the scalar curvature. The curva- ture tensor on a 3-dimensional almost contact B-metric manifold belonging to two main classes is studied. The corresponding curvatures are found and the respective geometric characteristics of the considered manifolds are obtained. Mathematics Subject Classifications 2000: Primary 53D15, 53C50; Secondary 53C15, 53C25. Key words: almost contact manifold, indefinite metric, curvatures. 1. Preliminaries Let (M,ϕ,ξ,η,g) be a (2n + 1)-dimensional almost contact manifold with B-metric, i.e. (ϕ,ξ,η) is an almost contact structure and g is a metric on M such that (1.1) ϕ 2 = -id + η ⊗ ξ ; η(ξ ) = 1; g(ϕ·,ϕ·)= -g(·, ·)+ η(·)η(·). Both metrics g and its associated ˜ g :˜ g = g * + η ⊗ η are indefinite metrics of signature (n, n + 1) [1], where it is denoted g * = g(·,ϕ·). In this paper we study the curvature properties of the almost contact B- metric manifolds of dimension three. This dimension is the lowest possible dimension of these manifolds. 51