Abstract. In this paper, we show new results on slant submanifolds of an almost contact metric manifold. We study and characterize slant submanifolds of Kcontact and Sasakian manifolds. We also study the special class of three-dimensional slant submanifolds. We give several examples of slant submanifolds.1991 Mathematics Subject Classi®cation. 53C15, 53C40.0. Introduction. Slant immersions in complex geometry were de®ned by B.-Y. Chen as a natural generalization of both holomorphic immersions and totally real immersions [2]. Examples of slant immersions into complex Euclidean spaces C 2 and C 4 were given by Chen and Tazawa [2, 4, 5], while slant immersions of KaÈ hler Cspaces into complex projective spaces were given by Maeda, Ohnita and Udagawa [9].In a recent paper [7], A. Lotta has introduced the notion of slant immersion of a Riemannian manifold into an almost contact metric manifold and he has proved some properties of such immersions. A. Lotta and A. M. Pastore have obtained examples of slant submanifolds in the Sasakian-space-form R 2m 1 as the leaves of a harmonic Riemannian 3-dimensional foliation [8]. Finally, A. Lotta has also studied some properties about the intrinsic geometry of 3-dimensional non-anti-invariant slant submanifolds of K-contact manifolds [6].The purpose of the present paper is to study slant immersions in K-contact and Sasakian manifolds. We ®rst review, in Section 1, basic formulas and de®nitions for almost contact metric manifolds and their submanifolds, which we shall use later. In Section 2, we recall the de®nition of a slant submanifold of an almost contact metric manifold and we show a ®rst characterization theorem. In Section 3, we give many interesting examples of slant submanifolds in almost contact metric manifolds and in Sasakian manifolds. Then, we characterize slant submanifolds by means of the covariant derivative of the square of the tangent projection T over the submanifold of the almost contact structure of a K-contact manifold. Later, we study the ®rst interesting class of slant submanifolds: the three-dimensional slant submanifolds. We show some results concerning the tangent T and the normal N projections. We also use the given examples in order to remark some facts concerning the main theorems of the paper. We study slant submanifolds in K-contact manifolds and threedimensional slant submanifolds in Sections 4 and 5 respectively.

Abstract. We study contact metric and trans-Sasakian generalized Sasakian-space-forms. We also give some interesting examples of generalized Sasakian-space-forms by using warped products and conformal changes of metric.

Given a Lie algebra of finite dimension, with a selected basis of it, we show in this paper that it is possible to associate it with a combinatorial structure, of dimension 2, in general. In some particular cases, this structure is reduced to a weighted graph. We characterize such graphs, according to they have 3-cycles or not.

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Abstract. We introduce and study generalized S-space-forms with two structure vector fields. We also present several examples of these manifolds such as certain hypersurfaces of Sasakianspace-forms, principal toroidal bundles and warped products. Moreover, we investigate generalized S-space-forms endowed with an additional structure and we obtain some obstructions for them to be S-manifolds.