We introduce the concept of ε -almost paracontact manifolds, and in particular, of ε -para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of ε -para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an εpara Sasakian structure. We show that, for an ε -para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike resp., timelike ε -para Sasakian manifold M n is locally isometric to a pseudohyperbolic space H n ν 1 resp., pseudosphere S n ν 1 . At last, it is proved that for an ε -para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

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Abstract. Generalized (κ, µ)-space forms are introduced and studied. We examine in depth the contact metric case and present examples for all possible dimensions. We also analyse the trans-Sasakian case.

We present Chen-Ricci inequality and improved Chen-Ricci inequality for curvature like tensors. Applying our improved Chen-Ricci inequality we study Lagrangian and Kaehlerian slant submanifolds of complex space forms, and C-totally real submanifolds of Sasakian space forms.2000 Mathematics Subject Classification. 53C40, 53C42, 53B25, 53C15, 53C25.Keywords and phrases: Curvature like tensor, Riemannian vector bundle, improved Chen-Ricci inequality, improved Chen-Ricci inequality, Lagrangian submanifold, Kaehlerian slant submanifold, C-totally real submanifold, complex space form and Sasakian space form.