Abstract. We study non-paraSasakian paracontact metric (κ, µ)-spaces with κ = −1 (equivalent to h 2 = 0 but h = 0). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of h. We will also give explicit examples of every possible constant rank of h.

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.

Abstract. Starting from g-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle T 1 M of a Riemannian manifold (M, , ), we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under D-homothetic deformations, and classify paraSasakian and paracontact (κ, µ)-spaces inside this class. We also present a way to build paracontact (κ, µ)-spaces from corresponding contact metric structures on T 1 M .

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.Mathematics Subject Classification (2010). Primary 53C15, Secondary 53C25.