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.

Starting from \(g\) -natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle \(T_1 M\) of a Riemannian manifold \((M,\langle ,\rangle )\) , we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under \(\mathcal {D}\) -homothetic deformations, and classify paraSasakian and paracontact \((\kappa ,\mu )\) -spaces inside this class. We also present a way to build paracontact \((\kappa ,\mu )\) -spaces from corresponding contact metric structures on \(T_1 M\) .

Generalized \({{(\kappa, \mu)}}\)-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 study paracontact metric (k,μ)-spaces with k = -1, equivalent to h2=0 but not h=0. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric (-1,2)-spaces and (-1,0)-spaces of arbitrary dimension with tensor h of every possible constant rank. We will also show explicit examples of paracontact metric (-1,μ)-spaces with tensor h of non-constant rank, which were not known to exist until now.

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