Geometric structures associated to a contact metric (k,μ)-space

We prove that any contact metric (κ, μ)-space (M, φ, ξ, η, g) admits a canonical paracontact metric structure that is compatible with the contact form η. We study this canonical paracontact structure, proving that it satisfies a nullity condition and induces on the underlying contact manifold (M, η) a sequence of compatible contact and paracontact metric structures satisfying nullity conditions. We then study the behavior of that sequence, which is related to the Boeckx invariant IM and to the bi-Legendrian structure of (M, φ, ξ, η, g). Finally we are able to define a canonical Sasakian structure on any contact metric (κ, μ)-space whose Boeckx invariant satisfies |IM|>1.