Geometric structures associated with a contact metric \((\kappa,\mu)\)-space

We prove that any contact metric \((\kappa,\mu)\)-space \((M,\xi,\phi,\eta,g)\) admits a canonical paracontact metric structure which is compatible with the contact form \(\eta\). We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold \((M,\eta)\) a sequence of compatible contact and paracontact metric structures verifying nullity conditions. The behavior of that sequence, related to the Boeckx invariant \(I_M\) and to the bi-Legendrian structure of \((M,\xi,\phi,\eta,g)\), is then studied. Finally we are able to define a canonical Sasakian structure on any contact metric \((\kappa,\mu)\)-space whose Boexkx invariant satisfies \(|I_M|>1\).