Paper detail

Geometric structures associated with a contact metric $(κ,μ)$-space

We prove that any contact metric $(κ,μ)$-space $(M,ξ,ϕ,η,g)$ admits a canonical paracontact metric structure which is compatible with the contact form $η$. We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold $(M,η)$ 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 $(M,ξ,ϕ,η,g)$, is then studied. Finally we are able to define a canonical Sasakian structure on any contact metric $(κ,μ)$-space whose Boexkx invariant satisfies $|I_M|>1$.