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

TL;DR

This paper demonstrates that contact metric $(ppa,mu)$-spaces inherently possess a canonical paracontact structure, revealing new geometric properties and establishing conditions for Sasakian structures based on the Boeckx invariant.

Contribution

It introduces a canonical paracontact structure on contact metric $(ppa,mu)$-spaces and explores its properties, including conditions for Sasakian structures linked to the Boeckx invariant.

Findings

Existence of a canonical paracontact structure compatible with the contact form.

The canonical structure satisfies a nullity condition and induces a sequence of compatible structures.

A canonical Sasakian structure exists when the Boeckx invariant satisfies |I_M|>1.

Abstract

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 $|I_M|>1$.