The foliated structure of contact metric $(\kappa,\mu)$-spaces

TL;DR

This paper explores the foliated structure of contact metric $(ppa,mu)$-spaces, providing geometric insights into their classification and conditions for compatible structures, including Sasakian and Tanaka-Webster geometries.

Contribution

It offers a geometric interpretation of Boeckx's classification and establishes conditions for the existence of compatible Sasakian or Tanaka-Webster structures based on the Boeckx invariant.

Findings

Boeckx's classification has a geometric interpretation via Legendre foliations.

Necessary conditions are identified for contact manifolds to admit compatible $(ppa,mu)$-structures.

Spaces with invariant $|I_M| eq 1$ admit compatible Sasakian or Tanaka-Webster structures.

Abstract

In this paper we study the foliated structure of a contact metric $(\kappa,\mu)$-space. In particular, using the theory of Legendre foliations, we give a geometric interpretation to the Boeckx's classification of contact metric $(\kappa,\mu)$-spaces and we find necessary conditions for a contact manifold to admit a compatible contact metric $(\kappa,\mu)$-structure. Finally we prove that any contact metric $(\kappa,\mu)$-space $M$ whose Boeckx invariant $I_M$ is different from $\pm 1$ admits a compatible Sasakian or Tanaka-Webster parallel structure according to the circumstance that $|I_M|>1$ or $|I_M|<1$, respectively.