# Canonical fibrations of contact metric $(\kappa,\mu)$-spaces

## TL;DR

This paper classifies complete, simply connected contact metric $(ppa,mu)$-spaces as homogeneous manifolds by analyzing their canonical fibrations, revealing their base spaces as complex or para-complexifications of spheres or hyperbolic spaces, and providing new homogeneous representations.

## Contribution

It offers a complete classification of contact metric $(ppa,mu)$-spaces via their canonical fibrations, including new homogeneous models for certain cases.

## Key findings

- Base spaces are complex or para-complexifications of spheres or hyperbolic spaces.
- Provides a new homogeneous representation for spaces with Boeckx invariant less than -1.
- Classifies these spaces based on the Boeckx invariant value.

## Abstract

We present a classification of the complete, simply connected, contact metric $(\kappa,\mu)$-spaces as homogeneous contact metric manifolds, by studying the base space of their canonical fibration. According to the value of the Boeckx invariant, it turns out that the base is a complexification or a para-complexification of a sphere or of a hyperbolic space. In particular, we obtain a new homogeneous representation of the contact metric $(\kappa,\mu)$-spaces with Boeckx invariant less than $-1$.