Classification results for three-dimensional (para)contact metric and almost (para)cosymplectic $(\kappa,\mu)$-spaces

TL;DR

This paper provides a comprehensive classification of three-dimensional contact metric, almost cosymplectic, and paracontact metric $(ppa,mu)$-spaces, detailing their structures and local classifications for all parameter values.

Contribution

It offers a complete local classification of three-dimensional paracontact and almost paracosymplectic $(ppa,mu)$-spaces, extending existing results to new geometric contexts.

Findings

Full classification of 3D contact metric and almost cosymplectic $(ppa,mu)$-spaces.

Local classification of 3D paracontact metric $(ppa,mu)$-spaces.

Local classification of 3D almost paracosymplectic $(ppa,mu)$-spaces for all $ppa$ values.

Abstract

It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(\kappa,\mu)$-manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic $(\kappa,\mu)$-spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic $(\kappa,\mu)$-spaces. There is obtained local classification of paracontact metric $(\kappa,\mu)$-spaces, and almost paracosymplectic $(\kappa,\mu)$-spaces, for every possible value of $\kappa$.