# The decomposition of almost paracontact metric manifolds in eleven classes revisited

## TL;DR

This paper revisits the classification of almost paracontact metric manifolds, expanding the classes to twelve, and provides concrete examples on 3-dimensional Lie groups, enhancing understanding of their structure.

## Contribution

It refines the classification of these manifolds by splitting one class into two and offers explicit examples on Lie groups for each class.

## Key findings

- The classification now includes twelve classes.
- 3-dimensional manifolds belong to four classes.
- Concrete Lie group examples are provided for each class.

## Abstract

This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of $\alpha$-para-Sasakian, $\alpha$-para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four basic classes from the considered classification. We define an almost paracontact metric structure on any 3-dimensional Lie group and give concrete examples of Lie groups belonging to each of the four basic classes, characterized by commutators on the corresponding Lie algebras.