Local classification and examples of an important class of paracontact metric manifolds

TL;DR

This paper investigates specific paracontact metric manifolds with -1, providing new examples and classifications of their tensor properties, including cases with non-constant rank of the tensor h.

Contribution

It offers an alternative proof of a key theorem and introduces new examples of paracontact metric spaces with various tensor ranks, expanding understanding of their structure.

Findings

Examples of paracontact metric (-1,2)-spaces and (-1,0)-spaces with arbitrary tensor h rank

Explicit examples of spaces with non-constant tensor h rank

Alternative proof of a significant theorem in the field

Abstract

We study paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We will also show explicit examples of paracontact metric $(-1, \mu)$-spaces with tensor $h$ of non-constant rank, which were not known to exist until now.