A Classification of 3-dimensional paracontact metric manifolds with $Q\varphi=\varphi Q$

TL;DR

This paper classifies 3-dimensional paracontact metric manifolds satisfying a specific commutation relation, revealing they are either flat, have certain curvature properties, or satisfy particular trace conditions.

Contribution

It provides a comprehensive classification of 3D paracontact manifolds with the condition $Q= Q$, identifying their geometric types and curvature characteristics.

Findings

Manifolds with $trh^2=0$

Flat manifolds

Manifolds with constant $\xi$-sectional curvature $k eq-1$ and $$-sectional curvature $-k eq 1$

Abstract

We show that a $3-$dimensional paracontact manifold on which $Q\varphi =\varphi Q$ is either a manifold with $trh^2=0$, flat or of constant $\xi-$sectional curvature $k\neq-1$ and constant $\varphi$-sectional curvature $-k\neq 1$.