Almost paracontact metric 3-dimensional Walker manifolds

Abstract

In this paper we construct and study almost paracontact metric structures $(\varphi ,\xi ,\eta ,g)$ on a 3-dimensional Walker manifold $(M,g)$ with respect to a local basis only by the coordinate functions of a unit space-like vector field $\xi $, globally defined on $M$ and a function $f$ on $M$, characterizing the Lorentzian metric $g$. Necessary and sufficient conditions are obtained for $M$, endowed with these structures, to fall in one of the following classes of 3-dimensional almost paracontact metric manifolds according to the classification given by S. Zamkovoy and G. Nakova: paracontact metric, normal, almost $\alpha $-paracosymplectic, almost paracosymplectic, paracosymplectic and $\mathbb{G}_{12}$-manifolds. Also, classes to which the studied manifolds do not belong are found. Special attention is paid to an $\eta $-Einstein manifold among the considered manifolds and its $\xi $-sectional, $\varphi $-sectional and scalar curvature are investigated. Examples of the examined manifolds are given.