Para-CR Structures on almost Paracontact Metric Manifolds

TL;DR

This paper investigates conditions under which almost paracontact metric manifolds are para-CR, establishing new results on their curvature, classification, and examples, with implications for special subclasses like para-Sasakian manifolds.

Contribution

It provides necessary and sufficient conditions for almost paracontact metric manifolds to be para-CR, including curvature identities and classifications for various subclasses.

Findings

Normal almost paracontact metric manifolds are para-CR.

Para-CR paracontact metric manifolds of constant curvature are para-Sasakian with curvature -1.

Conformally flat para-Sasakian manifolds have constant curvature -1.

Abstract

Almost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and suffcient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that the normal almost paracontact metric manifolds are para-CR. We establish necessary and suffcient conditions for paracontact metric manifolds as well as for almost paracosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greather than tree must be para-Sasakian and its curvature equal to minus one. The last assertion do not hold in dimension tree. Moreover, we show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to minus one. New classes of examples of para-CR manifolds are established.