$\eta$-Normality, CR-structures, para-CR structures on almost contact metric and almost paracontact metric manifolds

TL;DR

This paper explores the concept of $\\eta$-normality in almost contact and paracontact metric manifolds, establishing their correspondence with CR-structures, characterizing them via covariant derivatives, and analyzing associated connections and bi-Legendrian structures.

Contribution

It introduces the notion of $\eta$-normality, links it to CR-structures, and provides new characterizations and connections on these manifolds, including conditions for flatness and bi-Legendrian structures.

Findings

$\eta$-normal manifolds correspond to CR-structures.

Existence of Tanaka-like connections with autoparallel Reeb vector field.

Bi-Legendrian flatness characterizes normality.

Abstract

For almost contact metric or almost paracontact metric manifolds there is natural notion of $\eta$-normality. Manifold is called $\eta$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $\eta$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $\eta$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $\eta$-normal manifold with autoparallel Reeb vector field. In particular case contact metric CR-manifold it is usual Tanaka connection. Similar results are obtained for almost paracontact metric manifolds. For manifold with closed fundamental form we shall state uniqueness of this connection. In the last part is studied bi-Legendrian structure of almost paracontact metric manifold with contact characteristic form. It is established that such manifold is bi-Legendrian flat if and only if is normal. There are characterized semi-flat bi-Legendrian manifolds.