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

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.