Classification of Π-Manifolds Regarding the Pseudo-Riemannian Metric Associated Through the Structure

The study of the so-called Π-manifolds has been continued. This is a short name for almost paracontact almost paracomplex manifolds with a pair of metrics—Riemannian and pseudo-Riemannian—that are mutually related and compatible with the structure of the manifold. The well-known classification of these manifolds with respect to the Riemannian metric has been used to derive an alternative classification of the same manifolds with respect to the pseudo-Riemannian metric. The interrelations between the two classifications and the corresponding classification tensors are investigated. Finally, a three-dimensional explicit example on a Lie group is given, which illustrates the most interesting case of the transition from one classification to the other.