Lie groups as 4-dimensional Riemannian or pseudo-Riemannian almost product manifolds with nonintegrable structure

TL;DR

This paper explores 4-dimensional Lie groups endowed with almost product structures and Killing metrics, resulting in families of nonintegrable Riemannian and pseudo-Riemannian manifolds with specific geometric properties.

Contribution

It introduces two new classes of 4-dimensional Lie group manifolds with almost product structures and Killing metrics, expanding the understanding of their geometric characteristics.

Findings

Construction of nonintegrable Riemannian almost product manifolds

Development of pseudo-Riemannian counterparts

Characterization of these manifolds within a 4-parametric family

Abstract

A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable structure is obtained, and in the second case - a pseudo-Riemannian one. Each belongs to a 4-parametric family of manifolds, which are characterized geometrically.