On the geometry of para-hypercomplex 4-dimensional Lie groups

In this paper we first completely determine left-invariant generalized Ricci solitons and Einstein-like metrics on para-hypercomplex 4- dimensional Lie groups equipped with a left-invariant Riemannian metric g and a left-invariant Lorentzian metric ĝ1. Then we obtain the exact form of all harmonic maps and prove that contrary to the Lorentzian case, there exist spaces where their energy functional restricted to vector fields of the same length admit any left-invariant vector fields X as critical points. We also obtain the spaces on which all homogeneous Riemannian structures coincide with all homogeneous Lorentzian structures. Finally, we give the complete classification and explicitly describe totally geodesic hypersurfaces of these spaces in both Riemannian and Lorentzian cases. The existence of algebraic Ricci solitons which are neither Einstein nor Ricci and Yamabe solitons is proved. Also remarkable differences between the existence of Einstein, Einstein-like metrics, locally symmetric, conformal flatness and some equations in Riemannian and Lorentzian cases are given.