On the moduli spaces of left invariant metrics on cotangent bundle of Heisenberg group

Abstract

The focus of the paper is on the study of the moduli space of left-invariant pseudo- Riemannian metrics on the cotangent bundle of the Heisenberg group. We use algebraic approach to obtain orbits of the automorphism group acting in a natural way on the space of left invariant metrics. However, geometric tools such as the classification of hyperbolic plane conics are often needed. For the metrics obtained by the classification, we study geometric properties: curvature, Ricci tensor, sectional curvature, holonomy, and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as the classification of pseudo-Kähler and pp-wave metrics. We obtain description of parallel symmetric tensors for each metric and show that they are derived from parallel vector fields. Finally, we study the totally geodesic subalgebras and show that for each subalgebra of the observed algebra there is a metric which makes it totally geodesic.