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

TL;DR

This paper classifies and analyzes the geometric properties of left invariant pseudo-Riemannian metrics on the cotangent bundle of the Heisenberg group, exploring their curvature, symmetries, and special structures.

Contribution

It provides a comprehensive algebraic and geometric classification of these metrics, including Ricci solitons, pseudo-Kahler, and ppwave structures, and studies their geodesic subalgebras.

Findings

Classification of metrics based on automorphism orbits

Identification of algebraic Ricci solitons and special structures

Existence of metrics making subalgebras totally geodesic

Abstract

The main focus of the paper is the investigation of moduli space of left invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group. Consideration of orbits of the automorphism group naturally acting on the space of the left invariant metrics allows us to use the algebraic approach. However, the geometrical tools, such as classification of hyperbolic plane conics, will often be required. For metrics that we obtain in the classification, we investigate geometrical properties: curvature, Ricci tensor, sectional curvature, holonomy and parallel vector fields. The classification of algebraic Ricci solitons is also presented, as well as classification of pseudo-Kahler and ppwave metrics. We get the description of parallel symmetric tensors for each metric and showthat they are derived from parallel vector fields. Finally, we investigate the totally geodesic subalgebras by showing that for any subalgebra of the observed algebra there exists a metric that makes it totally geodesic.