Geometry of cotangent bundle of Heisenberg group

TL;DR

This paper classifies left-invariant Riemannian metrics on the cotangent bundle of the Heisenberg group, proves the uniqueness of a complex structure, describes Ricci-flat pseudoKähler metrics, and establishes the uniqueness of an ad-invariant metric.

Contribution

It provides a complete classification of metrics and complex structures on the cotangent bundle of the Heisenberg group, including uniqueness results and Ricci-flat metrics.

Findings

Classification of left-invariant Riemannian metrics

Uniqueness of the complex structure

Existence of Ricci-flat pseudoKähler metrics

Abstract

In this paper the classification of left-invariant Riemannian metrics, up to the action of the automorphism group, on cotangent bundle of (2n+1)-dimensional Heisenberg group is presented. Also, it is proved that the complex structure on that group is unique and the corresponding pseudoK\"ahler metrics are described and shown to be Ricci flat. It is well known that this algebra admits an ad-invariant metric of neutral signature. Here, the uniqueness of such metric is proved