On symmetries of the Heisenberg group

TL;DR

This paper classifies affine, Killing, Ricci, curvature, and matter collineations of the 3D Heisenberg group with any left-invariant metric, revealing the absence of nontrivial examples and clarifying their geometric symmetries.

Contribution

It provides a complete classification of symmetries and collineations of the 3D Heisenberg group with any left-invariant metric, including Lorentzian and Riemannian cases.

Findings

All affine vector fields are classified.

No nontrivial Ricci, curvature, or matter collineations exist.

Relationship between affine and Killing vector fields is clarified.

Abstract

We consider the three-dimensional Heisenberg group, equipped with any left-invariant metric, either Lorentzian or Riemannian. We completely classify their affine vector fields and investigate their relationship with Killing vector fields and their casual character. We also classify their Ricci, curvature and matter collineations, proving that there are no nontrivial examples.