Conformal Killing 2-forms on 4-dimensional manifolds

TL;DR

This paper classifies 4-dimensional simply connected Lie groups with invariant metrics that admit non-trivial conformal Killing 2-forms, revealing conditions under which the metric is half conformally flat or related to conformally Kähler structures.

Contribution

It provides a classification of such Lie groups and metrics, linking conformal Killing forms to conformally Kähler structures and half conformally flat geometries.

Findings

Either the conformal Killing 2-form's real line is invariant under the group action or the metric is half conformally flat.

The problem reduces to studying invariant conformally Kähler structures or specific metric Lie algebra families.

The Lie algebra belongs to a finite list of families up to homothety in the half conformally flat case.

Abstract

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces to the study of invariant conformally K\"ahler structures, whereas in the second case, the Lie algebra of $G$ belongs (up to homothety) to a finite list of families of metric Lie algebras.