Conformal Killing forms on $2$-step nilpotent Riemannian Lie groups

TL;DR

This paper classifies conformal Killing forms on 2-step nilpotent Riemannian Lie groups, showing that large centers imply all such forms are Killing, and explicitly describes the forms on specific low-dimensional cases.

Contribution

It provides a complete classification of conformal Killing 2- and 3-forms on 2-step nilpotent Lie groups based on the dimension of their center, including explicit descriptions.

Findings

If the center dimension ≥ 4, all conformal Killing forms are coclosed.

The only groups with non-coclosed forms are Heisenberg and certain free 2-step nilpotent groups.

Explicit descriptions of the conformal Killing forms are given for these cases.

Abstract

We study left-invariant conformal Killing $2$- or $3$-forms on simply connected $2$-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing $2$- and $3$-forms are: the Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing $2$- and $3$-forms is provided in each case.