# Killing forms on $2$-step nilmanifolds

## TL;DR

This paper classifies left-invariant Killing forms on 2-step nilpotent Lie groups, revealing their structure, relation to complex structures, and conditions for existence, with implications for geometric and algebraic properties.

## Contribution

It provides a classification of Killing forms on 2-step nilpotent Lie groups, linking them to complex structures and reductive geometries, and determines their dimensionality.

## Key findings

- Killing 2-forms decompose according to the de Rham factors.
- Non-zero Killing 2-forms induce bi-invariant complex structures.
- Killing 3-forms occur only in naturally reductive cases.

## Abstract

We study left-invariant Killing $k$-forms on simply connected $2$-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $k=2,3$, we show that every left-invariant Killing $k$-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing $2$-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing $3$-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $k=2$ or $k=3$, we show that the space of Killing $k$-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.