Generalized vector cross products and Killing forms on negatively curved manifolds

TL;DR

This paper introduces generalized vector cross products, classifies them, and proves that Killing 3-forms vanish on negatively curved compact manifolds of dimension four or higher.

Contribution

It defines and classifies generalized vector cross products and establishes the vanishing of Killing 3-forms on negatively curved compact manifolds in dimensions four and above.

Findings

Generalized vector cross products are classified.

Killing 3-forms vanish on negatively curved compact manifolds for dimensions ≥ 4.

New characterization of SU(3)-structures related to Killing forms.

Abstract

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on $\mathbb{R}^n$ and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of $\mathrm{SU}(3)$-structures in dimension $6$ whose associated $3$-form is Killing, we then show that every Killing $3$-form on a compact $n$-dimensional Riemannian manifold with negative sectional curvature vanishes if $n\ge 4$.