Parallel differential forms of codegree two, and three-forms in dimension six

TL;DR

This paper investigates conditions under which differential forms on manifolds are parallel, focusing on specific forms in six dimensions and their geometric properties, with implications for special holonomy and Lie groups.

Contribution

It proves the converse of a known implication for certain forms in specific dimensions and characterizes forms with constant components in terms of geometric properties.

Findings

Converse holds for (n-2)-forms and 3-forms in dimension six.

Counterexamples exist for Cartan 3-forms on simple Lie groups of dimension ≥8.

Provides geometric characterizations and independence results for forms with constant components.

Abstract

For a differential form on a manifold, having constant components in suitable local coordinates trivially implies being parallel relative to a torsion-free connection, and the converse implication is known to be true for $p$-forms in dimension $n$ when $p=0,1,2,n-1,n$. We prove the converse for $(n-2)$-forms, and for 3-forms when $n=6$, while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions $n\ge8$ as well as for $(n,p)=(7,3)$ and $(n,p)=(8,4)$, where the 3-forms and 4-forms arise in compact simply connected Riemannian manifolds with exceptional holonomy groups. We also provide geometric characterizations of 3-forms in dimension six and $(n-2)$-forms in dimension $n$ having the constant-components property mentioned above, and describe examples illustrating the fact that various parts of these geometric characterizations are logically independent.