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

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 \(n\ge 8\) 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.