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 p = 0, 1, 2, n - 1, n. We prove the converse for (n-2) (n - 2) -forms, and for 3-forms when n=6 n = 6, while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions n 8 n ≥ 8 as well as for (n, p) = (7, 3) (n, p) = (7, 3) and (n, p) = (8, 4) (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) (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.