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

2025/02/20 by Derdzinski, Andrzej, Piccione, Paolo, Terek, Ivo
#Differential Geometry (math.DG) #FOS: Mathematics #Primary 53B05 and Secondary 53A55

paper · doi:10.48550/arxiv.2502.15061

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≥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.

Related

• The geometry of three-forms in six and seven dimensions
• 3-forms and almost complex structures on 6-dimensional manifolds
• Almost Parallel Structures
• Closed 3-forms in five dimensions and embedding problems
• Parallel forms, co-Kähler Manifolds and their Models