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

Andrzej Derdzinski, Paolo Piccione, Ivo Terek

TL;DR

This work analyzes when algebraically constant $p$-forms are forced to be parallel, focusing on the two remaining borderline cases: $(n-2)$-forms in general dimension and $3$-forms in dimension six. It proves the converse implications in these cases, while constructing counterexamples (notably Cartan $3$-forms on simple Lie groups and certain $(n,p)$ pairs) where parallelism does not imply local constancy. The authors develop detailed geometric characterizations involving divisibility and kernel distributions, almost complex structures, and duality with nondegenerate $2$-forms, and they examine isotropy Lie algebras and Cartan forms to delineate when parallelism is rigid. The paper also establishes logical independence among the components of the characterizations, and it highlights the limitations of local constancy versus parallelism in higher dimensions, with implications for holonomy and exceptional geometries.

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.

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

TL;DR

This work analyzes when algebraically constant -forms are forced to be parallel, focusing on the two remaining borderline cases: -forms in general dimension and -forms in dimension six. It proves the converse implications in these cases, while constructing counterexamples (notably Cartan -forms on simple Lie groups and certain pairs) where parallelism does not imply local constancy. The authors develop detailed geometric characterizations involving divisibility and kernel distributions, almost complex structures, and duality with nondegenerate -forms, and they examine isotropy Lie algebras and Cartan forms to delineate when parallelism is rigid. The paper also establishes logical independence among the components of the characterizations, and it highlights the limitations of local constancy versus parallelism in higher dimensions, with implications for holonomy and exceptional geometries.

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 -forms in dimension when . We prove the converse for -forms, and for 3-forms when , while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions as well as for and , 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 -forms in dimension having the constant-components property mentioned above, and describe examples illustrating the fact that various parts of these geometric characterizations are logically independent.

Paper Structure

This paper contains 16 sections, 18 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some invariants of exterior forms
  4. Further invariants
  5. The Hodge star duality
  6. Exterior $\,3$-forms in dimension six
  7. The simplest invariants of differential forms
  8. Differential $\,3$-forms in dimension six
  9. Local constancy and parallelism of differential forms
  10. Proof of Theorem \ref{['tfsix']}
  11. Proof of Theorem \ref{['lcint']}
  12. Iso-tropy Lie algebras and connections
  13. The Car-tan $\,3$-forms of simple Lie algebras
  14. Parallelism without local constancy
  15. Logical independence in Theorems \ref{['tfsix']} and \ref{['lcint']}
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $\,\mathcal{D}\,$ be a $\,\nabla$-par-al-lel distribution on a manifold $\,M\,$ with a tor-sion-free connection $\,\nabla$. If a $\,(0,r)\,$ tensor field $\,\xi$ on $\,M$ annihilates $\,\mathcal{D}$, then so does $\,\nabla _v^{\space i} \xi$, for any vector field $\,v$, while $\,\nabla _v^{\spac

Figures (2)

  • Figure 1: The five types (\ref{['six']}) of $3$-forms in dimension six. Each maximal solid line segment corresponds to one summand in (\ref{['six']}), and so does the small inscribed $\bigtriangledown$ triangle in (a). They are all oriented as indicated by the arrows.
  • Figure 2: The octahedron version of (\ref{['six']}-a). Four mutually non-ad-ja-cent faces correspond to $\,\mu=\mathrm{Re}\, \omega\,$ and the remaining four to $\,\mathrm{Im}\, \omega$, where seven faces (all but one of the latter, namely, 531) represent the same orientation of the boundary surface. The four $\,\mu$-faces are also characterized by being coherently oriented by the ar-row-mark-ed orientations of their sides.

Theorems & Definitions (48)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 38 more