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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p=0,1,2,n-1,n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. We prove the converse for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(n-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-forms, and for 3-forms when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>, while pointing out that it fails to hold for Cartan 3-forms on all simple Lie groups of dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n\ge 8$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>8</mn> </mrow> </math></EquationSource> </InlineEquation> as well as for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(n,p)=(7,3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>7</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(n,p)=(8,4)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>8</mn> <mo>,</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, 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 <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(n-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-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

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2026年4月17日

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Andrzej Derdzinski,
Paolo Piccione,
Ivo Terek

摘要

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 $n\ge 8$ $n \geq 8$ 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.