Parallel differential forms of codegree two, and three-forms in dimension six
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915956561805312 |
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| author | Derdzinski, Andrzej Piccione, Paolo Terek, Ivo |
| author_facet | Derdzinski, Andrzej Piccione, Paolo Terek, Ivo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15061 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Parallel differential forms of codegree two, and three-forms in dimension six Derdzinski, Andrzej Piccione, Paolo Terek, Ivo Differential Geometry Primary 53B05 and Secondary 53A55 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. |
| title | Parallel differential forms of codegree two, and three-forms in dimension six |
| topic | Differential Geometry Primary 53B05 and Secondary 53A55 |
| url | https://arxiv.org/abs/2502.15061 |