Conformal structures with $G_2$-symmetric twistor distribution
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909376704413696 |
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| author | Nurowski, Pawel Sagerschnig, Katja The, Dennis |
| author_facet | Nurowski, Pawel Sagerschnig, Katja The, Dennis |
| contents | For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is $(2,3,5)$ when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type $\mathrm{G}_2$) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. Relaxing the rolling assumption, we establish a complete local classification result among those homogeneous 4D split-conformal structures for which the symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conformal structures with $G_2$-symmetric twistor distribution Nurowski, Pawel Sagerschnig, Katja The, Dennis Differential Geometry Primary: 53C30, 53Z05 For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is $(2,3,5)$ when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type $\mathrm{G}_2$) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. Relaxing the rolling assumption, we establish a complete local classification result among those homogeneous 4D split-conformal structures for which the symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives. |
| title | Conformal structures with $G_2$-symmetric twistor distribution |
| topic | Differential Geometry Primary: 53C30, 53Z05 |
| url | https://arxiv.org/abs/2411.01936 |