Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914568240889856 |
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| author | Russo, Giovanni Swann, Andrew |
| author_facet | Russo, Giovanni Swann, Andrew |
| contents | We study nearly parallel $\mathrm{G}_{2}$-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel $\mathrm{G}_{2}$-manifold yields a multi-moment map. The torus acts freely on its regular level sets, so they are torus bundles over smooth three-dimensional manifolds. We show that the geometry of the base spaces is specified by two triples of closed two-forms related by a Riemannian metric. We then describe an inverse construction producing invariant nearly parallel $\mathrm{G}_{2}$-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry Russo, Giovanni Swann, Andrew Differential Geometry 53C25, 53C26, 53D20, 57S25 We study nearly parallel $\mathrm{G}_{2}$-structures with a three-torus symmetry via multi-moment map techniques. An effective three-torus action on a nearly parallel $\mathrm{G}_{2}$-manifold yields a multi-moment map. The torus acts freely on its regular level sets, so they are torus bundles over smooth three-dimensional manifolds. We show that the geometry of the base spaces is specified by two triples of closed two-forms related by a Riemannian metric. We then describe an inverse construction producing invariant nearly parallel $\mathrm{G}_{2}$-structures from three-dimensional data. We observe that locally this may produce examples with four-torus symmetry. |
| title | Nearly Parallel $\mathrm{G}_{2}$-Structures with Torus Symmetry |
| topic | Differential Geometry 53C25, 53C26, 53D20, 57S25 |
| url | https://arxiv.org/abs/2508.21703 |