Closed $G_2$-Structures with Negative Ricci Curvature
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866914073276317696 |
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| author | Payne, Alec |
| author_facet | Payne, Alec |
| contents | We study existence problems for closed $G_2$-structures with negative Ricci curvature, and we prove the $G_2$-Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed $G_2$-structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed $G_2$-structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed $G_2$-structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19553 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Closed $G_2$-Structures with Negative Ricci Curvature Payne, Alec Differential Geometry 53C25, 53C29, 53C20 We study existence problems for closed $G_2$-structures with negative Ricci curvature, and we prove the $G_2$-Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed $G_2$-structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed $G_2$-structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed $G_2$-structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics. |
| title | Closed $G_2$-Structures with Negative Ricci Curvature |
| topic | Differential Geometry 53C25, 53C29, 53C20 |
| url | https://arxiv.org/abs/2310.19553 |