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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2209.02761 |
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| _version_ | 1866929615748988928 |
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| author | Alonso, Izar |
| author_facet | Alonso, Izar |
| contents | We consider two different $\text{SU}(2)^2$-invariant cohomogeneity one manifolds, one non-compact $M=\mathbb{R}^4 \times S^3$ and one compact $M=S^4 \times S^3$, and study the existence of coclosed $\text{SU}(2)^2$-invariant $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. For $\mathbb{R}^4 \times S^3$, we prove the existence of a family of coclosed (but not necessarily torsion-free) $G_2$-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $G_2$-structure constructed from a half-flat $\text{SU}(3)$-structure is in this family. For $S^4 \times S^3$, we prove that there are no $\text{SU}(2)^2$-invariant coclosed $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02761 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coclosed $G_2$-structures on $\text{SU}(2)^2$-invariant cohomogeneity one manifolds Alonso, Izar Differential Geometry 53C10 We consider two different $\text{SU}(2)^2$-invariant cohomogeneity one manifolds, one non-compact $M=\mathbb{R}^4 \times S^3$ and one compact $M=S^4 \times S^3$, and study the existence of coclosed $\text{SU}(2)^2$-invariant $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. For $\mathbb{R}^4 \times S^3$, we prove the existence of a family of coclosed (but not necessarily torsion-free) $G_2$-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed $G_2$-structure constructed from a half-flat $\text{SU}(3)$-structure is in this family. For $S^4 \times S^3$, we prove that there are no $\text{SU}(2)^2$-invariant coclosed $G_2$-structures constructed from half-flat $\text{SU}(3)$-structures. |
| title | Coclosed $G_2$-structures on $\text{SU}(2)^2$-invariant cohomogeneity one manifolds |
| topic | Differential Geometry 53C10 |
| url | https://arxiv.org/abs/2209.02761 |