The $G_2$ geometry of $3$-Sasaki structures
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866917731401465856 |
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| author | Nagy, Paul-Andi Semmelmann, Uwe |
| author_facet | Nagy, Paul-Andi Semmelmann, Uwe |
| contents | We initiate a systematic study of the deformation theory of the second Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure $φ_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with infinitesimal $G_2$ deformations for $φ_{1/\sqrt{5}}$. The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal $G_2$ deformations which are unobstructed to second order. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_04494 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The $G_2$ geometry of $3$-Sasaki structures Nagy, Paul-Andi Semmelmann, Uwe Differential Geometry 53C25, 58H15, 53C10, 58J50, 57R57 We initiate a systematic study of the deformation theory of the second Einstein metric $g_{1/\sqrt{5}}$ respectively the proper nearly $G_2$ structure $φ_{1/\sqrt{5}}$ of a $3$-Sasaki manifold $(M^7,g)$. We show that infinitesimal Einstein deformations for $g_{1/\sqrt{5}}$ coincide with infinitesimal $G_2$ deformations for $φ_{1/\sqrt{5}}$. The latter are showed to be parametrised by eigenfunctions of the basic Laplacian of $g$, with eigenvalue twice the Einstein constant of the base $4$-dimensional orbifold, via an explicit differential operator. In terms of this parametrisation we determine those infinitesimal $G_2$ deformations which are unobstructed to second order. |
| title | The $G_2$ geometry of $3$-Sasaki structures |
| topic | Differential Geometry 53C25, 58H15, 53C10, 58J50, 57R57 |
| url | https://arxiv.org/abs/2101.04494 |