The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915417861128192 |
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| author | Ivanov, Stefan Petkov, Alexander |
| author_facet | Ivanov, Stefan Petkov, Alexander |
| contents | It is shown that on compact $Spin(7)$--manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$--torsion connection $R\in S^2Λ^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$--torsion connections simultaneously.
Precise conditions for a compact $Spin(7)$--manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$--torsion connection. It is shown that a compact $Spin(7)$--manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$--manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$--structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06438 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons Ivanov, Stefan Petkov, Alexander Differential Geometry High Energy Physics - Theory It is shown that on compact $Spin(7)$--manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$--torsion connection $R\in S^2Λ^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$--torsion connections simultaneously. Precise conditions for a compact $Spin(7)$--manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$--torsion connection. It is shown that a compact $Spin(7)$--manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$--manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$--structure. |
| title | The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons |
| topic | Differential Geometry High Energy Physics - Theory |
| url | https://arxiv.org/abs/2307.06438 |