Two notes on Spin(7)-structures
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866910538292789248 |
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| author | Niedzialomski, Kamil |
| author_facet | Niedzialomski, Kamil |
| contents | We derive the explicit formula for the intrinsic torsion of a ${\rm Spin}(7)$-structure on a $8$--dimensional Riemannian manifold $M$. Here, the intrinsic torsion is a difference of the minimal ${\rm Spin}(7)$--connection and the Levi-Civita connection. Hence it is a a section of a bundle $T^{\ast}M\otimes\mathfrak{spin}^{\bot}(M)$. The formula relates the intrinsic torsion with the Lee form $θ$ and $Λ^3_{48}$--component $(δΦ)_{48}$ of a codifferential $δΦ$ of the $4$--form defining a given structure. Using the formula obtained, we compute the condition for a ${\rm Spin}(7)$ structure of type $\mathcal{W}_8$ to be (second order) nearly parallel. Moreover, applying the divergence formula obtained by the author for general Riemannian $G$--structure in another article, we rediscover the well known formula for the scalar curvature in terms of norms of $θ$, $(δΦ)_{48}$ and the divergence ${\rm div}θ$. We justify the formula on appropriate examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_13811 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Two notes on Spin(7)-structures Niedzialomski, Kamil Differential Geometry We derive the explicit formula for the intrinsic torsion of a ${\rm Spin}(7)$-structure on a $8$--dimensional Riemannian manifold $M$. Here, the intrinsic torsion is a difference of the minimal ${\rm Spin}(7)$--connection and the Levi-Civita connection. Hence it is a a section of a bundle $T^{\ast}M\otimes\mathfrak{spin}^{\bot}(M)$. The formula relates the intrinsic torsion with the Lee form $θ$ and $Λ^3_{48}$--component $(δΦ)_{48}$ of a codifferential $δΦ$ of the $4$--form defining a given structure. Using the formula obtained, we compute the condition for a ${\rm Spin}(7)$ structure of type $\mathcal{W}_8$ to be (second order) nearly parallel. Moreover, applying the divergence formula obtained by the author for general Riemannian $G$--structure in another article, we rediscover the well known formula for the scalar curvature in terms of norms of $θ$, $(δΦ)_{48}$ and the divergence ${\rm div}θ$. We justify the formula on appropriate examples. |
| title | Two notes on Spin(7)-structures |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2212.13811 |