论文标题
旋转（7）结构的两个注释
Two notes on Spin(7)-structures
论文作者
论文摘要
我们得出了$ 8 $ - 尺寸riemannian歧管$ m $的$ {\ rm spin}（7）$的内在扭转的明确公式。在这里，固有的扭转是最小$ {\ rm spin}（7）$ - 连接和levi-civita连接的差异。因此，它是捆绑包$ t^{\ ast} m \ otimes \ mathfrak {spin}^{\ bot}（m）$的一个部分。该公式将固有扭转与李的形式$θ$和$λ^3_ {48} $ - 组件$（Δφ）_ {48} $的编码差$Δφ$的$ 4 $ - 形式定义给定结构的形式。使用获得的公式，我们计算了$ {\ rm spin}（7）$类型$ \ MATHCAL {W} _8 $的结构（二阶）几乎并行。此外，应用作者获得的差异公式，以在另一篇文章中为一般的Riemannian $ g $ - 结构，我们以$θ$，$（Δφ）_ {48} $的规范为标量曲率的众所周知的公式和差异$ {\ rm {\ rm {\ rm {\ rm {\ rm {\ rm（Δφ）_ {48} $。我们在适当的示例中证明公式是合理的。
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.
