Two notes on Spin(7)-structures

Abstract

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 $\theta$ and $\Lambda^3_{48}$--component $(\delta\Phi)_{48}$ of a codifferential $\delta\Phi$ 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 $\theta$, $(\delta\Phi)_{48}$ and the divergence ${\rm div}\theta$. We justify the formula on appropriate examples.