Two notes on Spin(7)-structures

TL;DR

This paper derives an explicit formula for the intrinsic torsion of Spin(7)-structures on 8-manifolds, relating it to the Lee form and codifferential components, and applies it to characterize nearly parallel structures and scalar curvature.

Contribution

It provides a new explicit formula for the intrinsic torsion of Spin(7)-structures and applies it to characterize nearly parallel structures and compute scalar curvature.

Findings

Derived explicit formula for intrinsic torsion in terms of Lee form and codifferential components.

Characterized conditions for Spin(7) structures of type W8 to be nearly parallel.

Recomputed scalar curvature formula using the divergence of the Lee form.

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.