Intrinsic torsion and scalar curvature of Spin(7)-structure

Kamil Niedzialomski

Published 2022-12-28Version 1

We obtain explicit formula for the intrinsic torsion of a ${\rm Spin}(7)$-structure on an $8$--dimensional Riemannian manifold. The formula relates the intrinsic torsion with the Lee form $\theta$ and $\Lambda^3_{48}$--component $(\delta\Phi)_{48}$ of codifferential $\delta\Phi$ of the $4$--form defining given structure. Moreover, using the divergence formula obtained by the author for general Riemannian $G$--structure, we derive the 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.