Two notes on Spin(7)-structures

2022/12/28 by Kamil Niedziałomski, Niedzialomski, Kamil · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Noncommutative and Quantum Gravity Theories

paper · pdf · doi:10.48550/arxiv.2212.13811

openalex publication_date 2022/12/28 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28

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^∗M⊗\mathfrakspin^\bot(M). The formula relates the intrinsic torsion with the Lee form θ and Λ³₄₈--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 W₈ 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 divθ. We justify the formula on appropriate examples.

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