Spin(7)-manifolds with parallel torsion form

TL;DR

This paper classifies Spin(7)-manifolds with non-zero parallel torsion, showing they are either Riemannian products or homogeneous spaces with special spinor fields, expanding understanding of their geometric structures.

Contribution

It provides a classification of Spin(7)-manifolds with parallel torsion and non-Abelian isotropy algebra, identifying their geometric types and spinor field properties.

Findings

Manifolds are either Riemannian products or homogeneous spaces.

Each admits two -parallel spinor fields.

Classification extends understanding of Spin(7)-structures with torsion.

Abstract

Any Spin(7)-manifold admits a metric connection \nabla^c with totally skew-symmetric torsion T^c preserving the underlying structure. We classify those with \nabla^c-parallel T^c\neq0 and non-Abelian isotropy algebra iso(T^c)<spin(7). These are isometric to either Riemannian products or homogeneous naturally reductive spaces, each admitting two \nabla^c-parallel spinor fields.