Flows of Spin(7)-structures

TL;DR

This paper studies the evolution of Spin(7)-structures on 8-manifolds, deriving equations for their torsion, metric, and curvature, and providing explicit decompositions and identities relevant to Spin(7)-geometry.

Contribution

It introduces evolution equations for Spin(7)-structures, including torsion and metric dynamics, and establishes an analogue of the second Bianchi identity in this context.

Findings

Derived explicit evolution equations for torsion and metric.

Provided a decomposition of forms on Spin(7)-manifolds.

Established a Spin(7) analogue of the second Bianchi identity.

Abstract

We consider flows of Spin(7)-structures. We use local coordinates to describe the torsion tensor of a Spin(7)-structure and derive the evolution equations for a general flow of a Spin(7)-structure on an 8-manifold M. Specifically, we compute the evolution of the metric and the torsion tensor. We also give an explicit description of the decomposition of the space of forms on a manifold with Spin(7)-structure, and derive an analogue of the second Bianchi identity in Spin(7)-geometry. This identity yields an explicit formula for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion.