The canonical 8-form on manifolds with holonomy group Spin(9)

TL;DR

This paper provides an explicit formula for the canonical 8-form on manifolds with Spin(9) holonomy, completing the list of explicit forms for Berger's holonomy groups and exploring related geometric structures.

Contribution

It offers the first explicit expression of the 8-form for Spin(9)-structures and completes the catalog of canonical forms for Berger's holonomy groups.

Findings

Explicit expression of the 8-form in terms of local involutions

Completion of explicit forms for all Berger holonomy groups

Results on Spin(9)-structures as G-structures and curvature of Cayley planes

Abstract

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained.