Spin(9) and almost complex structures on 16-dimensional manifolds

Abstract

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its K\"ahler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in the special case of holonomy Spin(9).