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

Abstract

For a Spin(9)-structure on a Riemannian manifold M16 we write explicitly the matrix ψ of its Kahler 2-forms and the canonical 8-form ΦSpin(9). We then prove that ΦSpin(9) coincides up to a constant with the fourth coefficient of the characteristic polynomial of ψ. 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 ΦSpin(9) and \({\Phi_{\rm Spin(9)}^2}\) in the special case of holonomy Spin(9).