Weak $Spin(9)$-structures on 16-dimensional Riemannian manifolds

Abstract

The aim of the present paper is the investigation of 5'pm(9)-structures on 16dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a Spin( We compute the homotopy groups 7ri(X 84 ) of the space X 84 = SO(16)/Spin(9) for i < 14. Next we introduce different geometric types of Spm^-structures and derive the corresponding differential equation for the unique self-dual 8-form O 8 assigned to any type of Spm^j-structure. Finally we construct the twistor space of a 16-dimensional manifold with 5pm(9)-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.