Weak $Spin(9)$-Structures on 16-dimensional Riemannian Manifolds

The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional 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 $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.