On $Sp(2)$ and $sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles

Abstract

Let $\xi$ be an oriented 8-dimensional vector bundle. We prove that the structure group $SO(8)$ of $\xi$ can be reduced to $Sp(2)$ or $Sp(2)\cdot Sp(1)$ if and only if the vector bundle associated to $\xi$ via a certain outer automorphism of the group $Spin(8)$ has 3 linearly independent sections or contains a $3$-dimensional subbundle. Necessary and sufficient conditions for the existence of an $Sp(2)$-structure in $\xi$ over a closed connected spin manifold of dimension~$8$ are also given in terms of characteristic classes.