Martin \v{C}adek and Michael Crabb

A generalization of classical theorems on the existence of sections of real, complex and quaternionic Stiefel manifolds over spheres is proved. We obtain a complete list of Lie group homomorphisms $\rho : G \to G_n$, where $G_n$ is one of the groups $SO(n)$, $SU(n)$ or $Sp(n)$ and $G$ is one of the groups $SO(k)$, $SU(k)$ or $Sp(k)$, which reduce the structure group $G_n$ in the fibre bundle $G_n \to G_{n + 1} \to G_{n + 1} / G_n$.

2000 Mathematics Subject Classification: 55R25, 55R50 (primary), 53C10 (secondary).

E-mail:
cadek@math.muni.cz
m.crabb@maths.abdn.ac.uk