Equivariant Principal Bundles Over Spheres and Cohomogeneity One Manifolds

We classify smooth ${\rm SO}(n)$-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(\Pi, G)$-bundles over manifolds with cohomogeneity 1. 2000 Mathematical Subject Classification: 55R91.