A two-level Kaluza-Klein theory

Abstract

Let the Lie groups G and H act on the manifold P in such a way that P fibres as a principal G-bundle over P/G and as an H-bundle over P/H. We find that every pair (τ′,τ″) where τ′ is an H-invariant connection form in P→P/G and τ″ is a G-invariant connection form in P→P/H corresponds uniquely to a connection form in P→P/(H×G) and a cross-section of a vector bundle with base P/(H×G).