2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64192Let G = SL(n,R) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/L is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H'/L'. Consequently, there is a strong restriction on the groups K that can arise over this particular M.15 pages, Latex2e file, no figuresDifferential GeometryDynamical SystemsRepresentation Theory57S20; 22F50, 28D15text