Connection preserving actions are topologically engaging
  Topologically and geometrically engaging actions have proved to be useful to obtain rigidity results for semisimple Lie group actions. We show that the action of a simple noncompact Lie group on a compact manifold preserving a unimodular rigid geometric structure of algebraic type (e.g. a connection together with a volume density) is topologically engaging on an open conull dense set. 
A. Candel and R. Quiroga-Barranco
math.DG