Let $G$ be a Lie group, $X$ a smooth manifold, and $G$ act on $X$ smoothly. We have a smooth map $\rho:G\times X \to X$ satisfying the usual axioms of a group action.
In general, $\rho$ is different from just the projection on the second component, so $\rho$ is not the trivial bundle if the action is nontrivial. The fibers of the map are however still homeomorphic to $G$. So is $\rho$ still a principal bundle? What is its geometry? Also, are there known interesting examples of such a bundle, in geometry?