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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?

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Yes, it is a principal $G$-bundle.

To see that, let us write $Y:=G\times X$. The group $G$ acts on $Y$ by $$(h,(g,x))\mapsto(gh,h^{-1}x).$$(This is a right action). The action is free, and its orbits are the fibers of $\rho$, so we are done.

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