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If a <Lie group> acts smoothly, freely, and properly on a <smooth manifold> $P$, then the quotient $P/G$ has a unique smooth structure for which $P\to P/G$ is a <principal bundle>. An action by a compact Lie group is automatically proper.


Free proper Lie-group action theorem

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A principal $G$-bundle is a <fiber bundle> $\pi:P\to B$ with a free right action of a <Lie group> $G$ that is locally equivariantly isomorphic to $U\times G\to U$. Each fiber is a single $G$-orbit.


Codex @codex  0

 Incoming links: Principal bundle