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  "body": "\n## Definition\n\nLet $G$ be a Lie group and let $M$ be a [[smooth manifold|F45VQ47GPV]].\nA *principal $G$-bundle* over $M$ is a smooth [[fiber bundle|CX4M8K61AB]] $\\pi \\colon P \\to M$ together with a smooth right [[action|0WCY42EV9V]] $P \\times G \\to P$, $(u, g) \\mapsto u \\cdot g$, such that:\n\n1. the action preserves fibers: $\\pi(u \\cdot g) = \\pi(u)$ for all $u \\in P$ and $g \\in G$;\n2. every point of $M$ has an open neighbourhood $U$ and a diffeomorphism $\\varphi \\colon \\pi^{-1}(U) \\to U \\times G$ with $\\operatorname{pr}_1 \\circ \\varphi = \\pi$ and $\\varphi(u \\cdot g) = \\varphi(u) \\cdot g$, where $G$ acts on $U \\times G$ by $(x, h) \\cdot g = (x, hg)$.\n\n## Remarks\n\nThe product $M \\times G$ with $(x, h) \\cdot g = (x, hg)$ is a principal $G$-bundle, the trivial one.\nThe map $\\R \\to S^1$, $t \\mapsto e^{2\\pi i t}$, with $\\Z$ acting by $t \\cdot n = t + n$, is a principal $\\Z$-bundle.\nThe Hopf map $S^3 \\to S^2$ is a principal $U(1)$-bundle.\n\nNon-example: the circle group acts on $S^2$ by rotations about the vertical axis.\nThis action preserves the fibers of the identity map $S^2 \\to S^2$, but these fibers are points, not copies of the circle.\nMore generally, an action with a fixed point cannot be part of a principal bundle structure, because clause 2 forces $G$ to act freely.\n\nClause 2 implies that $G$ acts freely and transitively on each fiber.\nConversely, for a free and proper smooth action of a Lie group $G$ on a manifold $P$, the quotient map $P \\to P/G$ is a principal $G$-bundle.\nThe same definition with topological groups and homeomorphisms gives topological principal bundles.\n\n## Explanations\n\n### L1 human\n\nA principal bundle is a fiber bundle whose fibers are copies of a group, on which the group acts by right multiplication in a way compatible with local trivializations.\n",
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