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  "body": "\n## Definition\n\nLet $G$ be a [[Lie group|7JBRGCVJRB]] with identity $e$ and let $M$ be a [[smooth manifold|F45VQ47GPV]].\nA *(left) Lie group action* of $G$ on $M$ is a map $\\alpha \\colon G \\times M \\to M$, written $\\alpha(g, p) = g \\cdot p$, such that:\n\n1. $e \\cdot p = p$ for all $p \\in M$;\n2. $g \\cdot (h \\cdot p) = (gh) \\cdot p$ for all $g, h \\in G$ and $p \\in M$;\n3. $\\alpha$ is [[smooth|0V33BNKZ9S]].\n\n## Remarks\n\nClauses 1 and 2 say that $\\alpha$ is a [[group action|0WCY42EV9V]]; clause 3 is the smoothness condition.\nRight actions are defined in the same way with $(p \\cdot h) \\cdot g = p \\cdot (hg)$.\n\nExample: $SO(2)$ acts on $\\R^2$ by $A \\cdot v = Av$.\nThe identity matrix fixes every vector, $(AB)v = A(Bv)$, and the map $(A, v) \\mapsto Av$ is polynomial in the entries, hence smooth.\n\nNon-example: let $G = \\R$ act on $M = \\R$ by $t \\cdot x = (x^3 + t)^{1/3}$, with the real cube root.\nThis is a group action, since $s \\cdot (t \\cdot x) = (x^3 + t + s)^{1/3}$, and it is continuous.\nIt is not smooth: the map $t \\mapsto t \\cdot 0 = t^{1/3}$ is not differentiable at $t = 0$.\n\nEach $g$ acts by a diffeomorphism of $M$ with inverse given by $g^{-1}$.\n",
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