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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $k \\ge 0$ be an integer.\nLet $J^k_\\times(M)$ be the set of $k$-[[jets|1QE0Z0756D]] $j^k_x f$ of local [[diffeomorphisms|0KZZ70ZHXP]] $f$ of $M$ at points $x$ of their domains.\nIt is a [[Lie groupoid|HNVW0E732J]] over $M$, with composition of jets as multiplication.\nLet $\\Gamma$ be a [[pseudogroup|3017J7P6JD]] of local diffeomorphisms of $M$.\nIt is a *Lie pseudogroup of order $k$* if there is a subgroupoid $R \\subseteq J^k_\\times(M)$ with these properties.\n\n1. $R$ is a closed embedded submanifold of $J^k_\\times(M)$.\n2. A local diffeomorphism $f \\colon U \\to V$ of $M$ lies in $\\Gamma$ if and only if $j^k_x f \\in R$ for every $x \\in U$.\n\n## Remarks\n\nThe condition $j^k_x f \\in R$ for all $x$ is a system of partial differential equations of order $k$ for $f$, and $\\Gamma$ is its set of local invertible solutions.\nDefinitions in the literature differ in details, for example in requiring the equations to be analytic or involutive.\n\nExample: let $g$ be a Riemannian metric on $M$, and let $R \\subseteq J^1_\\times(M)$ be the set of $1$-jets whose derivative $T_x M \\to T_y M$ is a linear isometry from $g_x$ to $g_y$.\nIt is a closed subgroupoid and a submanifold, and a local diffeomorphism has all its $1$-jets in $R$ exactly when it is a local isometry.\nSo local isometries form a Lie pseudogroup of order $1$.\nIn the same way, area-preserving maps of $\\R^2$ are the local diffeomorphisms whose $1$-jets satisfy $\\det Df = 1$.\n\nNon-example: the pseudogroup of all local homeomorphisms of $\\R^n$ is not a Lie pseudogroup.\nIt contains maps that are not smooth, so it is not even a pseudogroup of local diffeomorphisms.\n\n## Explanations\n\n### L1 human\n\nA Lie pseudogroup is the set of all local invertible solutions of a system of differential equations.\nExamples are the maps that preserve a metric or an area form.\n",
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