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  "body": "\n## Summary\n\nLet $M$ be a compact manifold and $N$ a manifold.\nThe set $C^\\infty(M, N)$ of all smooth maps is itself an infinite-dimensional manifold, modelled on [[Fréchet spaces|3GAA614673]] of smooth sections.\nThe charts come from the exponential map of a Riemannian metric on $N$.\nA map near $f$ is written as $x \\mapsto \\exp_{f(x)}(s(x))$ for a small vector field $s$ along $f$.\nLoop spaces, [[diffeomorphism groups|WTCNKST134]] and spaces of embeddings are open subsets or submanifolds of such manifolds.\nMaps of class $C^k$, or of Sobolev class $H^s$ with $s \u003e \\dim M / 2$, form Banach or [[Hilbert manifolds|R8DMMW0E9M]] of maps, which are better suited to analysis.\n\n## Explanations\n\n### L1 human\n\nA map from a circle to a sphere is a closed curve on the sphere.\nAll such curves together form a huge space, with one point for each curve.\nNear a given curve, any nearby curve is obtained by pushing each point a little along a tangent arrow.\nSo nearby curves correspond to small fields of tangent arrows along the given curve.\n\nThese fields form a vector space, and this gives coordinates on the space of curves, making it a manifold of infinite dimension.\n",
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