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  "body": "\n## Definition\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary and $N$ a smooth manifold, and let $C^\\infty(M, N)$ be the set of smooth maps $M \\to N$.\nFix a [[Riemannian metric|Q61VFHCH2J]] on $N$, and let $\\pi \\colon TN \\to N$ be the [[tangent bundle|DQB1BVQJWD]].\nFix an open neighbourhood $V$ of the zero section in $TN$ on which the [[exponential map|Y7Z0AGY1A7]] is defined.\nChoose $V$ so that $(\\pi, \\exp) \\colon V \\to N \\times N$ is a diffeomorphism onto an open neighbourhood $W$ of the diagonal.\nFor $f \\in C^\\infty(M, N)$, let $\\Gamma(f^* TN)$ be the space of smooth [[sections|WWX7H5W1RP]] of the pullback bundle.\nThese are the smooth maps $s \\colon M \\to TN$ with $s(x) \\in T_{f(x)} N$ for all $x$.\nWith the topology of uniform convergence of all derivatives, $\\Gamma(f^* TN)$ is a Fréchet space.\nLet $U_f = \\set{g \\in C^\\infty(M, N) : (f(x), g(x)) \\in W \\text{ for all } x \\in M}$ and define $\\varphi_f \\colon U_f \\to \\Gamma(f^* TN)$ by\n$$\\varphi_f(g)(x) = (\\pi, \\exp)^{-1}(f(x), g(x)).$$\nThe *manifold of smooth mappings* is $C^\\infty(M, N)$ with the smooth [[Fréchet manifold|EBMCFQT0JN]] structure given by the atlas $(U_f, \\varphi_f)_{f \\in C^\\infty(M, N)}$.\n\n## Remarks\n\nEach $\\varphi_f$ is a bijection onto an open subset of $\\Gamma(f^* TN)$, with inverse $s \\mapsto \\exp \\circ s$, and the chart changes are smooth.\nThe resulting manifold structure does not depend on the choice of the metric, its topology is the $C^\\infty$ topology, and the tangent space at $f$ is $\\Gamma(f^* TN)$.\n\nExample: for $M = N = S^1 = \\R / 2\\pi\\Z$ with the flat metric, take $V$ to be the vectors of length less than $\\pi$.\nFor the identity map $f$, the chart sends $g$ near $f$ to the function $s$ with $g(x) = x + s(x)$ and $\\abs{s(x)} \u003c \\pi$.\nSo near the identity, the diffeomorphisms of the circle correspond to small smooth functions on the circle.\n\nNon-example: in the same setting, let $f$ be the constant map at a point $y$, and $g$ the constant map at the antipodal point $y + \\pi$.\nThen $(f(x), g(x)) = (y, y + \\pi)$ is a pair at distance $\\pi$, so it lies outside $W$, so $g$ is not in the chart domain $U_f$; it lies in a different chart.\n\n\nWhen $\\dim M \\ge 1$ and $\\dim N \\ge 1$, the model spaces are Fréchet spaces that are not normable, so this atlas is not an atlas of a Banach manifold.\n",
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