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  "body": "\n## Statement\n\nLet $n \\ge 3$, let $U \\subseteq \\R^n$ be a connected open set, and let $f \\colon U \\to \\R^n$ be a smooth [[conformal map|0YYT36R4YD]].\nThis means that at each $x \\in U$ the derivative is $Df(x) = \\lambda(x) A(x)$ with $\\lambda(x) \u003e 0$ and $A(x)$ an orthogonal matrix.\nThen $f$ is the restriction to $U$ of a Möbius transformation of $\\R^n \\cup \\set{\\infty}$.\nThat is, $f$ is a composition of translations, orthogonal linear maps, dilations $x \\mapsto c x$ with $c \u003e 0$, and at most one [[inversion|YXST7NGMTC]] $x \\mapsto x_0 + r^2 (x - x_0) / \\abs{x - x_0}^2$ in a sphere.\n\n## Remarks\n\nProof idea: the condition $Df^T Df = \\lambda^2 I$ is a system of partial differential equations for $f$.\nDifferentiating it and using $n \\ge 3$ forces the function $1/\\lambda$ to be either constant or of the form $a \\abs{x - x_0}^2$ with $a \u003e 0$.\nIn the first case $f$ is a similarity; in the second, composing with an inversion centred at $x_0$ reduces to the first case.\nThe hypothesis $n \\ge 3$ is necessary: in $\\R^2 = \\C$, every holomorphic function with nonzero derivative is conformal, for example $z \\mapsto e^z$, and it is not a Möbius transformation since it is not injective.\nThe smoothness hypothesis can be weakened a great deal.\nBy results of Gehring and of Reshetnyak, a map of Sobolev class $W^{1,n}_{\\mathrm{loc}}$ whose derivative is conformal almost everywhere is constant or a restricted Möbius transformation.\n\n## Explanations\n\n### L1 human\n\nIn the plane there are very many maps that keep angles, such as all the maps used by complex analysis.\nIn space of three or more dimensions, Liouville's theorem says there are very few.\nEvery angle-preserving map of a region of space is built from moves, rotations, scalings and reflections in spheres.\nSo in three dimensions there is no freedom to make conformal maps of arbitrary shapes, unlike in the plane.\n\n### L2 human\n\nThe theorem explains why conformal geometry in dimension $n \\ge 3$ is rigid: the conformal maps form the finite-dimensional group of Möbius transformations, of dimension $(n+1)(n+2)/2$.\nIn dimension $2$, by contrast, the Riemann mapping theorem gives a conformal map between any two simply connected proper open subsets of the plane.\nThe result is the local form of the fact that the conformal group of the round sphere $S^n$, $n \\ge 3$, is the Möbius group.\nIt is a starting point for the theory of quasiconformal maps in higher dimensions.\n",
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