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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n \\ge 2$.\nTwo [[Riemannian metrics|Q61VFHCH2J]] $g$ and $g'$ on $M$ are *conformally equivalent* when $g' = e^{2f} g$ for some smooth function $f \\colon M \\to \\R$.\nA *conformal structure* on $M$ is an equivalence class $[g]$ for this relation, and $(M, [g])$ is a *conformal manifold*.\nA [[diffeomorphism|0KZZ70ZHXP]] $\\varphi \\colon (M, [g]) \\to (N, [h])$ is *conformal* when $\\varphi^* h \\in [g]$.\n*Conformal geometry* is the study of the properties of conformal manifolds that are invariant under conformal diffeomorphisms.\n\n## Remarks\n\nThe angle between two tangent vectors at a point is the same for all metrics in $[g]$, so angles are defined in a conformal manifold, while lengths are not.\nAccordingly, conformal diffeomorphisms are exactly the diffeomorphisms that are [[conformal maps|0YYT36R4YD]], that is, that preserve angles.\nExample: stereographic projection from the north pole identifies $S^n$ minus a point with $\\R^n$, and the round metric becomes $\\dfrac{4}{(1 + \\abs{x}^2)^2} \\abs{dx}^2$.\nThis is $e^{2f}$ times the flat metric with $f(x) = \\ln 2 - \\ln(1 + \\abs{x}^2)$, so the round sphere minus a point is conformally equivalent to Euclidean space.\nNon-example: the map $(x, y) \\mapsto (2x, y)$ of $\\R^2$ pulls back $dx^2 + dy^2$ to $4\\, dx^2 + dy^2$, which is not a multiple of $dx^2 + dy^2$; it does not preserve the angle between the vectors $(1, 1)$ and $(1, -1)$.\nOn an oriented surface, a conformal structure is the same thing as a structure of Riemann surface.\n",
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