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  "body": "\n## Summary\n\nConformal geometry studies spaces in which angles can be measured but lengths cannot.\nThere, a Riemannian metric is known only up to multiplication by a positive function.\nIts maps are the [[conformal maps|0YYT36R4YD]], those that preserve angles.\nIn dimension $2$ there are very many conformal maps, such as all holomorphic functions with nonzero derivative.\nIn dimension at least $3$, [[Liouville's theorem|NH7J4YQMQC]] reduces them to the Möbius transformations.\nThe conformal geometry of the sphere is Möbius geometry, the subject of [[inversive geometry|HFBW3SXQVM]].\n\n## Explanations\n\n### L1 human\n\nImagine a map of the world that may stretch some regions more than others, but always keeps the angles between roads correct.\nA conformal geometry remembers exactly this: angles, not distances.\nThe Mercator map is an example: it enlarges Greenland a lot, yet every angle between two streets is drawn true.\nTwo shapes count as the same in conformal geometry when one can be bent onto the other by such an angle-keeping map.\n",
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