Conformal geometry

Summary

Conformal geometry studies spaces in which angles can be measured but lengths cannot. There, a Riemannian metric is known only up to multiplication by a positive function. Its maps are the conformal maps, those that preserve angles. In dimension $2$ there are very many conformal maps, such as all holomorphic functions with nonzero derivative. In dimension at least $3$, Liouville's theorem reduces them to the Möbius transformations. The conformal geometry of the sphere is Möbius geometry, the subject of inversive geometry.