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Conformal geometry

Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a "Euclidean-like space with a point added at infinity" or a "Minkowski-like space with a couple of points added at infinity". That is, it works with compactifications of familiar spaces. In higher dimensions this geometry is quite rigid; it is the low dimensions that exhibit extensive symmetry

For the former case (Euclidean space), 2-dimensional conformal geometry is simply the Riemann sphere. For three or more dimensions, this is simply inversive geometry (with or without reflections (i.e. inversions!)).

For the latter (Minkowski space) case, in two dimensions, it is simply (taking the universal cover of the compactification), if the space is assumed to be oriented (see Virasoro algebra). This is the default assumption in conformal field theory, the primary field which studies Minkowskilike conformal geometries. For three or more dimensions, its automorphism group is just the SO(n,2).

See also Erlanger program.