Conformal geometry
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In mathematics, 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, the setting is a compactification of a familiar space; the geometry is concerned with the implications of preserving angles.
In higher dimensions this geometry is quite rigid; it is the low dimensions that exhibit extensive symmetry.
For the Euclidean space case, the two-dimensional conformal geometry is that of the Riemann sphere.
The elements of the Riemann sphere are described by a complex number z, which can be infinite and the conformal transformations are given by the Möbius transformations
- <math>z \rightarrow \frac{az+b}{cz+d}<math>
where a d − b c is nonzero.
The n-dimensional conformal geometry with reflections (also known as inversions) is simply the n-dimensional inversive geometry.
For the other, Minkowski space, case, in two dimensions, it is simply
- <math>(\mathbb{Z}<math>
<math>\bold{Diff}(S^1))\times(\mathbb{Z}<math><math>\bold{Diff}(S^1)) <math>
(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 Minkowski-like conformal geometries. For three or more dimensions, its automorphism group is SO(n,2).