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Encyclopedia > Conformal geometry

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, two-dimensional conformal geometry is that of the Riemann sphere. n-dimensional conformal geometry with reflections (also known as inversions) is simply n-dimensional inversive geometry.

For the other, 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 Minkowski-like conformal geometries. For three or more dimensions, its automorphism group is SO(n,2).

Results from FactBites:

 Conformal map - Wikipedia, the free encyclopedia (821 words) The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation. The importance of conformal transformations for electromagnetism was brought to light by Harry Bateman in 1910. Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable but that exhibit inconvenient geometries.
 Conformal geometry - Wikipedia, the free encyclopedia (557 words) In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Riemannian manifold or pseudo-Riemannian manifold. For the Euclidean space case, the two-dimensional conformal geometry is that of the Riemann sphere. Conformally curved geometry (referred to by its practitioners simply as conformal geometry) is the study of a Riemannian manifold or pseudo-Riemannian manifold M with metric g.
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