Conformal geometry

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In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Riemannian manifold or pseudo-Riemannian manifold.


Conformally flat geometry

Conformally flat geometry is the study of "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. The Euclidean case is also known as Möbius geometry.

In higher dimensions this geometry is quite rigid; it is the low dimensions that exhibit extensive symmetry. Technically speaking, the conformal algebra - the Lie algebra that generates all infinitesimal angle-preserving transformations - is infinite-dimensional in two dimensions (however, the conformal group itself is only PSL(2,C) in a Riemann surface of genus 0; most of the elements of the conformal algebra don't generate finite conformal transformations because of singularities) but finite-dimensional in higher dimensions. In fact, it is isomorphic to SO(d+1,1) in d dimensions for Euclidean space and SO(d,2) in d dimensions for Minkowski space.

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

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where a db c is nonzero.

The n-dimensional conformal geometry with reflections (also known as inversions) is the n-dimensional inversive geometry.

For the other, Minkowski space, case, in two dimensions, it is

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(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).

Conformally curved geometry

Conformally curved geometry (referred to its practitioners simply as conformal geometry) is the study of a Riemannian manifold or pseudo-Riemannian manifold M with metric g. However, unlike in (psuedo-)Riemannian geometry, the metric is only defined up to scale at each point. In other words, the metric is only defined up to changes of the form

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where λ>0 is a smooth positive function. So a conformal structure consists of the equivalence class of all positive multiples of the metric.

Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can. Another feature is that there is no Levi-Civita connection because if g and λg are two representatives of the conformal structure, then the Christoffel symbols of g and λg would not agree. Those associated with λg would involve derivatives of the function λ whereas those associated with g would not.

Despite these differences, conformal geometry is still tractable. The Levi-Civita connection and curvature tensor, although only being defined once a particular representative of the conformal structure has been singled out, do satisfy certain transformation laws involving the λ and its derivatives when a different representative is chosen. In particular, (in dimension higher than 3) the Weyl tensor turns out not to depend on λ, and so it is a conformal invariant. Moreover, even though there is no Levi-Civita connection on a conformal manifold, there is a Cartan connection on a higher-order frame bundle. This allows one to define conformal curvature, as well as other invariants of the conformal structure.

See also