Conformal geometry Totally Explained

Everything about Conformal Geometry totally explained

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Riemannian manifold or pseudo-Riemannian manifold. In particular conformal geometry in two (real) dimensions is the geometry of Riemann surfaces.

Conformally flat geometry

Conformally flat geometry is the study of "Euclidean space with a point added at infinity", or a "Minkowski (or pseudo-Euclidean) 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.
At an abstract level, the Euclidean and pseudo-Euclidean spaces can be handled in much the same way, except in the case of dimension two. The compactified two dimensional Minkowski plane exhibits extensive conformal symmetry. Formally, its group of conformal transformations is infinite dimensional. By contrast, the group of conformal transformations of the compactified Euclidean plane is only 6 dimensional.

Two dimensions

Minkowski space

The conformal group for the Minkowski quadratic form q(x,y) = 2xy in the plane is the abelian Lie group: ť

CSO(1,1)=left
.

Alternatively, this decomposition agrees with a natural Lie algebra structure defined on Rⁿ ⊕ cso(p,q) ⊕ (Rⁿ)^*.
The stabilizer of the null ray pointing up the last coordinate vector is given by the Borel subalgebra ť h = g₀ ⊕ g₁.

Weyl geometries and the Killing form

Conformally curved geometry

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. However, unlike in (pseudo-)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 ť

gmapsto lambda g

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 can't be defined, but the angle between two vectors still can. Another feature is that there's no Levi-Civita connection because if g and λg are two representatives of the conformal structure, then the Christoffel symbols of g and λg wouldn't 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's a conformal invariant. Moreover, even though there's no Levi-Civita connection on a conformal manifold, there's 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.

Further Information

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