Inmathematics,conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

In a real two dimensional space, conformal geometry is precisely the geometry ofRiemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study ofconformal transformations of what are called "flat spaces" (such asEuclidean spaces orspheres), or to the study ofconformal manifolds which areRiemannian orpseudo-Riemannian manifolds with a class ofmetrics that are defined up to scale. Study of the flat structures is sometimes termedMöbius geometry, and is a type ofKlein geometry.

Aconformal manifold is aRiemannian manifold (orpseudo-Riemannian manifold) equipped with anequivalence class ofmetric tensors, in which two metricsg andh are equivalent if and only if

${\displaystyle h={\lambda }^{2}g,}$

whereλ is a real-valuedsmooth function defined on the manifold and is called theconformal factor. An equivalence class of such metrics is known as aconformal metric orconformal class. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.

A conformal metric isconformally flat if there is a metric representing it that is flat, in the usual sense that theRiemann curvature tensor vanishes. It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point. When it is necessary to distinguish these cases, the latter is calledlocally conformally flat, although often in the literature no distinction is maintained. Then-sphere is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is (globally) conformally flat in this sense. A locally conformally flat manifold is locally conformal to aMöbius geometry, meaning that there exists an angle preservinglocal diffeomorphism from the manifold into a Möbius geometry. In two dimensions, every conformal metric is locally conformally flat. In dimensionn > 3 a conformal metric is locally conformally flat if and only if itsWeyl tensor vanishes; in dimensionn = 3, if and only if theCotton tensor vanishes.

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 atangent vector cannot be defined, but the angle between two vectors still can. Another feature is that there is noLevi-Civita connection because ifg andλ²g are two representatives of the conformal structure, then theChristoffel symbols ofg andλ²g would not agree. Those associated withλ²g would involve derivatives of the function λ whereas those associated withg would not.

Despite these differences, conformal geometry is still tractable. The Levi-Civita connection andcurvature tensor, although only being defined once a particular representative of the conformal structure has been singled out, do satisfy certain transformation laws involving theλ and its derivatives when a different representative is chosen. In particular, (in dimension higher than 3) theWeyl tensor turns out not to depend onλ, and so it is aconformal invariant. Moreover, even though there is no Levi-Civita connection on a conformal manifold, one can instead work with aconformal connection, which can be handled either as a type ofCartan connection modelled on the associated Möbius geometry, or as aWeyl connection. This allows one to defineconformal curvature and other invariants of the conformal structure.

Möbius geometry is the study of "Euclidean space with a point added at infinity", or a "Minkowski (or pseudo-Euclidean) space with anull cone added at infinity". That is, the setting is acompactification of a familiar space; thegeometry is concerned with the implications of preserving angles.

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-dimensionalMinkowski plane exhibits extensive conformalsymmetry. 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.

Theconformal group for the Minkowski quadratic formq(x,y) = 2xy in the plane is theabelianLie group

withLie algebracso(1, 1) consisting of all real diagonal2 × 2 matrices.

Consider now the Minkowski plane,${\displaystyle {\mathbb{R}}^2}$ equipped with the metric

${\displaystyle g=2dxdy.}$

A 1-parameter group of conformal transformations gives rise to a vector fieldX with the property that theLie derivative ofg alongX is proportional tog. Symbolically,

L_Xg =λg   for someλ.

In particular, using the above description of the Lie algebracso(1, 1), this implies that

1. L_X dx =a(x)dx
2. L_X dy =b(y)dy

for some real-valued functionsa andb depending, respectively, onx andy.

Conversely, given any such pair of real-valued functions, there exists a vector fieldX satisfying 1. and 2. Hence theLie algebra of infinitesimal symmetries of the conformal structure, theWitt algebra, isinfinite-dimensional.

The conformal compactification of the Minkowski plane is a Cartesian product of two circlesS¹ ×S¹. On theuniversal cover, there is no obstruction to integrating the infinitesimal symmetries, and so the group of conformal transformations is the infinite-dimensional Lie group

${\displaystyle (\mathbb{Z}⋊\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^1))\times (\mathbb{Z}⋊\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^1)),}$

where Diff(S¹) is thediffeomorphism group of the circle.^[1]

The conformal groupCSO(1, 1) and its Lie algebra are of current interest intwo-dimensional conformal field theory.

The group of conformal symmetries of the quadratic form

${\displaystyle q(z,\overline{z})=z\overline{z}}$

is the groupGL₁(C) =C^×, themultiplicative group of the complex numbers. Its Lie algebra isgl₁(C) =C.

Consider the (Euclidean)complex plane equipped with the metric

${\displaystyle g=dzd\overline{z}.}$

The infinitesimal conformal symmetries satisfy

1. ${\displaystyle {\mathbf{L}}_{X}dz=f(z)dz}$
2. ${\displaystyle {\mathbf{L}}_{X}d\overline{z}=f(\overline{z})d\overline{z},}$

wheref satisfies theCauchy–Riemann equation, and so isholomorphic over its domain. (SeeWitt algebra.)

The conformal isometries of a domain therefore consist of holomorphic self-maps. In particular, on the conformal compactification – theRiemann sphere – the conformal transformations are given by theMöbius transformations

${\displaystyle z\mapsto \frac{az+b}{cz+d}}$

wheread −bc is nonzero.

In two dimensions, the group of conformal automorphisms of a space can be quite large (as in the case of Lorentzian signature) or variable (as with the case of Euclidean signature). The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained. In Lorentzian signature, the freedom is in a pair of real valued functions. In Euclidean, the freedom is in a single holomorphic function.

In the case of higher dimensions, the asymptotic developments of infinitesimal symmetries are at most quadratic polynomials.^[2] In particular, they form a finite-dimensionalLie algebra. The pointwise infinitesimal conformal symmetries of a manifold can be integrated precisely when the manifold is a certain modelconformally flat space (up to taking universal covers and discrete group quotients).^[3]

The general theory of conformal geometry is similar, although with some differences, in the cases of Euclidean and pseudo-Euclidean signature.^[4] In either case, there are a number of ways of introducing the model space of conformally flat geometry. Unless otherwise clear from the context, this article treats the case of Euclidean conformal geometry with the understanding that it also applies,mutatis mutandis, to the pseudo-Euclidean situation.

The inversive model of conformal geometry consists of the group of local transformations on theEuclidean spaceEⁿ generated by inversion in spheres. ByLiouville's theorem, any angle-preserving local (conformal) transformation is of this form.^[5] From this perspective, the transformation properties of flat conformal space are those ofinversive geometry.

The projective model identifies the conformal sphere with a certainquadric in aprojective space. Letq denote the Lorentzianquadratic form onRⁿ⁺² defined by

${\displaystyle q(x_0,x_1,\dots ,x_{n+1})=-2x_0{x}_{n+1}+x_1^2+x_2^2+\cdots +x_n^2.}$

In the projective spaceP(Rⁿ⁺²), letS be the locus ofq = 0. ThenS is the projective (or Möbius) model of conformal geometry. A conformal transformation onS is aprojective linear transformation ofP(Rⁿ⁺²) that leaves the quadric invariant.

In a related construction, the quadricS is thought of as thecelestial sphere at infinity of thenull cone in the Minkowski spaceR^n+1,1, which is equipped with the quadratic formq as above. The null cone is defined by

${\displaystyle N=\left\{(x_0,\dots ,x_{n+1})\mid -2x_0{x}_{n+1}+x_1^2+\cdots +x_n^2=0\right\}.}$

This is the affine cone over the projective quadricS. LetN⁺ be the future part of the null cone (with the origin deleted). Then the tautological projectionR^n+1,1 \ {0} →P(Rⁿ⁺²) restricts to a projectionN⁺ →S. This givesN⁺ the structure of aline bundle overS. Conformal transformations onS are induced by theorthochronous Lorentz transformations ofR^n+1,1, since these are homogeneous linear transformations preserving the future null cone.

Intuitively, the conformally flat geometry of a sphere is less rigid than theRiemannian geometry of a sphere. Conformal symmetries of a sphere are generated by the inversion in all of itshyperspheres. On the other hand, Riemannianisometries of a sphere are generated by inversions ingeodesic hyperspheres (see theCartan–Dieudonné theorem.) The Euclidean sphere can be mapped to the conformal sphere in a canonical manner, but not vice versa.

The Euclidean unit sphere is the locus inRⁿ⁺¹

${\displaystyle z^2+x_1^2+x_2^2+\cdots +x_n^2=1.}$

This can be mapped to the Minkowski spaceR^n+1,1 by letting

${\displaystyle x_0=\frac{z+1}{\sqrt{2}},x_1=x_1,\dots ,x_n=x_n,x_{n+1}=\frac{z-1}{\sqrt{2}}.}$

It is readily seen that the image of the sphere under this transformation is null in the Minkowski space, and so it lies on the coneN⁺. Consequently, it determines a cross-section of the line bundleN⁺ →S.

Nevertheless, there was an arbitrary choice. Ifκ(x) is any positive function ofx = (z,x₀, ...,x_n), then the assignment

${\displaystyle x_0=\frac{z+1}{\kappa (x)\sqrt{2}},x_1=x_1,\dots ,x_n=x_n,x_{n+1}=\frac{(z-1)\kappa (x)}{\sqrt{2}}}$

also gives a mapping intoN⁺. The functionκ is an arbitrary choice ofconformal scale.

A representativeRiemannian metric on the sphere is a metric which is proportional to the standard sphere metric. This gives a realization of the sphere as aconformal manifold. The standard sphere metric is the restriction of the Euclidean metric onRⁿ⁺¹

${\displaystyle g=d{z}^2+d{x}_1^2+d{x}_2^2+\cdots +d{x}_n^2}$

to the sphere

${\displaystyle z^2+x_1^2+x_2^2+\cdots +x_n^2=1.}$

A conformal representative ofg is a metric of the formλ²g, whereλ is a positive function on the sphere. The conformal class ofg, denoted [g], is the collection of all such representatives:

${\displaystyle [g]=\left\{{\lambda }^{2}g\mid \lambda >0\right\}.}$

An embedding of the Euclidean sphere intoN⁺, as in the previous section, determines a conformal scale onS. Conversely, any conformal scale onS is given by such an embedding. Thus the line bundleN⁺ →S is identified with the bundle of conformal scales onS: to give a section of this bundle is tantamount to specifying a metric in the conformal class [g].

Another way to realize the representative metrics is through a specialcoordinate system onR^{n+1, 1}. Suppose that the Euclideann-sphereS carries astereographic coordinate system. This consists of the following map ofRⁿ →S ⊂Rⁿ⁺¹:

${\displaystyle \mathbf{y}\in {\mathbf{R}}^n\mapsto \left(\frac{2\mathbf{y}}{{\left|\mathbf{y}\right|}^2+1},\frac{{\left|\mathbf{y}\right|}^2-1}{{\left|\mathbf{y}\right|}^2+1}\right)\in S\subset {\mathbf{R}}^{n+1}.}$

In terms of these stereographic coordinates, it is possible to give a coordinate system on the null coneN⁺ in Minkowski space. Using the embedding given above, the representative metric section of the null cone is

${\displaystyle x_0=\sqrt{2}\frac{{\left|\mathbf{y}\right|}^2}{1+{\left|\mathbf{y}\right|}^2},x_i=\frac{y_i}{{\left|\mathbf{y}\right|}^2+1},x_{n+1}=\sqrt{2}\frac{1}{{\left|\mathbf{y}\right|}^2+1}.}$

Introduce a new variablet corresponding to dilations upN⁺, so that the null cone is coordinatized by

${\displaystyle x_0=t\sqrt{2}\frac{{\left|\mathbf{y}\right|}^2}{1+{\left|\mathbf{y}\right|}^2},x_i=t\frac{y_i}{{\left|\mathbf{y}\right|}^2+1},x_{n+1}=t\sqrt{2}\frac{1}{{\left|\mathbf{y}\right|}^2+1}.}$

Finally, letρ be the following defining function ofN⁺:

${\displaystyle \rho =\frac{-2x_0{x}_{n+1}+x_1^2+x_2^2+\cdots +x_n^2}{t^2}.}$

In thet,ρ,y coordinates onR^n+1,1, the Minkowski metric takes the form:

${\displaystyle t^2{g}_{ij}(y)d{y}^{i}d{y}^j+2\rho d{t}^2+2tdtd\rho ,}$

whereg_ij is the metric on the sphere.

In these terms, a section of the bundleN⁺ consists of a specification of the value of the variablet =t(yⁱ) as a function of theyⁱ along the null coneρ = 0. This yields the following representative of the conformal metric onS:

${\displaystyle t(y{)}^2{g}_{ij}d{y}^{i}d{y}^j.}$

Consider first the case of the flat conformal geometry in Euclidean signature. Then-dimensional model is thecelestial sphere of the(n + 2)-dimensional Lorentzian spaceR^n+1,1. Here the model is aKlein geometry: ahomogeneous spaceG/H whereG = SO(n + 1, 1) acting on the(n + 2)-dimensional Lorentzian spaceR^n+1,1 andH is theisotropy group of a fixed null ray in thelight cone. Thus the conformally flat models are the spaces ofinversive geometry. For pseudo-Euclidean ofmetric signature(p,q), the model flat geometry is defined analogously as the homogeneous spaceO(p + 1,q + 1)/H, whereH is again taken as the stabilizer of a null line. Note that both the Euclidean and pseudo-Euclidean model spaces arecompact.

To describe the groups and algebras involved in the flat model space, fix the following form onR^p+1,q+1:

whereJ is a quadratic form of signature(p,q). ThenG = O(p + 1,q + 1) consists of(n + 2) × (n + 2) matrices stabilizingQ :^tMQM =Q (the superscriptt means transpose). The Lie algebra admits aCartan decomposition

${\displaystyle \mathbf{g}={\mathbf{g}}_{-1}\oplus {\mathbf{g}}_0\oplus {\mathbf{g}}_1}$

where

Alternatively, this decomposition agrees with a natural Lie algebra structure defined onRⁿ ⊕cso(p,q) ⊕ (Rⁿ)^∗.

The stabilizer of the null ray pointing up the last coordinate vector is given by theBorel subalgebra

h =g₀ ⊕g₁.

• Conformal geometric algebra
• Conformal gravity
• Conformal Killing equation
• Erlangen program
• Möbius plane

• Kobayashi, Shoshichi (1970).Transformation Groups in Differential Geometry (First ed.). Springer.ISBN 3-540-05848-6.
• Slovák, Jan (1993).Invariant Operators on Conformal Manifolds. Research Lecture Notes, University of Vienna (Dissertation).
• Sternberg, Shlomo (1983).Lectures on differential geometry. New York: Chelsea.ISBN 0-8284-0316-3.

Wikimedia Commons has media related toConformal geometry.

• G.V. Bushmanova (2001) [1994],"Conformal geometry",Encyclopedia of Mathematics,EMS Press
• http://www.euclideanspace.com/maths/geometry/space/nonEuclid/conformal/index.htm

• Conformal geometry
• Differential geometry

• Articles with short description
• Short description is different from Wikidata
• Commons category link is on Wikidata