Editing Conformal geometry (section)

===Higher dimensions===
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.<ref>Kobayashi (1972).</ref>  In particular, they form a finite-dimensional [[Lie algebra]].  The pointwise infinitesimal conformal symmetries of a manifold can be integrated precisely when the manifold is a certain model ''conformally flat'' space ([[up to]] taking universal covers and discrete group quotients).<ref>Due to a general theorem of Sternberg (1962).</ref>

The general theory of conformal geometry is similar, although with some differences, in the cases of Euclidean and pseudo-Euclidean signature.<ref>Slovak (1993).</ref>  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====
The inversive model of conformal geometry consists of the group of local transformations on the [[Euclidean space]] '''E'''<sup>''n''</sup> generated by inversion in spheres.  By [[Liouville's theorem (conformal mappings)|Liouville's theorem]], any angle-preserving local (conformal) transformation is of this form.<ref>{{springer|id=L/l059680|title=Liouville theorems|author=S.A. Stepanov}}.  {{cite book|chapter=''Extension au case des trois dimensions de la question du tracé géographique, Note VI'' (by J. Liouville)|pages=609–615|author=G. Monge|title=Application de l'Analyse à la géometrie|url=https://archive.org/details/applicationdela00monggoog|publisher=Bachelier, Paris|year=1850}}.</ref>  From this perspective, the transformation properties of flat conformal space are those of [[inversive geometry]].

====The projective model====
The projective model identifies the conformal sphere with a certain [[quadric]] in a [[projective space]].  Let ''q'' denote the Lorentzian [[quadratic form]] on '''R'''<sup>''n''+2</sup> defined by

:<math>q(x_0,x_1,\ldots,x_{n+1}) = -2x_0x_{n+1}+x_1^2+x_2^2+\cdots+x_n^2.</math>

In the projective space '''P'''('''R'''<sup>''n''+2</sup>), let ''S'' be the locus of {{nowrap|1=''q'' = 0}}.  Then ''S'' is the projective (or Möbius) model of conformal geometry.  A conformal transformation on ''S'' is a [[projective linear group|projective linear transformation]] of '''P'''('''R'''<sup>''n''+2</sup>) that leaves the quadric invariant.

In a related construction, the quadric ''S'' is thought of as the [[celestial sphere]] at infinity of the [[null cone]] in the Minkowski space {{nowrap|'''R'''<sup>''n''+1,1</sup>}}, which is equipped with the quadratic form ''q'' as above.  The null cone is defined by

:<math> N = \left\{ ( x_0 , \ldots , x_{n+1} ) \mid -2 x_0 x_{n+1} + x_1^2 + \cdots + x_n^2 = 0 \right\} .</math>

This is the affine cone over the projective quadric ''S''.  Let ''N''<sup>+</sup> be the future part of the null cone (with the origin deleted).  Then the tautological projection {{nowrap|'''R'''<sup>''n''+1,1</sup> \ {0} → '''P'''('''R'''<sup>''n''+2</sup>)}} restricts to a projection {{nowrap|''N''<sup>+</sup> → ''S''}}.  This gives ''N''<sup>+</sup> the structure of a [[line bundle]] over ''S''.  Conformal transformations on ''S'' are induced by the [[Lorentz transformation|orthochronous Lorentz transformation]]s of {{nowrap|'''R'''<sup>''n''+1,1</sup>}}, since these are homogeneous linear transformations preserving the future null cone.

====The Euclidean sphere====
Intuitively, the conformally flat geometry of a sphere is less rigid than the [[Riemannian geometry]] of a sphere.  Conformal symmetries of a sphere are generated by the inversion in all of its [[hypersphere]]s.  On the other hand, Riemannian [[isometry|isometries]] of a sphere are generated by inversions in ''[[geodesic]]'' hyperspheres (see the [[Cartan–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 in '''R'''<sup>''n''+1</sup>

:<math>z^2+x_1^2+x_2^2+\cdots+x_n^2=1.</math>

This can be mapped to the Minkowski space {{nowrap|'''R'''<sup>''n''+1,1</sup>}} by letting

:<math>x_0 = \frac{z+1}{\sqrt{2}},\, x_1=x_1,\, \ldots,\, x_n=x_n,\, x_{n+1}=\frac{z-1}{\sqrt{2}}.</math>

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 cone ''N''<sup>+</sup>.  Consequently, it determines a cross-section of the line bundle {{nowrap|''N''<sup>+</sup> → ''S''}}.

Nevertheless, there was an arbitrary choice.  If ''κ''(''x'') is any positive function of {{nowrap|1=''x'' = (''z'', ''x''<sub>0</sub>, ..., ''x''<sub>''n''</sub>)}}, then the assignment

:<math>x_0 = \frac{z+1}{\kappa(x)\sqrt{2}}, \, x_1=x_1,\, \ldots,\, x_n=x_n,\, x_{n+1}=\frac{(z-1)\kappa(x)}{\sqrt{2}}</math>

also gives a mapping into ''N''<sup>+</sup>.  The function ''κ'' is an arbitrary choice of ''conformal scale''.

====Representative metrics====
A representative [[Riemannian metric]] on the sphere is a metric which is proportional to the standard sphere metric.  This gives a realization of the sphere as a [[conformal geometry#Conformal manifolds|conformal manifold]].  The standard sphere metric is the restriction of the Euclidean metric on '''R'''<sup>''n''+1</sup>

:<math>g=dz^2+dx_1^2+dx_2^2+\cdots+dx_n^2</math>

to the sphere

:<math>z^2+x_1^2+x_2^2+\cdots+x_n^2=1.</math>

A conformal representative of ''g'' is a metric of the form ''λ''<sup>2</sup>''g'', where ''λ'' is a positive function on the sphere.  The conformal class of ''g'', denoted [''g''], is the collection of all such representatives:

:<math> [ g ] = \left\{ \lambda ^2 g \mid \lambda > 0 \right\} .</math>

An embedding of the Euclidean sphere into ''N''<sup>+</sup>, as in the previous section, determines a conformal scale on ''S''.  Conversely, any conformal scale on ''S'' is given by such an embedding.  Thus the line bundle {{nowrap|''N''<sup>+</sup> → ''S''}} is identified with the bundle of conformal scales on ''S'': to give a section of this bundle is tantamount to specifying a metric in the conformal class [''g''].

====Ambient metric model====
{{see also|Ambient construction}}
Another way to realize the representative metrics is through a special [[coordinate system]] on {{nowrap|'''R'''<sup>''n''+1, 1</sup>}}.  Suppose that the Euclidean ''n''-sphere ''S'' carries a [[stereographic projection|stereographic coordinate system]].  This consists of the following map of {{nowrap|'''R'''<sup>''n''</sup> → ''S'' ⊂ '''R'''<sup>''n''+1</sup>}}:

:<math> \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 \sub \mathbf{R} ^{n+1} .</math>

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

:<math> 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 } .</math>

Introduce a new variable ''t'' corresponding to dilations up ''N''<sup>+</sup>, so that the null cone is coordinatized by

:<math>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 } .</math>

Finally, let ''ρ'' be the following defining function of ''N''<sup>+</sup>:

:<math> \rho = \frac{ - 2 x _0 x _{n+1} + x _1^2 + x _2^2 + \cdots + x _n^2 }{ t ^2 } .</math>

In the ''t'', ''ρ'', ''y'' coordinates on {{nowrap|'''R'''<sup>''n''+1,1</sup>}}, the Minkowski metric takes the form:

:<math> t ^2 g _{ij} ( y ) \, dy ^i \, dy ^j + 2 \rho \, dt ^2 + 2 t \, dt \, d \rho , </math>

where ''g''<sub>''ij''</sub> is the metric on the sphere.

In these terms, a section of the bundle ''N''<sup>+</sup> consists of a specification of the value of the variable {{nowrap|1=''t'' = ''t''(''y''<sup>''i''</sup>)}} as a function of the ''y''<sup>''i''</sup> along the null cone {{nowrap|1=''ρ'' = 0}}.  This yields the following representative of the conformal metric on ''S'':

:<math> t ( y ) ^2 g _{ij} \, d y ^i \, d y ^j .</math>

====The Kleinian model====
Consider first the case of the flat conformal geometry in Euclidean signature.  The ''n''-dimensional model is the [[celestial sphere]] of the {{nowrap|(''n'' + 2)}}-dimensional Lorentzian space '''R'''<sup>''n''+1,1</sup>.  Here the model is a [[Klein geometry]]: a [[homogeneous space]] ''G''/''H'' where {{nowrap|1=''G'' = SO(''n'' + 1, 1)}} acting on the {{nowrap|(''n'' + 2)}}-dimensional Lorentzian space '''R'''<sup>''n''+1,1</sup> and ''H'' is the [[isotropy group]] of a fixed null ray in the [[light cone]].  Thus the conformally flat models are the spaces of [[inversive geometry]].  For pseudo-Euclidean of [[metric signature]] {{nowrap|(''p'', ''q'')}}, the model flat geometry is defined analogously as the homogeneous space {{nowrap|O(''p'' + 1, ''q'' + 1)/''H''}}, where ''H'' is again taken as the stabilizer of a null line.  Note that both the Euclidean and pseudo-Euclidean model spaces are [[Compact space|compact]].

====The conformal Lie algebras====
To describe the groups and algebras involved in the flat model space, fix the following form on {{nowrap|'''R'''<sup>''p''+1,''q''+1</sup>}}:

:<math>
Q=\begin{pmatrix}
0&0&-1\\
0&J&0\\
-1&0&0
\end{pmatrix}
</math>
where ''J'' is a quadratic form of signature {{nowrap|(''p'', ''q'')}}.  Then {{nowrap|1=''G'' = O(''p'' + 1, ''q'' + 1)}} consists of {{nowrap|(''n'' + 2) × (''n'' + 2)}} matrices stabilizing {{nowrap|1=''Q'' : <sup>t</sup>''MQM'' = ''Q''}} (the superscript ''t'' means transpose).  The Lie algebra admits a [[Cartan decomposition]]

:<math>\mathbf{g}=\mathbf{g}_{-1}\oplus\mathbf{g}_0\oplus\mathbf{g}_1</math>

where

:<math>
\mathbf{g}_{-1} = \left\{\left.
\begin{pmatrix}
0&^tp&0\\
0&0&J^{-1}p\\
0&0&0
\end{pmatrix}\right| p\in\mathbb{R}^n\right\},\quad 
\mathbf{g}_{-1} = \left\{\left.
\begin{pmatrix}
0&0&0\\
^tq&0&0\\
0&qJ^{-1}&0
\end{pmatrix}\right| q\in(\mathbb{R}^n)^*\right\}
</math>

:<math>
\mathbf{g}_0 = \left\{\left.
\begin{pmatrix}
-a & 0 & 0\\
0 & A & 0\\
0 & 0 & a
\end{pmatrix} \right| A \in \mathfrak{so} ( p , q ) , a \in \mathbb{R} \right\} .</math>

Alternatively, this decomposition agrees with a natural Lie algebra structure defined on {{nowrap|'''R'''<sup>''n''</sup> ⊕ '''cso'''(''p'', ''q'') ⊕ ('''R'''<sup>''n''</sup>)<sup>∗</sup>}}.

The stabilizer of the null ray pointing up the last coordinate vector is given by the [[Borel subalgebra]]

:'''h''' = '''g'''<sub>0</sub> ⊕ '''g'''<sub>1</sub>.