Editing Conformal geometry (section)

====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>