Editing Conformal geometry (section)

====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''.