Editing Conformal geometry (section)

====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.