Editing Conformal geometry (section)

====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]].