The conformal geometry of complex quadrics and the fractional-linear form of Möbius transformations

Abstract

A new derivation is given of the Vahlen (1902) form of the local conformal transformations of Cn, v ■ (av+b)(cv+d)−1, where v∈Cn and a, b, c, d are suitable elements of the complex Clifford algebra Cl(n). The derivation is based on the homomorphism of groups Spin(n+2)→SO(n+2), the isomorphism of algebras Cl(n+2)≂C(2)⊗Cl(n), and the action of the Möbius group SO(n+2) on the quadric Qn, the conformal compactification of Cn. It is shown how the conformal geometry of Qn lifts, for every n=1,2,..., to a unique conformal spin structure. The Hermite–Sylvester interpolation method is used to represent the map exp: spin(n)→Spin(n) in such a manner that exp a becomes a Clifford polynomial in a∈Λ2Cn.