Anti-Möbius transformation

Definition

An anti-Möbius transformation is a map of the Riemann sphere of the form

T(z)=\frac{a\overline z+b}{c\overline z+d}, \qquad ad-bc\ne0,

with the usual extensions at a zero of the denominator and at $\infty$ . Equivalently, it is a Möbius transformation composed with complex conjugation.

Conformal orientation

Anti-Möbius transformations are antiholomorphic, conformal, and orientation reversing. They are not holomorphic maps of Riemann surfaces. Every orientation-reversing conformal automorphism of the round $2$ -sphere is anti-Möbius.

Composition

The composite of two anti-Möbius transformations is Möbius, while composing a Möbius and an anti-Möbius transformation in either order is anti-Möbius. Together the two types form the full conformal automorphism group of the round sphere; the Möbius group is its orientation-preserving subgroup of index $2$ .

Circle geometry

Like Möbius transformations, anti-Möbius transformations preserve generalized circles. A circle-preserving bijection therefore need not be holomorphic; orientation or cross-ratio conjugation distinguishes the two cases.

References

Alan F. Beardon, The Geometry of Discrete Groups, Springer, 1983. Publisher record. Relevant: Chapters 3–4.