Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 6 / a

For a holomorphic map, conformality means preservation of oriented angles locally; it holds at every point of $G$ exactly when $f^{\prime }(z)\ne 0$ there. A Möbius transformation is $M(z)=(az+b)/(cz+d)$ with $ad-bc\ne 0$, extended to the Riemann sphere at its pole and at infinity.

For three distinct finite points definewhich sends $z_1,z_2,z_3$ to $0,1,\infty$. If one point is infinite, take the corresponding limit: for $z_3=\infty$ use $(z-z_1)/(z_2-z_1)$; for $z_1=\infty$ use $(z_2-z_3)/(z-z_3)$; for $z_2=\infty$ use $(z-z_1)/(z-z_3)$. Construct $T_w$ similarly. Then $M=T_w^{-1}\circ T_z$ has the required three images.

For uniqueness, normalize any two proposed maps by these same transformations. Composing one normalized map with the inverse of the other fixes $0,1,\infty$. A Möbius map fixing infinity has $c=0$, one fixing zero then has $b=0$, and fixing one forces $a=d$ up to the irrelevant common nonzero scalar. It is the identity. This proves sharp three-transitivity of Möbius transformations.

Treat a straight line together with infinity as a generalized circle on the Riemann sphere. Its finite points have a real equationTranslations and nonzero complex scalings preserve circles and lines. Under $z=1/w$, multiplying the equation by $|w{|}^2$ givesagain the equation of a circle or line. Nondegeneracy is preserved because the reciprocal map is a bijection of the sphere. The zero and infinite points are interpreted there, rather than discarded when multiplying denominators.

Every Möbius map with $c\ne 0$ factors asif $c=0$ it is already affine. Thus Möbius translation-scaling-inversion factorization and the reciprocal transformation of a generalized circle prove every generalized circle maps to another generalized circle. A circle becomes a line exactly when its image contains infinity, that is, when the original circle contains the map's pole.