 Solution

A Möbius transformation of the Riemann sphere is a mapwith its natural values at the pole and at infinity.

For three distinct points $z_1,z_2,z_3$, defineThe usual limiting conventions cover an infinite $z_i$. This Möbius transformation sends $(z_1,z_2,z_3)$ to $(1,0,\infty )$. Defining $T_w$ similarly, the mapsends $z_i$ to $w_i$. If two Möbius transformations do so, their quotient fixes $0,1,\infty$. A Möbius transformation fixing infinity is affine, and fixing zero and one then makes it the identity. This proves uniqueness.

With this convention, the cross-ratio isSubstitution shows that translations, nonzero scalings, and inversion preserve it; since these generate the Möbius group, every Möbius transformation preserves cross-ratios.

Conversely, suppose a bijection $f$ of the Riemann sphere preserves every cross-ratio. Let $m$ be the unique Möbius transformation agreeing with $f$ at three chosen points $z_1,z_2,z_3$. For any other $z$, preservation by $f$ and $m$ givesThe last coordinate in a cross-ratio with three fixed distinct entries is injective, so $f(z)=m(z)$. The equality already holds at the three base points, hence $f=m$ everywhere and $f$ is Möbius.

Finally, the map $z\mapsto a\overline{z}+b$ is constant when $a=0$ and therefore is not Möbius. If $a\ne 0$, it is bijective and fixes infinity. Were it Möbius, it would have the affine form $\alpha z+\beta$. Equality on real $z$ forces $\alpha =a$ and $\beta =b$, whereas equality at $z=i$ would require $ai=-ai$, contradicting $a\ne 0$. Thus