= Antipodal fixed points do not characterize sphere rotations
The <Möbius transformation> $M(z)=(3z-1)/(3-z)$ fixes the <antipodal pair> $1,-1$ under <stereographic projection>. But $M'(1)=2$, so it multiplies infinitesimal spherical lengths at that fixed point by two, as follows from the round line element $2|dz|/(1+|z|^2)$. It is not an <isometry> and hence not a sphere <rotation>. Fixing an <antipodal pair> is therefore necessary for a nontrivial rotational <Möbius transformation>, but is not sufficient.
Back to article page
