= Equality in hyperbolic Schwarz-Pick contraction
A <holomorphic function> between hyperbolic <Riemann surfaces> lifts to a disk map. If the lift is not an automorphism, strict <Schwarz-Pick lemma> derivative inequality makes every nonconstant geodesic shorter. Equality of distance for one distinct pair therefore forces the map to be a <covering map> and a local <isometry>. Covering maps attain equality for sufficiently close pairs in a single lift of a small convex metric ball; they need not preserve all global distances. A punctured disk covers a thrice-punctured sphere by including a cyclic parabolic subgroup into $\Gamma(2)$, so equality is possible in that example.
// Destination: analysis.bigb
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