The Hyperbolic Disk

Taking Schwarz’s Lemma as a starting point, the hyperbolic metric on the unit disk is constructed as a Riemannian metric so that the automorphisms of the disk are isometries. The automorphisms of the half-plane are characterised and the hyperbolic metric in the disk transferred to the half-plane via an explicit conformal map. Geodesics in the hyperbolic geometry of the disk and of the half-plane are described, and the geometry is shown to be non-Euclidean in that Euclid’s Parallel Postulate fails. The Schwarz-Pick Lemma, which states that self-maps of the disk are contractions in the hyperbolic metric, is proved.