(i) C is a maximal curve of zero curvature; (ii) C is a noncompact maximal curve of constant curvature. Now we turn to the hyperbolic metric ds = (1 − |z|2)−1|dz| in the unit disk D. Let C : w(s), s ∈ I, be a curve of class C 2 in D that is parameterized by the hyperbolic arc-length s; that is, |w ′| = 1− |w|2, w ′′ is continuous in I. (1.1) Then the hyperbolic (= geodesic) curvature κ satisfies the differential equation w ′′ w ′ + 2ww ′ 1− |w|2 = iκ. (1.2)
