A metric invariant of mobius transformations

Abstract

© TUBITAK. The complex unit disk D = z ∈ C: |z| < 1 is endowed with Mobius addition ⊕M defined by We prove that the metric dT defined on D by dT (w, z) = tan -1 |-w ⊕M z| is an invariant of Mobius transformations carrying D onto itself. We also prove that (D, dT) and (D, dp), where dp denotes the Poincare metric, have the same isometry group and then classify the isometries of (D, dT).