A metric invariant of Möbius transformations

Abstract

The complex unit disk D = {z ∈ C: |z| < 1} is endowed with Möbius addition ⊕M defined by $woplus M;;z=frac{w+z}{1+wz}$ We prove that the metric dT defined on D by $d_t(omega,z)=tan^{-1}left|-omegaoplus M;zright|$ is an invariant of Möbius transformationscarrying D onto itself. We also prove that (D, dT ) and (D, dP ) , where dP denotes the Poincaré metric, have the sameisometry group and then classify the isometries of (D, dT ) .