Schwarz Reflection Geometry II: Local and Global Behavior of the Exponential Map

Abstract

A local normal form is obtained for geodesics in the space $\Lambda=\{\Gamma\}$ of analytic Jordan curves in the extended complex plane with symmetric space multiplication $\Gamma_1\cdot\Gamma_2$ defined by Schwarzian reflection of $\Gamma_2$ in $\Gamma_1$. Local geometric features of $(\Lambda, \cdot)$ will be seen to reflect primarily the structure of the Witt algebra, while issues of global behavior of the exponential map will be viewed in the context of conformal mapping theory.