A binary infinitesimal form of the Teichmüller metric

Abstract

Let S be a hyperbolic Riemann surface of finite analytic type. We give a binary infinitesimal form of the Teichmullermetric on the Teichmuller space T(S) of S by use of the fundamental Reich-Strebel inequalities. As an application, we reduce the Teichmuller disk problem to the infinitesimal setting. Moreover, we define the “angle” between geodesic rays in T(S) and give an explicit formula.