A binary infinitesimal form of Teichmuller metric

Abstract

Let $S$ be a Riemann surface of analytic finite type or the unit disk in the complex plane. Let $[\mu]$ denote the Teichm\"uller equivalence classes of Beltrami differentials $\mu $. We apply the Fundamental Inequalities to obtain a binary infinitesimal form of Teichm\"uller metric. Using this form, we define "\emph{angle}" between two geodesics originating from a point and conjecture that the sum of the angles of a triangle in $T(S)$ should be less than $\pi$ if $S$ is of analytic finite type. As a consequence, the well-known necessary condition for two geodesics coinciding is derived immediately.