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 Teichmüller equivalence classes of Beltrami differentials $\mu$. We apply the Fundamental Inequalities to obtain a binary infinitesimal form of Teichmüller 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.

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