A binary infinitesimal form of Teichmuller metric

TL;DR

This paper introduces a new binary infinitesimal form of the Teichmüller metric on Riemann surfaces, enabling the definition of angles between geodesics and proposing a conjecture about triangle angles in Teichmüller space.

Contribution

It develops a novel binary infinitesimal form of the Teichmüller metric and explores geometric properties like angles and triangle sums in Teichmüller space.

Findings

Defined an angle between geodesics using the new form

Conjectured triangle angle sum less than π in finite type surfaces

Derived a necessary condition for geodesic coincidence

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.