On the Moduli Space of Flat Tori Having Unit Area

Year 2021, Volume: 14 Issue: 1, 59 - 65, 15.04.2021

İsmail Sağlam

https://doi.org/10.36890/iejg.754478

Abstract

Inspiring from Thurston's asymmetric metric on Teichmüller spaces, we define and study a natural (weak) metric on the Teichmüller space of the torus. We prove that this weak metric is indeed a metric: it separates points and it is symmetric. We relate this metric with the hyperbolic metric on the upper half-plane. We define another metric which measures how much length of a closed geodesic changes when we deform a flat structure on the torus. We show that these two metrics coincide.

Keywords

weak metric, asymmetric metric, Teichmüller space, torus, flat metric, hyperbolic metric

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