On the Moduli Space of Flat Tori Having Unit Area

Inspiring from Thurston's asymmetric metric on Teichmilller spaces, we define and study a natural (weak) metric on the Teichmilller space of the torus. We prove that this weak metric is indeed a metric: it separates points and it is symmetric. Our main strategy to do this is to compute the metric explicitly. 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.