On spaces of Euclidean triangles and triangulated Euclidean surfaces

Abstract In this paper, we introduce an asymmetric distance function on the space of marked Euclidean triangles of normalised area, and we prove several properties of this metric, which turns out to be (a restriction of) a non-symmetric version of the classical Thompson distance. We give a description of the geodesics of this metric, we show that it is Finsler, and we give a formula for its infinitesimal Finsler structure. We then introduce and study a Finsler metric on the space of singular Euclidean structures on a surface adapted to an underlying fixed triangulation, and we also study its geodesics and its Finsler infinitesimal structure. We then develop a theory of completeness and completion of asymmetric metrics which is adapted to our setting, and we use this theory in the study of the completeness of the metric we introduced on the space of triangles. In doing so, we establish a bridge between one aspect of Thurston’s theory of metrics on spaces of surfaces and Thompson’s metrics.