Total excess and Tits metric for piecewise Riemannian 2-manifolds

A piecewise Riemannian 2-manifold is a combinatorial 2-manifold with a triangulation such that each 2-simplex is a geodesic triangle of some Riemannian 2-manifold. In this paper, we study the total excess e(X) of a simply connected nonpositively curved piecewise Riemannian 2-manifold X in connection with the Tits metric on the boundary at infinity X(∞).