Total excess and Tits metric for piecewise Riemannian 2-manifolds (Q1296288)

This paper studies the total excess \(e(X)\) of a simply-connected nonpositively curved piecewise Riemannian \(2\)-manifold. In the smooth case the total excess is equal to the total curvature. The second author proved in [\textit{F. Ohtsuka}, Bull. Fac. Sci., Ibaraki Univ., Ser. A 20, 5-8 (1988; Zbl 0666.53026)] that \(e(X)=2\pi-2 \text{diam}_{Td}(X(\infty))\) where \(\text{diam}_{Td}(X(\infty))\) is the diameter of the ideal boundary in the Tits metric. This formula is generalized here to the piecewise Riemannian case. It is also proved that the excess is finite iff the Tits metric induces the same topology at \(X(\infty)\) as the cone topology. Finally, it is shown that any decomposition of a circle into points and intervals is realized as the Tits boundary of some simply connected nonpositively curved piecewise Riemannian \(2\)-manifold.