Curvature Testing in 3-Dimensional Metric Polyhedral Complexes

Abstract

In a previous article, the authors described an algorithm to determine whether a finite metric polyhedral complex satisfied various local curvature conditions such as being locally CAT(0). The proof made use of Tarski's theorem about the decidability of first order sentences over the reals in an essential way, and thus it was not immediately applicable to a specific finite complex. In this article, we describe an algorithm restricted to 3-dimensional complexes which uses only elementary 3-dimensional geometry. After describing the procedure, we include several examples involving Euclidean tetrahedra which were run using an implementation of the algorithm in GAP.