Decomposition of closed orientable geometric surfaces into acute geodesic triangles
The authors consider differentiable closed orientable geometric surfaces \(S_g\) of genus \(g\geq 2\) in \(\mathbb{R}^3\) and decompositions of such surfaces into acute geodesic triangles. These decompositions are slightly more general than triangulations because they allow two triangles to intersect in two vertices. It is shown that there exists a decomposition of \(S_g\) into \(16g-16\) acute geodesic triangles. An example of decomposition of the torus into eight acute geodesic triangles, which is not a triangulation, is also given. The proof uses results on geometric surfaces from Chapter 9 of [\textit{J. G. Ratcliffe}, Foundations of hyperbolic manifolds. 2nd ed. New York, NY: Springer (2006; Zbl 1106.51009)] that allow to write \(S_g\) as a disjoint union of \(3g-3\) essential, simple, and closed curves in \(S_g\) from different homotopy classes and of \(2g-2\) surfaces each homeomorphic to a 2-sphere minus three disjoint closed disks. Each of the latter in turn is written as the union of two convex right-angled hexagons, each of which is triangulated by four triangles. This yields a locally geodesic decomposition of \(S_g\) into \(16g-16\) triangles. The main part of the proof is then to show that there are motions of vertices along geodesic arcs that change all angles into acute angles.
scientific article; zbMATH DE number 6827948
scientific article