A characterization of shortest geodesics on surfaces

Max Neumann-Coto
2001 Algebraic and Geometric Topology  
Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.
doi:10.2140/agt.2001.1.349 fatcat:adfus2abjrgatlepqaxei3bpay