  We consider geometric triangulations of surfaces, i.e., triangulations whose
edges can be realized by disjoint locally geodesic segments. We prove that the
flip graph of geometric triangulations with fixed vertices of a flat torus or a
closed hyperbolic surface is connected. We give upper bounds on the number of
edge flips that are necessary to transform any geometric triangulation on such
a surface into a Delaunay triangulation.

Computational Geometry;Geometric Topology