Shortest paths on surfaces are called geodesics. Their behavior very much depends on curvature. Amazingly, the total curvature of a surface is always quantized and does not change if the surface is deformed. Curvature also very much influences how a body under no influence of external forces moves. In this picture, the curvature is encoded in color. The yellow paths are light rays. The light source in front started in a point of positive curvature, the light source in the back at a point of negative curvature.

The problem of analyzing wave fronts on surfaces is exciting and very much unexplored. Here are some problems about wave fronts. We expect that except for very special cases like a sphere, the wave fronts fill out the surface more and more.