Liouville Surfaces with K=p(U)q(V)

We can also visualize the geodesic flow if superimpose a graph of the Gauss curvature on a torus. Because the graph is over a bounded domain, each time we cross one of the bounding lines, we need to return within our domain from the opposite edge. But since opposite edges coincide if we use a torus, the smoothness of the geodesic is mainained.

The first case corresponds to the geodesics with parameter 1 <= C < 2 on a ellipsoid:

If C = 2 the geodesic passes through the critical points. The first image shows the geodesic on the ellipsoid and the second one shows the geodesic on the torus:

The last case corresponds to the case in which 2 < C <= 3: