Geodesic motion on closed spaces: Two numerical examples

The geodesic structure is very closely related to the trace of the Laplace operator, involved in the calculation of the expectation value of the energy–momentum tensor in Universes with non-trivial topology. The purpose of this work is to provide concrete numerical examples of geodesic flows. Two manifolds with genus g=0 are given. In one the chaotic regions, form sets of negligible or zero measure. In the second example the geodesic flow shows the presence of measurable chaotic regions. The approach is "experimental", numerical, and there is no attempt to an analytical calculation. -- Highlights: ► Elementary differential geometry of surfaces and the Gauss–Bonnet theorem. ► The geodesic equation is numerically solved for two metrics on the sphere. ► With the Poincare surface, chaotic and regular regions are identified. ► Chaotic regions increase as the curvature fluctuation of the manifold increases.