  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.
