Geodesic Motion on Closed Spaces: Two Numerical Examples

TL;DR

This paper provides numerical examples of geodesic flows on two different closed manifolds, illustrating the presence or absence of measurable chaotic regions, and explores their relation to the Laplace operator in universes with non-trivial topology.

Contribution

It offers concrete numerical illustrations of geodesic behavior on closed surfaces, highlighting differences in chaotic regions without analytical calculations.

Findings

One manifold exhibits negligible or zero measure chaotic regions.

The other manifold shows measurable chaotic regions.

Numerical approach demonstrates differences in geodesic flow behavior.

Abstract

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.