Characterization and Computation of Closed Geodesics on Toroïdal Surfaces

Abstract

The aim of the present study is to characterize and compute closed geodesics on toroïdal surfaces. We show that a closed geodesic must make a number of rotations about the equatorial part ($k$ rotations) and the axis of revolution ($k'$ rotations) of the surface. We give the relation that exists between the numbers $k$ and $k'$, and the Clairaut's constant $C$ corresponding to the geodesic. Moreover, we prove that the numbers $k$ and $k'$ are relatively prime. We validate our findings by constructing closed geodesics on some examples of toroïdal surfaces using MAPLE. Finally, using experimental data on cardiac fiber direction, we show that fibers run as geodesics in the left ventricle whose geometrical shape looks like a toroïdal surface.