On the triaxial ellipsoid there are various classical dynamical systems. In the talk we shall report on joint work of Ronald Garcia on curvature lines and geodesics. We are dealing with the question under which conditions these curves are closed or not. In the case of geodesics we give a complete answer if the ellipsoid is defined over a number field. In the case of curvature lines we show that there is a countable set of so-called $\alpha$ - curvature lines which are not closed. These are classical problems in the theory of dynamical systems and there were almost no results in this direction. We also discuss the cases of ellipsoids of revolution and ellipsoids in Minkowski space.

The proofs make basic use of the analytic subspace theorem in the case of elliptic periods and periods on abelian surfaces coming up naturally. In the elliptic case we solve an extended problem of Th. Schneider dealing with periods of differentials of the third kind.