Algebraically closed real geodesics on n-dimensional ellipsoids are dense in the parameter space and related to hyperelliptic tangential coverings

The closedness condition for real geodesics on n-dimensional ellipsoids is in general transcendental in the parameters (semiaxes of the ellipsoid and constants of motion). We show that it is algebraic in the parameters if and only if both the real and the imaginary geodesics are closed and we characterize such double--periodicity condition via real hyperelliptic tangential coverings. We prove the density of algebraically closed geodesics on n-dimensional ellipsoids with respect to the natural topology in the (2n)-dimensional real parameter space. In particular, the approximating sequence of algebraic closed geodesics on the approximated ellipsoids may be chosen so to share the same values of the length and of the real period vector as the limiting closed geodesic on the limiting ellipsoid. Finally, for real doubly-periodic geodesics on triaxial ellipsoids, we show how to evaluate algebraically the period mapping and we present some explicit examples of families of algebraically closed geodesics.