Non-compact curvature foliations on the triaxial ellipsoid

On the triaxial ellipsoid we consider curvature lines and geodesics and null-geodesics. We discuss when the curvature lines define a foliation with non-compact leaves. We also show that compact geodesics correspond to $2$-fold coverings of an elliptic curve by a curve of genus $2$ and extend this to null-geodesics. These problems lead to determine the dimension of elliptic and abelian period spaces which is related to Schneider's $3$rd and $4$th problem in his monograph on transcendental numbers.