The structure of the conjugate locus of a general point on ellipsoids and certain Liouville manifolds

Jin-ichi Itoh, Kazuyoshi Kiyohara

Published 2019-01-18Version 1

It is well known since Jacobi that the geodesic flow of the ellipsoid is "completely integrable", which means that the geodesic orbits are described in a certain explicit way. However, it does not directly indicate that any global behavior of the geodesics becomes easy to see. In fact, it happened quite recently that a proof for the statement "The conjugate locus of a general point in two-dimensional ellipsoid has just four cusps" in Jacobi's Vorlesungen \"uber dynamik appeared in the literature. In this paper, we consider Liouville manifolds, a certain class of Riemannian manifolds which contains ellipsoids. We solve the geodesic equations; investigate the behavior of the Jacobi fields, especially the positions of the zeros; and clarify the structure of the conjugate locus of a general point. In particular, we show that the singularities arising in the conjugate loci are only cuspidal edges and $D_4^+$ Lagrangian singularities, which would be the higher dimensional counterpart of Jacobi's statement.