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

TL;DR

This paper analyzes the structure of conjugate loci on ellipsoids and Liouville manifolds, revealing that singularities are limited to cuspidal edges and specific Lagrangian types, extending classical results.

Contribution

It provides a detailed analysis of conjugate loci on Liouville manifolds, including ellipsoids, identifying the types of singularities present and solving geodesic equations.

Findings

Conjugate loci have only cuspidal edges and D4+ Lagrangian singularities.

The paper solves geodesic equations explicitly for Liouville manifolds.

It extends classical results on ellipsoids to higher-dimensional Liouville manifolds.

Abstract

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.