Jacobi’s last geometric statement extends to a wider class of Liouville surfaces

Abstract:

Numerical evidence is presented which strongly suggests that “Jacobi’s last geometric statement"—that the conjugate locus from a point has exactly four cusps and the corresponding cut locus consists of only one topological segment—holds for compact real analytic Liouville surfaces diffeomorphic to $S^2$ if the Gaussian curvature is everywhere positive and has exactly six critical points, these being two saddles, two global minima, and two global maxima (as is the case for an ellipsoid). Our experiments suggest that this is a sufficient rather than a necessary condition. Furthermore, for compact real analytic Liouville surfaces diffeomorphic to $S^2$ upon which the Gaussian curvature can be negative but has exactly six critical points, these being two saddles, two global minima, and two global maxima, it appears that the cut locus is always a subarc of a line given by $x_1=\mathrm {const}$ or $x_2=\mathrm {const}$, where $(x_1,x_2)$ are canonical coordinates with respect to which the metric has the form $(f_1(x_1)+f_2(x_2))(dx_1^2+dx_2^2)$. In the case of an ellipsoid, these curves are lines of curvature.