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

Abstract

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 = const or x 2 = 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 2 1 + dx 2 2 ). In the case of an ellipsoid, these curves are lines of curvature.The point of this paper is to present a conjecture, already contained in the abstract, as a contribution to pure mathematical research in global Riemannian geometry. The overwhelming bulk of the paper will however necessarily be devoted to a description of the computational methods used to motivate the conjecture, and, in particular, the checks which have been performed to verify that the software (which has been written specifically for this study) is indeed performing correctly. Such checks are vital in any experimental work. This paper is not intended to be a contribution to computational science as such, since the algorithms we have used, although perhaps combined in an unusual manner, are standard.The conjugate locus from a point p (which we will often refer to as the starting point) on a Riemannian manifold is the envelope of the geodesics emanating from p. In two dimensions, it can also be defined analogously to the nodes of a vibrating string as follows. A geodesic passing through p can be uniquely defined by the angle θ it makes at p with one fixed (reference) geodesic also passing through p. Varying this angle by a small amount (δθ) will cause the geodesic to deviate from its original path. The distance between points on the geodesic with angle θ (say g θ ) and the geodesic curve with angle θ + δθ (write g θ+δθ ) can be written as a real function ξ δθ of arclength s along g θ , where s = 0 corresponds to the point p. Clearly ξ δθ (0) = 0, since we are only considering geodesics which pass through p. The limit of ξ δθ (s)/δθ as δθ approaches zero may have further zeros (the nodes of the string, where, to

Bifurcations of the conjugate locus

“…Of particular relevance to this paper is the so-called "last geometric statement of Jacobi", which asserts (among other things) that the conjugate locus of a non-umbilic point on the triaxial ellipsoid has precisely 4 cusps (see [20] for a historical sketch and list of references). This conjecture was recently proved by Itoh and Kiyohara [13], and a renewed interest in the conjugate and cut locus can be seen in the recent papers providing formal studies ( [11], [24], [14], [15]), simulations ( [23], [8], [21], [22], [20]) and applications ( [6], [5], [3], [4], [7]). It is no surprise that the papers which focused on the triaxial ellipsoid and surfaces of revolution made heavy use of the fact that the geodesic flow on those surfaces is (Liouville) integrable.…”

Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces

“…In the analytic case, the cut locus is a finite tree and the extremity of each branch is a cusp point. But the explicit computation of the number of branches and cusps points is a very complicated problem and only very recently was proved the four cusps Jacobi conjecture on ellipsoids [7], [12].…”

Geodesic bifurcation on smooth surfaces

“…Computing cut loci and related concepts such as the medial axis or Voronoi diagrams based [20,21] on discrete respectively smooth surface representations has received some attention in the literature, see e.g. [5,6,9,16], respectively [30,31]. The two and three-dimensional smooth Riemannian setting has been considered in [11,19,26] and [38,39], the latter two being historical overviews on the respective computational methods.…”