The Cut Loci on Ellipsoids and Certain Liouville Manifolds

Abstract

Abstract. We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.Key words. Cut locus, ellipsoid, Liouville manifold, integrable geodesic flow.AMS subject classifications. Primary 53C22, Secondary 53A051. Introduction. On a complete riemannian manifold, any geodesic γ(t) starting at a point γ(0) = p has the property that any segment {γ(t) | 0 ≤ t ≤ T } is minimal, i.e., the length of the segment is equal to the distance between the points p and γ(T ), if T > 0 is small. If the supremum t 0 of the set of such T is finite, then the point γ(t 0 ) is called the cut point of p along the geodesic γ(t) (t ≥ 0). The cut locus of the point p is then defined as the set of all cut points of p along the geodesics starting at p. For the general properties of cut loci, we refer to [19], [26].The study of cut locus was started at 1905 by H. Poincaré [22] in the case of convex surfaces, and there are several classical results, for example, [21], [35], [36]. From its definition, the cut locus of a point p on a compact manifold M is homotopically equivalent to M − {p}, but it can be very complicated, see [5], [9]. The structure of cut locus was studied in connection with the singularity theory, see It is well known that the cut locus of any point on the sphere of constant curvature consists of a single point, and it is also known that this property characterizes the sphere of constant curvature (an affirmatively solved case of the Blaschke conjecture, see [1] ([29] is an experimental work). Especially in higher dimensional case there are not many results without symmetric spaces and some singular spaces [14], even if using computational approximations.In the earlier paper [10], we proved that the cut locus of a non-umbilic point on a tri-axial ellipsoid is a segment of the curvature line containing the antipodal point, inspired by an experimental work [12]. Also, we gave the complete proof of Jacobi's last geometric statement ([15], [16], see also [28], which contains historical remarks). Furthermore, we have seen in [11] that there are many surfaces possessing such simple cut loci. Surfaces we considered in [11] are so-called Liouville surfaces, i.e., surfaces whose geodesic flows possess first integrals which are fiberwise quadratic

Small time heat kernel asymptotics at the cut locus on surfaces of revolution

“…In [5], the foreseen conjugate and cut loci were given as a conjecture and the first proof appeared only recently in [14]. For other results about the conjugate and cut locus on surfaces of revolution (and some generalizations) one can see also [7,15,21].…”

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.…”

The conjugate locus on convex surfaces

“…The conjugate locus is a classical topic in Differential Geometry (see Jacobi [16], Poincaré [22], Blaschke [3] and Myers [21]) which is undergoing renewed interest due to the recent proof of the famous Last Geometric Statement of Jacobi by Itoh and Kiyohara [13] (Figure 5, see also [14], [15], [24]) and the recent development of interesting applications (for example [27], [28], [4], [5]). Recent work by the author [30] showed how, as the base point is moved in the surface, the conjugate locus may spontaneously create or annihilate pairs of cusps; in particular when two new cusps are created there is also a new "loop" of a particular type, and this suggests a relationship between the number of cusps and the number of loops.…”