Decomposition of Cut Loci

Abstract

Abstract. If p is a point in a complete riemannian manifold, the points of the cut locus of p are designated as singular or ordinary according to whether there is just one or more minimizing geodesies from p. It is proved that the ordinary cut-points are dense in the cut locus.1 Introduction. Throughout this paper we will consider a complete riemannian manifold M of dimension d. For m E M, a cut-point of m is a point/» such that every extension of a distance-minimizing segment from m to p beyond p is no longer minimizing. In the tangent space Mm, a vector x E Mm is a tangent cut-point if the ray t -» tx, 0 < / < 1, is mapped by expm to a minimizing segment from m to a cut-point expmx. If there are two or more minimizing segments from' m to p, we define p to be an ordinary cut-point. If there is just one minimizing segment, we call p a singular cut-point. Ordinary and singular tangent cut-points are defined so that they correspond to the same kind of points in M under expm.

Optimality results in orbit transfer

“…The separating line, L(x 0 ), is the set of points where at least two minimizing geodesics intersect. When the metric is complete, the cut locus is the closure of the separating line [2], and one has just to add the first conjugate points: conjugate points are critical values of the exponential mapping, and the set of first such points along geodesics issuing from x 0 is the conjugate locus, C(x 0 ).…”

“…Moreover, averaging is performed on a simplified optimal control problem so as to provide tractable computations. We first replace the L 1 cost by an L 2 one (the so-called energy criterion), and do not take into account the variation of the mass (2). Though the two criterions are distinct, they are close enough to be connected using a continuation procedure, see [12].…”

On the measure of the cut locus of a Fréchet mean

“…Points with the latter property do occur (cf. [5]). However, the set of cut points of x to which there are at least two minimal geodesics is dense in C(x) (cf.…”

“…However, the set of cut points of x to which there are at least two minimal geodesics is dense in C(x) (cf. [5]) and may include first conjugate points as, for example, it does in standard spheres.…”

The radial part of Brownian motion II. Its life and times on the cut locus

“…We say that the geodesic {Expp(tv): 0 _< t <_ s} is conjugate precisely when the differential d Expp at sv ~ TpM is not of full rank 9 For fixed p define a function c on the unit sphere bundle Se-IM c TM, Because there are at least two distinct minimal geodesics connecting p to q there are, as observed by Bishop (1977), at least two candidates for grad r at q ~ Co(p), provided by the negative of the tangent vector for each of the incoming minimal geodesics. Consequently r is not C 1 anywhere on Co(p), and so Co(p) is contained in the region of gradient discontinuity of r. This set Co(p) has a relatively simple structure described in Ozols (1974).…”