Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Abstract

Abstract. We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. Analyse nonlinéaire 13, p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize these classes of distributions we determine the dimensions of the manifolds on which generic germs of distributions of given rank are respectively 2-generating or medium fat.Résumé. Onétudie des métriques sous-Riemanniennes (des Carnot-Carathéodory) définies par les distributions non involutives sur les variétés Riemanniennes analytiques réelles. Onétablie la connexion entre les propriétés de la régularité de ces métriques et l'absence des géodésiques anormales de longueur minimale. En utilisant les résultats desétudes précédentes sur les minimiseurs anormaux accomplies par les auteurs dans [Annales IHP. Analyse nonlinéaire 13, p. 635-690], on décrit dans cet article, pour certains types des germes, des distributions (appelées 2-générées et d'une croissance moyenne) telles que les métriques sous-Riemanniennes correspondantes sont sous-analytiques. Pour caractériser la classe de ces types des distributions, on détermine les dimensions des variétés sur lesquelles les germes génériques des distributions du rang donné sont 2-générées ou d'une croissance moyenne.

Mass Transportation on Sub-Riemannian Manifolds

“…SR is locally Lipschitz on Ω, and let φ be the c-concave function provided by Theorem 2.3. Then: 5 The factor 1 2 appearing in front of dφ(x) is due to the fact that we are considering the cost function…”

“…Medium-fat distribution were introduced by Agrachev and Sarychev in [5] (we refer the interested reader to that paper for a detailed study of this kind of distributions). It can easily be shown that medium-fat distributions do not admit nontrivial Goh paths.…”

Resonance of minimizers forn-level quantum systems with an arbitrary cost

“…After elimination of the drift (see Sect. 2), the problem (P), with this cost is a sub-Riemannian problem or a singular-Riemannian problem depending on the number of controls (see (H1)) and was studied in [10,11], in the isotropic case (µ j,k = µ), for n = 2, 3 and in the case in which the Hamiltonian is given by (2). For sub-Riemannian and singular-Riemannian geometry see for instance [5,19,27].…”

Subriemannian geodesics of Carnot groups of step 3

“…Recently experts developed necessary/sufficient second order conditions of optimality, i.e, Goh condition and generalized Legendre-Jacobi condition, see e.g. [4,5,6,7,8]. These conditions were derived from the finiteness of the generalized Morse index of critical points of the end-point mapping.…”

“…These conditions were derived from the finiteness of the generalized Morse index of critical points of the end-point mapping. We refer to [6,8,5] for the finiteness of the generalized Morse index.…”