Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Andrei A. Agrachev; Andrei V. Sarychev

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 377-403
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Agrachev, Andrei A., and Sarychev, Andrei V.. "Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 377-403. <http://eudml.org/doc/197295>.

@article{Agrachev2010,
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. },
author = {Agrachev, Andrei A., Sarychev, Andrei V.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Sub-Riemannian metrics; subanalitycity; abnormal length minimizers.; abnormal length minimizer; Carnot-Carathéodory metrics; noninvolutive distributions; abnormal geodesics; sub-Riemannian metrics; subanalytic},
language = {eng},
month = {3},
pages = {377-403},
publisher = {EDP Sciences},
title = {Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity},
url = {http://eudml.org/doc/197295},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Agrachev, Andrei A.
AU - Sarychev, Andrei V.
TI - Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 377
EP - 403
AB - 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.
LA - eng
KW - Sub-Riemannian metrics; subanalitycity; abnormal length minimizers.; abnormal length minimizer; Carnot-Carathéodory metrics; noninvolutive distributions; abnormal geodesics; sub-Riemannian metrics; subanalytic
UR - http://eudml.org/doc/197295
ER -

References

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Citations in EuDML Documents

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  1. Grégoire Charlot, Métriques sous-riemanniennes de quasi-contact : forme normale et caustique
  2. Ugo Boscain, Grégoire Charlot, Resonance of minimizers for n-level quantum systems with an arbitrary cost
  3. Ugo Boscain, Grégoire Charlot, Resonance of minimizers for -level quantum systems with an arbitrary cost
  4. Kanghai Tan, Xiaoping Yang, Subriemannian geodesics of Carnot groups of step 3
  5. Andrei Agrachev, Jean-Paul Gauthier, On the subanalyticity of Carnot–Caratheodory distances
  6. Emmanuel Trélat, Global subanalytic solutions of Hamilton–Jacobi type equations

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