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ESAIM: Control, Optimisation and Calculus of Variations July 1999, Vol. 4, p. 377403 URL: http://www.emath.fr/cocv/
 

Summary: ESAIM: Control, Optimisation and Calculus of Variations July 1999, Vol. 4, p. 377­403
URL: http://www.emath.fr/cocv/
SUB-RIEMANNIAN METRICS: MINIMALITY
OF ABNORMAL GEODESICS VERSUS SUBANALYTICITY
Andrei A. Agrachev1, 2
and Andrei V. Sarychev3
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 nonlinŽeaire 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 suban-
alytic. 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.
RŽesumŽe. On Žetudie des mŽetriques sous-Riemanniennes (des Carnot-CarathŽeodory) dŽefinies par les
distributions non involutives sur les variŽetŽes Riemanniennes analytiques rŽeelles. On Žetablie la connexion
entre les propriŽetŽes de la rŽegularitŽe de ces mŽetriques et l'absence des gŽeodŽesiques anormales de longueur
minimale. En utilisant les rŽesultats des Žetudes prŽecŽedentes sur les minimiseurs anormaux accomplies
par les auteurs dans [Annales IHP. Analyse nonlinŽeaire 13, p. 635-690], on dŽecrit dans cet article,
pour certains types des germes, des distributions (appelŽees 2-gŽenŽerŽees et d'une croissance moyenne)

  

Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)

 

Collections: Engineering; Mathematics