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Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Published online by Cambridge University Press:  15 August 2002

Andrei A. Agrachev  and
Andrei V. Sarychev
Andrei A. Agrachev
Affiliation:
Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina ul. 8, 117966 Moscow, Russia; andrei@agrachev.mian.su. SISSA, via Beirut 2-4, 34014 Trieste, Italy; agrachev@sissa.it.
Andrei V. Sarychev
Affiliation:
Department of Mathematics, University of Aveiro 3810-193 Aveiro, Portugal; ansar@mat.ua.pt.
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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.

Keywords

Sub-Riemannian metricssubanalitycityabnormal length minimizers.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 377 - 403
Copyright
© EDP Sciences, SMAI, 1999

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