Sub-Riemannian  Metrics: Minimality of  Abnormal  Geodesics versus
Subanalyticity

A. A. Agrachev  A. V. Sarychev

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
nonoccurrence of abnormal length-minimizers. Utilizing the results of the
previous study of abnormal minimizers accomplished by the authors in [Ann.
IHP. Analyse nonlineaire, Vol.13, no. 6, pp.635-690] we describe in this
paper some classes of germs of so-called 2-generating and medium fat
distributions for which the corresponding sub-Riemannian metrics are
subanalytic. To caracterize the vastness of these classes we determine the
dimensions of manifolds on which generic germs of rank $r$ distributions are
respectively 2-generating or medium fat.