Andrei A. Agrachev, Davide Barilari, Luca Rizzi, Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), The first author has been supported by the grant of the Russian Federation for the state support of research, Agreement No 14 B25 31 0029. The second author has been supported by the European Research Council, ERC StG 2009 'GeCoMethods', contract number 239748, by the ANR Project GCM, program 'Blanche', project number NT09-504490. The third author has been supported by INdAM (GDRE CONEDP) and Institut Henri Poincaré, Paris, where part of this research has been carried out. We warmly thank Richard Montgomery and Ludovic Rifford for their careful reading of the manuscript. We are also grateful to Igor Zelenko and Paul W.Y. Lee for very stimulating discussions., and European Project: 239748,EC:FP7:ERC,ERC-2009-StG,GECOMETHODS(2010) more...
The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces., 120 pages, 12 figures, (v2) minor revision; (v3) new sections on Finsler manifolds, slow growth distributions, Heisenberg group; (v4) major revision, new extended section on 3D contact structures, constant curvature, improved results about existence of ample geodesics on SR structures, 2 new appendices, many minor revisions; (v5) 1 new appendix, minor revisions more...