Sub-riemannian geometry

Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely: control theory classical mechanics Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) diffusion on manifolds analysis of hypoelliptic operators Cauchy-Riemann (or CR) geometry Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub-Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists, namely Andre Bellaiche: The tangent space in sub-Riemannian geometry Mikhael Gromov: Carnot-Caratheodory spaces seen from within Richard Montgomery: Survey of singular geodesics Hector J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers Jean-Michel Coron: Stabilization of controllable systems Table of Contents: The Tangent Space in Sub-Riemannian Geometry /Andre Bellaiche Sub-Riemannian manifolds Accessibility Two examples Privileged coordinates The tangent nilpotent Lie algebra The algebraic structure of the tangent space Gromov's notion of tangent space Estimating the distances between the distances in the manifold and in its tangent space Why is the tangent space a group? References Carnot-Caratheodory Spaces seen from within /Michael Gromov Basic definitions, examples and problems Horizontal curves and small C-C balls Hypersurfaces in C-C spaces Carnot-Caratheodory geometry of contact manifolds Pfaffian geometry in the internal light Anisotropic connections References Survey of Singular Geodesics /Richard Montgomery Introduction The example and its properties Some open questions Acknowledgments Note in proof References A Cornucopia of Four-Dimensional Abnormal Sub-Riemannian Minimizers /Hector J. Sussmann Introduction Sub-Riemannian manifolds and abnormal extremals Abnormal extremals in dimension 4 Optimality An optimality lemma End of the proof Strict abnormality Conclusion References Stabilization of Controllable Systems /Jean-Michel Coron Introduction Local controllability Sufficient conditions for local stabilizability of local controllable systems by means of stationary feedback laws Necessary conditions for local stabilizability by means of stationary feedback laws Stabilization by means of time-varying feedback laws Return method and controllability References Index Series: Progress in Mathematics, Volume 144