A comprehensive introduction to sub-Riemannian geometry

Sub-Riemannian geometry is geometry of the world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. Mathematically this is a smooth manifold equipped with a noninvolutive vector distribution and an Euclidean structure on it. The unconstrained Riemannian world is an important special case. In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with motivations and ramifications in several areas of pure and applied mathematics. The book gives a comprehensive presentation of sub-Riemannian geometry and can be used by graduate students and experienced researchers. It could serve as a basis for various graduated courses: from an introductory course in Riemannian geometry to an advanced course in sub-Riemannian one, passing through elements of Hamiltonian dynamics, integrable systems and Lie theory.