A Comprehensive Introduction to Sub-Riemannian Geometry

Description

Sub-Riemannian geometry is the geometry of a 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. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines. 1 Geometry of Surfaces in R3 2 Vector Fields 3 Sub-Riemannian Structures 4 Pontryagin Extremals\: Characterization and\rLocal Minimality 5 First Integrals and Integrable Systems 6 Chronological Calculus 7 Lie Groups and Left-Invariant\rSub-Riemannian Structures 8 Endpoint Map and Exponential Map 9 2D Almost-Riemannian Structures 10 Nonholonomic Tangent Space 11 Regularity of the Sub-Riemannian Distance 12 Abnormal Extremals and Second Variation 13 Some Model Spaces 14 Curves in the Lagrange Grassmannian 15 Jacobi Curves\r 16 Riemannian Curvature 17 Curvature in 3D Contact\rSub-Riemannian Geometry 18 Integrability of the Sub-Riemannian Geodesic\rFlow on 3D Lie Groups 19 Asymptotic Expansion of the 3D Contact\rExponential Map 20 Volumes in Sub-Riemannian Geometry 21 The Sub-Riemannian Heat Equation Appendix. Geometry of Parametrized Curves in Lagrangian Grassmannians\r