A Comprehensive Introduction to Sub-Riemannian Geometry

by Andrei Agrachev, Davide Barilari, Ugo Boscain

Overview

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.

Product Details

ISBN-13:	9781108476355
Publisher:	Cambridge University Press
Publication date:	10/31/2019
Series:	Cambridge Studies in Advanced Mathematics , #181
Edition description:	Comprehens
Pages:	762
Product dimensions:	6.14(w) x 9.25(h) x 1.77(d)

About the Author

Andrei Agrachev is currently a full professor at Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste. His research interests are: sub-Riemannian geometry, mathematical control theory, dynamical systems, differential geometry and topology, singularity theory and real algebraic geometry.

Davide Barilari is Maître de Conférence at Université de Paris VII (Denis Diderot). His research interests are: sub-Riemannian geometry, hypoelliptic operators, curvature and optimal transport.

Ugo Boscain is Research Director at Centre National de la Recherche Scientifique (CNRS), Paris. His research interests are: sub-Riemannian geometry, hypoelliptic operators, quantum mechanics, singularity theory and geometric control.

Table of Contents

Introduction; 1. Geometry of surfaces in R^3; 2. Vector fields; 3. Sub-Riemannian structures; 4. Pontryagin extremals: characterization and local minimality; 5. First integrals and integrable systems; 6. Chronological calculus; 7. Lie groups and left-invariant sub-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; 16. Riemannian curvature; 17. Curvature in 3D contact sub-Riemannian geometry; 18. Integrability of the sub-Riemannian geodesic flow on 3D Lie groups; 19. Asymptotic expansion of the 3D contact exponential map; 20. Volumes in sub-Riemannian geometry; 21. The sub-Riemannian heat equation; Appendix. Geometry of parametrized curves in Lagrangian Grassmannians with Igor Zelenko; References; Index.

Editorial Reviews

‘It is wonderful to have a wide swath of the work of this school explained clearly and set down in one place. I am understanding some of the concepts described for the first time. I am grateful to the three authors for their efforts in putting this book together.’ Richard Montgomery, Bulletin of the American Mathematical Society

‘This textbook is a valuable reference in sub-Riemannian geometry, providing a systematic and firm foundation to the theory … It is my opinion that this textbook will serve as a solid reference for many researchers in the field, and will contribute to the development of the subject in the forthcoming years.’ Luca Rizzi, Mathematical Reviews

‘The book can be used for either an introductory or advanced course on sub-Riemannian geometry (the authors suggest which chapters to use for each case), but it also constitutes a state-of-the-art reference for most of the topics that it treats and will be an essential work for researchers active in sub-Riemannian geometry.’ Robert Neel, MAA Reviews

‘… an excellent resource for a broad range of topics in Riemannian and sub-Riemannian geometry. I am strongly convinced that this will become one of the main references for people interested in these topics, ranging from students to specialists.’ Alpár R. Mészáros, zbMATH

From the Publisher