A Comprehensive Introduction to Sub-Riemannian Geometry

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Andrei Agrachev (Scuola Internazionale Superiore di Studi Avanzati, Trieste) Davide Barilari (Université de Paris VII (Denis Diderot))

A Comprehensive Introduction to Sub-Riemannian Geometry

Andrei Agrachev (Scuola Internazionale Superiore di Studi Avanzati, Trieste), Davide Barilari (Université de Paris VII (Denis Diderot)), Ugo Boscain (Centre National de la Recherche Scientifique (CNRS), Paris)

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English

Cambridge University Press
31 October 2019

Differential calculus & equations; Differential & Riemannian geometry; Algebraic topology

Series: Cambridge Studies in Advanced Mathematics

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.

By: Andrei Agrachev (Scuola Internazionale Superiore di Studi Avanzati Trieste), Davide Barilari (Université de Paris VII (Denis Diderot)), Ugo Boscain (Centre National de la Recherche Scientifique (CNRS), Paris)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 235mm, Width: 156mm, Spine: 45mm
Weight: 1.190kg
ISBN: 9781108476355
ISBN 10: 110847635X
Series: Cambridge Studies in Advanced Mathematics
Pages: 762
Publication Date: 31 October 2019
Audience: College/higher education , Further / Higher Education
Format: Hardback
Publisher's Status: Active

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.

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.

Reviews for A Comprehensive Introduction to Sub-Riemannian Geometry

'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