A Comprehensive Introduction to Sub-Riemannian Geometry
A Comprehensive Introduction to Sub-Riemannian Geometry
Davide Barilari, Université de Paris VII (Denis Diderot)
Ugo Boscain, Centre National de la Recherche Scientifique (CNRS), Paris
AUD$298.18
exc GSTSub-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.
- Provides a comprehensive and systematic presentation of sub-Riemannian geometry
- Accessible to graduate students with no prior knowledge of the subject
- Contains useful models and tools for researchers working in various areas of application, including robotics, quantum control and image processing
Reviews & endorsements
'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
Product details
October 2019Adobe eBook Reader
9781108754514
0 pages
66 b/w illus. 73 exercises
This ISBN is for an eBook version which is distributed on our behalf by a third party.
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.
