A Comprehensive Introduction to Sub-Riemannian Geometry

Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.

Andrei Agrachev (Author), Davide Barilari (Author), Ugo Boscain (Author)

9781108476355, Cambridge University Press

Hardback, published 31 October 2019

762 pages, 66 b/w illus. 73 exercises
23.5 x 15.6 x 4.5 cm, 1.19 kg

'… 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

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.

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.

Subject Areas: Algebraic topology [PBPD], Differential & Riemannian geometry [PBMP], Differential calculus & equations [PBKJ]