A Comprehensive Introduction to Sub-Riemannian Geometry

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.

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.