# A Tour of Subriemannian Geometries, Their Geodesics and Applications

### About this Title

**Richard Montgomery**, *University of California, Santa Cruz, CA*

Publication: Mathematical Surveys and Monographs

Publication Year:
2002; Volume 91

ISBNs: 978-0-8218-4165-5 (print); 978-1-4704-1318-7 (online)

DOI: https://doi.org/http://dx.doi.org/10.1090/surv/091

MathSciNet review: MR1867362

MSC: Primary 53C17; Secondary 37J99, 53C60, 58E10, 70G45, 70H05

### Table of Contents

**Front/Back Matter**

**Part 1. Geodesies in subriemannian manifolds **

- 1. Dido meets Heisenberg
- 2. Chow’s theorem: Getting from A to B
- 3. A remarkable horizontal curve
- 4. Curvature and nilpotentization
- 5. Singular curves and geodesics
- 6. A zoo of distributions
- 7. Cartan’s approach
- 8. The tangent cone and Carnot groups
- 9. Discrete groups tending to Carnot geometries
- 10. Open problems

**Part 2. Mechanics and geometry of bundles **

- 11. Metrics on bundles
- 12. Classical particles in yang-mills fields
- 13. Quantum phases
- 14. Falling, swimming, and orbiting

**Part 3. Appendices **

- Appendix A. Geometric mechanics
- Appendix B. Bundles and the Hopf fibration
- Appendix C. The Sussmann and Ambrose-Singer Theorems
- Appendix D. Calculus of the endpoint map and existence of geodesies