Sub-Riemannian manifolds are manifolds with the Heisenberg principle built in. This comprehensive text and reference begins by introducing the theory of sub-Riemannian manifolds using a variational approach in which all properties are obtained from minimum principles, a robust method that is novel in this context. The authors then present examples and applications, showing how Heisenberg manifolds (step 2 sub-Riemannian manifolds) might in the future play a role in quantum mechanics similar to the role played by the Riemannian manifolds in classical mechanics. Sub-Riemannian Geometry: General Theory and Examples is the perfect resource for graduate students and researchers in pure and applied mathematics, theoretical physics, control theory, and thermodynamics interested in the most recent developments in sub-Riemannian geometry.
Part I. General Theory: 1. Introductory chapter
2. Basic properties
3. Horizontal connectivity
4. Hamilton-Jacobi theory
5. Hamiltonian formalism
6. Lagrangian formalism
7. Connections on sub-Riemannian manifolds
8. Gauss' theory of sub-Riemannian manifolds
Part II. Examples and Applications: 9. Heisenberg manifolds
10. Examples of Heisenberg manifolds
11. Grushin manifolds
12. Hormander manifolds
Appendix A: local non-solvability
Appendix B: fibre bundles.