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Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely:
control theory classical mechanics Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) diffusion on manifolds analysis of hypoelliptic operators Cauchy-Riemann (or CR) geometry.
Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics.
This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists:
André Bellaïche: The tangent space in sub-Riemannian geometry Mikhael Gromov: Carnot-Carathéodory spaces seen from within Richard Montgomery: Survey of singular geodesics Héctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers Jean-Michel Coron: Stabilization of controllable systems
List of contents
The tangent space in sub-Riemannian geometry.-
1. Sub-Riemannian manifolds.-
2. Accessibility.-
3. Two examples.-
4. Privileged coordinates.-
5. The tangent nilpotent Lie algebra and the algebraic structure of the tangent space.-
6. Gromov's notion of tangent space.-
7. Distance estimates and the metric tangent space.-
8. Why is the tangent space a group?.- References.- Carnot-Carathéodory spaces seen from within.- 0. Basic definitions, examples and problems.-
1. Horizontal curves and small C-C balls.-
2. Hypersurfaces in C-C spaces.-
3. Carnot-Carathéodory geometry of contact manifolds.-
4. Pfaffian geometry in the internal light.-
5. Anisotropic connections.- References.- Survey of singular geodesics.-
1. Introduction.-
2. The example and its properties.-
3. Some open questions.-
4. Note in proof.- References.- A cornucopia of four-dimensional abnormal sub-Riemannian minimizers.-
1. Introduction.-
2. Sub-Riemannian manifolds and abnormal extremals.-
3. Abnormal extremals in dimension 4.-
4. Optimality.-
5. An optimality lemma.-
6. End of the proof.-
7. Strict abnormality.-
8. Conclusion.- References.- Stabilization of controllable systems.-
0. Introduction.-
1. Local controllability.-
2. Sufficient conditions for local stabilizability of locally controllable systems by means of stationary feedback laws.-
3. Necessary conditions for local stabilizability by means of stationary feedback laws.-
4. Stabilization by means of time-varying feedback laws.-
5. Return method and controllability.- References.
