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
Inhaltsverzeichnis
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.
