Sub-Riemannian Geometry by Andre BellaicheSub-Riemannian Geometry by Andre Bellaiche

Sub-Riemannian Geometry

byAndre BellaicheEditorJean-Jaques Risler

Paperback | October 18, 2011

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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

Title:Sub-Riemannian GeometryFormat:PaperbackDimensions:398 pagesPublished:October 18, 2011Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034899467

ISBN - 13:9783034899468


Table 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.