Bellaiche

Sub-Riemannian Geometry

Herausgegeben:Bellaiche, Andre; Risler, Jean-Jaques

• Gebundenes Buch

Produktbeschreibung

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

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Produktdetails

• Progress in Mathematics 144
• Verlag: Birkhäuser / Birkhäuser Basel / Springer, Basel
• Artikelnr. des Verlages: 978-3-7643-5476-3
• 1996.
• Seitenzahl: 412
• Erscheinungstermin: 26. September 1996
• Englisch
• Abmessung: 241mm x 160mm x 28mm
• Gewicht: 753g
• ISBN-13: 9783764354763
• ISBN-10: 3764354763
• Artikelnr.: 22925029

Inhaltsangabe

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.