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" Sub-Riemannian Geometry "
edited by André Bellaïche, Jean-Jacques Risler.
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BL
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| Record Number
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577098
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b406317
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| Main Entry
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Bellaïche, André.
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| Title & Author
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Sub-Riemannian Geometry\ edited by André Bellaïche, Jean-Jacques Risler.
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| Publication Statement
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Basel :: Birkhäuser Basel,, 1996.
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| Series Statement
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Progress in Mathematics ;; 144
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| ISBN
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9783034892100
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: 9783034899468
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| Contents
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The tangent space in sub-Riemannian geometry -- {sect} 1. Sub-Riemannian manifolds -- {sect} 2. Accessibility -- {sect} 3. Two examples -- {sect} 4. Privileged coordinates -- {sect} 5. The tangent nilpotent Lie algebra and the algebraic structure of the tangent space -- {sect} 6. Gromov's notion of tangent space -- {sect} 7. Distance estimates and the metric tangent space -- {sect} 8. Why is the tangent space a group? -- References -- Carnot-Carathéodory spaces seen from within -- {sect} 0. Basic definitions, examples and problems -- {sect} 1. Horizontal curves and small C-C balls -- {sect} 2. Hypersurfaces in C-C spaces -- {sect} 3. Carnot-Carathéodory geometry of contact manifolds -- {sect} 4. Pfaffian geometry in the internal light -- {sect} 5. Anisotropic connections -- References -- Survey of singular geodesics -- {sect} 1. Introduction -- {sect} 2. The example and its properties -- {sect} 3. Some open questions -- {sect} 4. Note in proof -- References -- A cornucopia of four-dimensional abnormal sub-Riemannian minimizers -- {sect} 1. Introduction -- {sect} 2. Sub-Riemannian manifolds and abnormal extremals -- {sect} 3. Abnormal extremals in dimension 4 -- {sect} 4. Optimality -- {sect} 5. An optimality lemma -- {sect} 6. End of the proof -- {sect} 7. Strict abnormality -- {sect} 8. Conclusion -- References -- Stabilization of controllable systems -- {sect} 0. Introduction -- {sect} 1. Local controllability -- {sect} 2. Sufficient conditions for local stabilizability of locally controllable systems by means of stationary feedback laws -- {sect} 3. Necessary conditions for local stabilizability by means of stationary feedback laws -- {sect} 4. Stabilization by means of time-varying feedback laws -- {sect} 5. Return method and controllability -- References.
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| Abstract
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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
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Mathematics.
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Global analysis.
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Global differential geometry.
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Risler, Jean-Jacques.
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SpringerLink (Online service)
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