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Sub-Riemannian Geometry

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概要 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続きを見る
目次 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?
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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
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Survey of singular geodesics
{sect} 1. Introduction
{sect} 2. The example and its properties
{sect} 3. Some open questions
{sect} 4. Note in proof
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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
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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
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登録日 2020.06.27
更新日 2020.06.28