Sub-Riemannian Geometry (Google eBook)

Front Cover
André Bellaïche, Jean-Jaques Risler
Springer Science & Business Media, Sep 26, 1996 - Mathematics - 393 pages
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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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Contents

The tangent space
1
Accessibility
10
Two examples
23
Privileged coordinates
30
The tangent nilpotent Lie algebra and the algebraic structure
43
Gromovs notion of tangent space
54
Why is the tangent space a group?
73
MIKHAEL GROMOV
82
Anisotropic connections
302
Survey of singular geodesics
325
The example and its properties
331
Note in proof
337
abnormal subRiemannian minimizers
341
Abnormal extremals in dimension 4
351
An optimality lemma
357
Conclusion
363

Basic definitions examples and problems
85
Horizontal curves and small CC balls
112
Hypersurfaces in CC spaces
152
CarnotCaratheodory geometry of contact manifolds
196
Pfaffian geometry in the internal light
234
Necessary conditions for local stabilizability by means
371
Return method and controllability
380
Index
389
Copyright

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