Blick ins Buch

Sub-Riemannian Geometry

Andre Bellaiche, Jean-Jaques Risler (Herausgeber)

Buch | Hardcover

VIII, 398 Seiten

1996 | 1996
Springer Basel (Verlag)
978-3-7643-5476-3 (ISBN)

Lese- und Medienproben

Blick ins Buch (midvox)

Artikel merken

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

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.

Erscheint lt. Verlag	26.9.1996
Reihe/Serie	Progress in Mathematics
Zusatzinfo	VIII, 398 p.
Verlagsort	Basel
Sprache	englisch
Maße	155 x 235 mm
Gewicht	753 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Algebra • Boundary element method • Control • Control Theory • Extrema • Feedback • Field • Fusion • Geometrie • Geometry • Group • Hardcover, Softcover / Mathematik/Geometrie • HC/Mathematik/Geometrie • manifold • Mathematics • Mathematik • metrics • NATURAL • Riemannian Geometry
ISBN-10	3-7643-5476-3 / 3764354763
ISBN-13	978-3-7643-5476-3 / 9783764354763
Zustand	Neuware