Sub-Riemannian Geometry
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Series Editors H. Bass J. Oesterle A. Weinstein
Sub-Riemannian Geometry Andre Bellalche J ean-J acques Risler Editors
Birkhauser Verlag Basel . Boston . Berlin
Editors: Andre Bellalche Departement de Mathematiques Universite Paris 7 - De nis Diderot 2, place Jussieu F-75251 Paris 5e
Jean-Jacques Risler Universite Paris VI - Pierre et Marie Curie F-75252 Paris 5e
1991 Mathematics Subject Classification 53C99, 58E25, 93B29, 49L99
A C IP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Biblio thek Cataloging-in-Publication Data Sub-Riemannian geometry I Andre Bellaiche ; Jean-Jacques Risler ed. - Basel ; Boston ; Be rlin : Birkhliuser, 1996 (Progress in mathematics; Vol. 144) ISBN- 13: 978-3-0348-9946-8 e-ISBN-13: 978-3-0348-9210-0 DOl : 10.1 0071978-3-0348-9210-0 NE: Bellaiche, Andre IH rsg.]; GT
This work is subject to copyright. All righ ts are reserved, whether the whole or part of the material is concerned, specificall y the rights of translation, reprinting, re- use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 1996 Birkhliuser Ve rlag, P. O. Box 133, CH-40JO Basel, Switzerland Softcover reprin t of the hardcover 1st edition 1996 Printed on acid-free paper produced of chlorine-free pulp. TCF 00
ISBN-13: 978-3-0348-9946-8 9876543 21
Preface Following a suggestion by Hector J. Sussmann we organized, in the summer of 1992 in Paris, a satellite meeting of the first European Congress of Mathematics. The topic of the meeting was "Nonholonomy" , and officially titled: JOURNEES NONHOLONOMES
Geometrie sous-riemannienne, theorie du controle, robotique It was held at Universite Paris VI-Pierre et Marie Curie (Jussieu), on June 30th and July 1st, 1992.
Sub-Riemannian Geometry (also known as Carnot Geometry in France, and Nonholonomic Riemannian Geometry in Russia) has been a fullyfledged 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); • Gauge theories; • Diffusions on manifolds; • Analysis of hypoelliptic operators; and • 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 (e.g., the conference paper by Agrachev at the 1994 International Mathematical Congress in Zurich). To illustrate this fact, it should be noted that the first editor of this volume was interested in nonholonomy, following encouragement by Robert Azencott to provide a geometric frame for the study of non-elliptic diffusions. The second editor, a specialist in real algebraic geometry, c
