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CONTEMPORARY MATHEMATICS 288 Global Differential Geometry: The Mathematical Legacy ·of Alfred Gray International Congress on Differential Geometry September 18-23, 2000 Bilbao, Spain Marisa Fernandez Joseph A. Wolf Editors http://dx.doi.org/10.1090/conm/288 Selected Titles In This Series 288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001 287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001 286 Edward L. Green, Serkan HO!~ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001 285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001 284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001 283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001 282 Arlan Ramsay and Jean Renault, Editors, Gruupoids in analysiR, gcnmetry, and physics, 2001 281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001 280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001 279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001 278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001 277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001 276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. Noel, Earl Barnes, and Sonya A. F. Stephens, Editors, Council for African American researchers in the mathematical sciences: Volume III, 2001 274 Ken-ichi Maruyama and John W. Rutter, Editors, Groups of homotopy self-equivalences and related topics, 2001 273 A. V. Kelarev, R. Gobel, K. M. Rangaswamy, P. Schultz, and C. Vinsonhaler, Editors, Abelian groups, rings and modules, 2001 272 Eva Bayer-Fluckiger, David Lewis, and Andrew Ranicki, Editors, Quadratic forms and their applications, 2000 271 J. P. C. Greenlees, Robert R. Bruner, and Nicholas Kuhn, Editors, Homotopy methods in algebraic topology, 2001 270 Jan Denef, Leonard Lipschitz, Thanases Pheidas, and Jan Van Geel, Editors, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, 2000 269 Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, Editors, Laminations and foliations in dynamics, geometry and topology, 2001 268 Robert Gulliver, Walter Littman, and Roberto Triggiani, Editors, Differential geometric methods in the control of partial differential equations, 2000 267 Nicolas Andruskiewitsch, Walter Ricardo Ferrer Santos, and Hans-Jiirgen Schneider, Editors, New trends in Hopf algebra theory, 2000 266 Caroline Grant Melles and Ruth I. Michler, Editors, Singularities in algebraic and analytic geometry, 2000 265 Dominique Arlettaz and Kathryn Hess, Editors, Une degustation topologique: Homotopy theory in the Swiss Alps, 2000 264 Kai Yuen Chan, Alexander A. Mikhalev, Man-Keung Siu, Jie-Tai Yu, and Efim I. Zelmanov, Editors, Combinatorial and computational algebra, 2000 (Continued in the back of this publication) Global Differential Geometry: The Mathematical Legacy of Alfred Gray Alfred Gray, Turin, 24 May 1994 Reprinted with permission from the Department of Mathematics, University of Turin. CoNTEMPORARY MATHEMATICS 288 Global Differential Geometry: The Mathematical Legacy of Alfred Gray International Congress on Differential Geometry September 18-23, 2000 Bilbao, Spain Marisa Fernandez Joseph A. Wolf Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor Andreas Blass Andy R. Magid Michael Vogelius This volume contains the proceedings of the International Congress on Differential Geometry held in memory of Professor Alfred Gray in Bilbao, Spain, September 18-23, 2000. 2000 Mathematics Subject Classification. Primary 22E15, 53A10, 53A30, 53B35, 53C15, 53C20, 53C23, 53C25, 53C55, 53D05. Library of Congress Cataloging-in-Publication Data International Congress on Differential Geometry (2000 : Bilbao, Spain) Global differential geometry: the mathematical legacy of Alfred Gray: International Congress on Differential Geometry, September 18-23, 2000, Bilbao, Spain / Marisa Fernandez, Joseph A. Wolf, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; v. 288) Includes bibliographical references. ISBN 0-8218-2750-2 (alk. paper) 1. Global differential geometry-Congresses. I. Gray, Alfred, 1939- II. Fernandez, Marisa III. Wolf, Joseph Albert IV. Title. V. Contemporary mathematics (American Mathematical Society) : v. 288. QA670 .155 2000 516.3'62--dc21 2001053300 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Assistant to the Publisher, American Mathematical Society, P. 0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionl!lams. org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) @ 2001 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http: I /www. ams. org/ 10 9 8 7 6 5 4 3 2 1 06 05 04 03 02 01 Contents Foreword xi Preface xiii Photograph of Alfred Gray Memorial Bronze XIV Acknowledgements XV List of publications of Alfred Gray XVII List of participants xxvii Invited contributions On the formality problem for symplectic manifolds IVAN K. BABENKO AND ISKANDER A. TAIMANOV 1 Osculating tubes and self-linking for curves on the three-sphere THOMAS F. BANCHOFF 10 Isoparametric functions and harmonic and minimal unit vector fields E. BOECKX AND L. VANHECKE 20 The Seiberg-Witten equations and almost-Hermitian geometry S. K. DONALDSON 32 Sculpture inspired by work with Alfred Gray: Kepler elliptic curves and minimal surface sculptures of the planets HELAMAN FERGUSON 39 Mesoscopic curvature and hyperbolicity MISHA GROMOV 58 Stable forms and special metrics NIGEL HITCHIN 70 A CR-Momentum Ansatz for nilpotent groups A. HUCKLEBERRY AND M. VOLLER 90 Can tangent sphere bundles over Riemannian manifolds have strictly positive sectional curvature? 0. KowALSKI, M. SEKIZAWA, AND Z. Vul.sEK 110 vii viii CONTENTS Volumes of geodesic balls and spheres associated to a metric connection with torsion VICENTE MIQUEL 119 Special isothermic surfaces and solitons EMILIO MUSSO AND LORENZO NICOLODI 129 The isoperimetric and Willmore problems ANTONIO Ros 149 Almost parallel structures SIMON SALAMON 162 Technical computing with Mathematica PAUL WELLIN 182 Complex geometry and representations of Lie groups JOSEPH A. WOLF 202 Communications On the volume of compact spacelike hypersurfaces with hyperbolic boundary in Minkowski space JUAN A. ALEDO AND JOSE A. PASTOR 239 Stability of zero mean curvature surfaces in flat Lorentzian 4-manifolds LUIS J. ALiAS AND BENNETT PALMER 244 Hitchin-Kobayashi correspondence for equivariant bundles on X x lP'1 LUIS ALVAREZ-CONSUL AND OSCAR GARciA-PRADA 249 Hopf vesicles in S3 (1) J. ARROYO AND 0. J. GARAY 258 Invariant nearly Kahler f -structures on homogeneous spaces VITALY V. BALASHCHENKO 263 An extension of A. Gray's investigations on small geodesic balls N. BOKAN, M. DJORIC, AND U. SIMON 268 An energy minimizing family of unit vector fields on odd-dimensional spheres VINCENT BORRELLI, FABIANO BRITO, AND OLGA GIL-MEDRANO 273 Is it admissible to linearize the Einstein equation in the presence of matter? LLUIS BRUNA AND JOAN GIRBAU 277 Solutions of some conformally invariant equations of the first order JAROLIM BURES 281 On biharmonic maps RENZO CADDEO, STEFANO MONTALDO, AND PAOLA PIU 286 Characterizations of Pfaffian systems. Consequences in dimension five MARIA A. CANADAS-PINEDO AND CEFERINO RUIZ 291 On a new class of contact Riemannian manifolds JONG TAEK CHO 296 CONTENTS ix Description of (1, 2)-symplectic metrics on flag manifolds NIR COHEN, CAIO J. C. NEGREIROS, AND LUIZ A. B. SAN MARTIN 300 Complex surfaces with Betti numbers b1 =1, b2 > 0 and finite quotients GEORGES DLOUSSKY 305 Hypercomplex nilpotent Lie groups ISABEL G. DOTTI AND ANNA FINO 310 Q:-spaces of Iwasawa type and Damek-Ricci spaces MARIA J. DRUETTA 315 Deformations of constant mean curvature surfaces in half space models M. ELISA E. L. GALVAO AND CELIA c. GOES 320 The geometry of the skew-symmetric