CONTEMPORARY MATHEMATICS 196 Finsler Geometry Joint Summer Research Conference on Finsler Geometry July 16-20, 1995 Seattle, Washington David Boo Shiing-shen Chern Zhongmin Shen Editors http://dx.doi.org/10.1090/conm/196 Selected Titles in This Series 196 David Bao, Shiing-shen Chern, and Zhongmin Shen, Editors, Finsler geometry, 1996 195 Warren Dicks and Enric Ventura, The group fixed by a family of injective endomorphisms of a free group, 1996 194 Seok-Jin Kang, Myung-Hwan Kim, and Insok Lee, Editors, Lie algebras and their representations, 1996 193 Chongying Dong and Geoffrey Mason, Editors, Moonshine, the Monster, and related topics, 1996 192 Tomek Bartoszynski and Marion Scheepers, Editors, Set theory, 1995 191 Tuong Ton-That, Kenneth I. Gross, Donald St. P. Richards, and Paul J. Sally, Jr., Editors, Representation theory and harmonic analysis, 1995 190 Mourad E. H. Ismail, M. Zuhair Nashed, Ahmed I. Zayed, and Ahmed F. Ghaleb, Editors, Mathematical analysis, wavelets, and signal processing, 1995 189 S. A. M. Marcantognini, G. A. Mendoza, M. D. Moran, A. Octavio, and W. 0. Urbina, Editors, Harmonic analysis and operator theory, 1995 188 Alejandro Adem, R. James Milgram, and Douglas C. Ravenel, Editors, Homotopy theory and its applications, 1995 187 G. W. Brumfiel and H. M. Hilden, SL(2) representations of finitely presented groups, 1995 186 Shreeram S. Abhyankar, Walter Feit, Michael D. Fried, Yasutaka Ihara, and Helmut Voelklein, Editors, Recent developments in the inverse Galois problem, 1995 185 Raul E. Curto, Ronald G. Douglas, Joel D. Pincus, and Norberto Salinas, Editors, Multivariable operator theory, 1995 184 L. A. Bokut', A. I. Kostrikin, and S. S. Kutateladze, Editors, Second International Conference on Algebra, 1995 183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995 182 Selman Akbulut, Editor, Real algebraic geometry and topology, 1995 181 Mila Cenkl and Haynes Miller, Editors, The Cech Centennial, 1995 180 David E. Keyes and Jinchao Xu, Editors, Domain decomposition methods in scientific and engineering computing, 1994 179 Yoshiaki Maeda, Hideki Omoro, and Alan Weinstein, Editors, Symplectic geometry and quantization, 1994 178 Helene Barcelo and Gil Kalai, Editors, Jerusalem Combinatorics '93, 1994 177 Simon Gindikin, Roe Goodman, Frederick P. Greenleaf, and Paul J. Sally, Jr., Editors, Representation theory and analysis on homogeneous spaces, 1994 176 David Ballard, Foundational aspects of "non" standard mathematics, 1994 175 Paul J. Sally, Jr., Moshe Flato, James Lepowsky, Nicolai Reshetikhin, and Gregg J. Zuckerman, Editors, Mathematical aspects of conformal and topological field theories and quantum groups, 1994 174 Nancy Childress and John W. Jones, Editors, Arithmetic geometry, 1994 173 Robert Brooks, Carolyn Gordon, and Peter Perry, Editors, Geometry of the spectrum, 1994 172 Peter E. Kloeden and Kenneth J. Palmer, Editors, Chaotic numerics, 1994 171 Riidiger Gobel, Paul Hill, and Wolfgang Liebert, Editors, Abelian group theory and related topics, 1994 170 John K. Beem and Krishan L. Duggal, Editors, Differential geometry and mathematical physics, 1994 169 William Abikoff, Joan S. Birman, and Kathryn Kuiken, Editors, The mathematical legacy of Wilhelm Magnus, 1994 (Continued in the back of this publication) CoNTEMPORARY MATHEMATICS 196 Finsler Geometry Joint Summer Research Conference on Finsler Geometry July 16-20, 1995 Seattle, Washington David Boo Shiing-shen Chern Zhongmin Shen Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andy Magid Michael Vogelius Clark Robinson Peter M. Winkler This volume features proceedings from the 1995 Joint Summer Research Con- ference on Finsler Geometry held in Seattle, WA, July 16~20, 1995. Supported by NSF grant DMS-9221892-003. 1991 Mathematics Subject Classification. Primary 53B40, 53C60. Library of Congress Cataloging-in-Publication Data Joint Summer Research Conference on Finsler Geometry (1995 : Seattle, Wash.) Finsler geometry: Joint Summer Research Conference on Finsler Geometry, July 16-20, 1995, Seattle, Washington / David Bao, Shiing-shen Chern, Zhongmin Shen, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 196) Includes bibliographical references. ISBN 0-8218-0507-X (alk. paper) 1. .Finsler spaces-Congresses. 2. Geometry, Differential-Congresses. I. Bao, David Dai- Wai. II. Chern, Shiing-Shen, 1911-. III. Shen, Zhongmin, 1963-. IV. Title. V. Series: Con- temporary mathematics (American Mathematical Society); v. 196. QA689.J65 1995 516.3'75-dc20 96-21779 CIP Copying and reprinting. Material in this book may be reproduced by any means for educational 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 acknowledgment 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-permission!Dams. 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.) © Copyright 1996 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. 0 Printed on recycled paper. 10987654321 01 00 99 98 97 96 Contents Foreword ............................................................... vii Paul Finsler and friends ............................................... ix Section 1 Finsler Geometry over the Reals 0. Preface ................................................................ 3 1. D. Bao, S.S. Chern and Z. Shen: On the Gauss-Bonnet integrand for 4-dimensional Landsberg spaces ... 15 2. R. Bryant: Finsler structures on the 2-sphere satisfying K = 1 .................... 27 3. G.D. Chakerian: Integral geometry in Minkowski spaces ................................ 43 4. S.S. Chern: Riemannian geometry as a special case of Finsler geometry ............ 51 5. F.J. Flaherty: Dirac operators for Finsler spaces ..................................... 59 6. D. Bao and B. Lackey: Special eigenforms on the sphere bundle of a Finsler manifold .......... 67 7. M. Matsumoto: Remarks on Berwald and Landsberg spaces ............................ 79 8. Z. Shen: Finsler manifolds of constant positive curvature ....................... 83 Section 2 Complex Finsler Geometry 0. Preface ............................................................... 97 1. M. Abate and G. Patrizio: Finsler metrics of constant curvature and the characterization of tube domains ............................. 101 2. T. Aikou: Some remarks on locally conformal complex Berwald spaces .......... 109 3. J. Bland and M. Kalka: Variations of holomorphic curvature for Kahler Finsler metrics ....... 121 4. J. Faran: The equivalence problem for complex Finsler Hamiltonians ........... 133 5. S. Kobayashi: Complex Finsler vector bundles ...................................... 145 v vi CONTENTS Section 3 Generalized Finsler Metrics 0. Preface .............................................................. 157 1. M. Anastasiei: Certain generalizations of Finsler metrics ............................. 161 2. M. Anastasiei: A historical remark on the connections of Chern and Rund ........... 171 3. L. Kozma: On Landsberg spaces and holonomy of Finsler manifolds ............. 177 4. R. Miron: Some problems in the geometries of Finsler type ...................... 187 Section 4 Applications to Biology, Engineering, and Physics 0. Preface .............................................................. 199 1. P. Antonelli, M. Matsumoto and T. Zastawniak: On y-Berwald spaces of dimension two and associated heterochronic systems with Finslerian noise ................ 203 2. R. lngarden: On physical applications of Finsler geometry ......................... 213 Section 5 Applications to Control Theory 0. Preface .............................................................. 227 1. R. Gardner and G. Wilkens: A pseudo-group isomorphism between control systems and certain generalized Finsler structures ............. 231 2. G. Wilkens: Finsler geometry in low dimensional control theory ................... 245 Section 6 Applications to Relativistic Field Theory 0. Preface .............................................................. 261 1. R. Beil: Finsler geometry and a unified field theory ........................... 265 2. H. Brandt: Finslerian spacetime ................................................. 273 3. S. Rutz: Symmetry in Finsler spaces .......................................... 289 4. J. Vargas and D. Torr: Elementary geometries underlying the theory of Euclidean connections on Finsler metrics of Lorentzian signature ..... 301 Foreword Ever since Riemann's 1854 address and Finsler's 1918 thesis, the