Finslerian Geometries Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board: LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, u.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland GUNTER LUDWIG, Philipps-Universitiit, Marburg, Germany NATHAN ROSEN, Israel Institute of Technology, Israel ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, , MENDEL SACHS, State University of New York at Buffalo, U.S.A. ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy HANS-JURGEN TREDER, Zentralinstitut fur Astrophysik der Akademie der Wissenschaften, Germany

Volume 109 Finslerian Geometries A Meeting of Minds

edited by

P.L. Antonell i DeJXlrlmem of Mathematical Sciences, University ofAlberta, EdmonlOn. , Canada

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C .I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-5838-4 ISBN 978-94-011-4235-9 (eBook) DOI 10.1007/978-94-011-4235-9

Printed an arid1ree paper

AH Rights Reserved © 2000 Springer Science+Business Media Dordrecht Originally published by K!uwer Academic Publishers in 2000 Soticovcr repri Il! of the hardcover l s( cd ition in 20()(} No part of the material protected by this copyright natice may be reproduced Of utilized in any farm Of by any means, electronic or mechanical, induding pholocopying, recording or by any informat ion slorage and relrieval system, withoUI wrinen permissian from the copyright owner. TABLE OF CONTENTS

Preface vii

SECTION I. PEDAGOGY 1 Generalizations of Finsler Geometry 3 M. Anastasiei and D. Hrimiuc Finsler Geometry Inspired 9 L. Kozma and L. Tamassy Finsler Geometry 15 H. Shimada and V.S. Sabiiu SECTION II. SUMMARY AND OVERVIEW 25 Summary and Overview 27 P.L. Antonelli

SECTION III. MEETING OF MINDS 33 Some Remarks On the Conformal Equivalence of Complex Finsler Structures 35 T. Aikou Deformations of Finsler Metrics 53 M. Anastasiei and H. Shimada The Constant Sprays of Classical Ecology and Noisy Finsler Perturbations 67 P.L. Antonelli On the Geometry of a Homogeneous Contact 'fransformation 79 P.L. Antonelli and D. Hrimiuc On Finsler Spaces of Douglas Type III 89 S. Bacso and M. Matsumoto

Equations of Motion from Finsler Geometric Methods 95 R.C. Beil

v vi Antonelli

On the Theory of Finsler Submanifolds 111 A. Bejancu Finslerian Fields 131 H.E. Brandt On the Inverse Problem of the Calculus of Variations for Systems of Second-Order Ordinary Differential Equations 139 M. Crampin Complex Finsler Geometry Via the Equivalence Problem on the Tangent Bundle 151 J.J. Faran, V Levy Concentration of Metric Measure Manifolds 169 W. Gu and Z. Shen Hypersurfaces in Generalised Lagrange Spaces 179 M. Kitayama The Notion of Higher Order Finsler Space. Theory and Applications 193 R. Miron Generalized Complex Lagrange Spaces 209 G. Munteanu Gravity in Finsler Spaces 223 S.F. Rutz and F.M. Paiva Higher Order Ecological Metrics 245 V.S. Sabau Area and Metrical Connections in Finsler Space 263 L. Tamassy Problem 281 L. Tamassy Finslerian Convexity and Optimization 283 C. Udri§te On Projective Transformations and Conformal Transformations of the Tangent Bundles of Riemannian Manifolds 297 K. Yamauchi PREFACE

The International Conference on Finsler and Lagrange Geometry and its Applications: A Meeting of Minds, took place August 13-20, 1998 at the in , Canada. The main objective of this meeting was to help acquaint North American geometers with the extensive modern literature on Finsler geometry and Lagrange geometry of the Japanese and European schools, each with its own venerable history, on the one hand, and to communicate recent advances in stochastic theory and Hodge theory for Finsler manifolds by the younger North American school, on the other. The intent was to bring together practitioners of these schools of thought in a Canadian venue where there would be ample opportunity to exchange information and have cordial personal interactions. The present set of refereed papers begins ·with the Pedagogical Sec• tion I, where introductory and brief survey articles are presented, one from the Japanese School and two from the European School ( and Hungary). These have been prepared for non-experts with the intent of explaining basic points of view. The Section III is the main body of work. It is arranged in alphabetical order, by author. Section II gives a brief account of each of these contribu• tions with a short reference list at the end. More extensive references are given in the individual articles. Whereas, there is only one paper (Antonelli's) in the present collec• tion dealing with stochastic Finsler theory, the larger body of work in this area and the related Hodge theory has already appeared in [1], [2]. These two volumes together with the present one will give readers a reasonably complete picture of recent developments in this field.

References

[11 Antonelli, P.L. and Zastawniak, T.J. (1999) Fundamentals 0/ Finsle• nan Diffusion with Applications, Fundamental Theories of Physics Series, Kluwer, Dordrecht.

vii viii Antonelli

[2] Antonelli, P.L. and Lackey, B.C. (eds.) (1998) The Theory of Finslerian Laplacians and Applications, Series on Mathemati~ and its Applications, Kluwer, Dordrecht.

Acknowledgements Funding for the Meeting of Minds Conference was provided by the U ni• versity of Alberta and by NSERC. The editor would like to thank Dra• gos Hrimiuc and Brad Lackey for their invaluable assistance in coordinating the conference. He would also like to thank Vivian Spak for her typically excellent typesetting of this book.

P. L. Antonelli Edmonton

March1999