FINSLER AND LAGRANGE GEOMETRIES

Finsler and Lagrange Geometries Proceedings of a Conference held on August 26-31, Ia~i, Romania

Edited by

M. Anastasiei Faculty of Mathematics, University "AI. I.Cuza" la§i, la§i, Romania

and

P.L. Antonelli Department of Mathematical Sciences, University of Alberta, Edmonton, Canada

Springer Science+Business Media, LLC

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6325-0 ISBN 978-94-017-0405-2 (eBook) DOl 10.1007/978-94-017-0405-2

Printed on acid-free paper

All Rights Reserved © 2003 Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 2003. Softcover reprint of the hardcover 1 st edition 2003

No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

TABLE OF CONTENTS

Foreword ............ . Professor Peter Louis Antonelli at sixty Preface Section 1 . . . . . . . . . .

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SECTION 1. LAGRANGE AND HAMILTON GEOMETRY AND AND APPLICATIONS IN CONTROL

Curvature tensors on complex Lagrange spaces Aldea Nicoleta .......... . 3

Symplectic structures and Lagrange geometry Mihai A nastasiei ...................... 9

A geometrical foundation for Seismic ray theory based on modern Finsler geometry

P.L. Antonelli, S.F. Rutz and M.A. Slawinski . . . . . . . 17 On a problem of M. Matsumoto and Z.Shen

Sandor Bacso . . . . . . . . . . . . . . . . . . . . . . . . 55 Metrical homogeneous 2 - 7r structures determined by a Finsler metric in tangent bundle

Victor Blanu/a and B. T. Hassan . . . . . . . . . . . . . 63 Nonholonomic frames for Finsler spaces with (0:,;3) metrics

loan Bucataru . . . . . . . . . . . . 69 Invariant submanifolds of a Kenmotsu manifold

Constantin Calin The Gaussian curvature for the indicatrix of a generalized Lagrange space

....... 77

Mircea Cra§mareanu ............. ...... 83 Infinitesimal projective transformations on tangent bundles

Izumi Hasegawa and Kazunari Yamauchi . . . . . . . . . . . . 91 Conformal transformations in Finsler geometry

B. T. Hassan and Fatma Mesbah . . . . . . . . . . . . . 99 Induced vector fields in a hypersurface of Riemannian tangent bundles

Masashi Kitayama ........... . On a normal cosymplectic manifold

Ion Mihai and Radu Ro§ca

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109

113

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The almost Hermitian structures determined by the Riemannian structures on the tangent bundle

Radu Miron and Koji Matsumoto .......... . On the semispray of nonlinear connections in rheonomic Lagrange geometry

Florian Munteanu . . . . . . . . . . . .c-dual complex Lagrange and Hamilton spaces

Gheorghe Munteanu ................... . Dirac operators on holomorphic bundles

Ovidiu Munteanu .................... . The generalised singular Finsler spaces

Tetsuya Nagano . . . . . . . . . . . . . . . . . . . . n- order dynamical systems and associated geometrical structures

M. Neamtu and V. ObCideanu ........... . The variational problem for Finsler spaces with (0, (3)- metric

Marcel Roman ..................... . On projectively flat Finsler spheres (Remarks on a theorem of R.L.Bryant)

Vasile Sorin SabCiu . . . . . . . . . . . . . . . . . . . . . On the corrected form of an old result:necessary and sufficient conditions of a Randers space to be of constant curvature

Hideo Shimada and Vasile Sorin SabCiu . . . . . . On the almost Finslerian Lagrange space of second order with (0, (3) metric

Co,Ui,lin Sterbeti and Brandu§a Nicolaescu . . . . Remarkable natural almost parakaehlerian structures on the tangent bundle

Emil Stoica Intrinsic geometrization of the variational Hamiltonian calculus

119

129

139

149

155

163

171

181

193

197

203

Ovidiu flie $andru .................... 213 Finsler spaces of Riemann- Minkowski type

L. Tamassy . . . . . . . . . . . . . . . . . . . . . .. 225 Finsler- Lagrange- Hamilton structures associated to control systems

Constantin Udri§te 233

Preface Section 2 . . . 245

SECTION 2. APPLICATIONS TO PHYSICS

Contraforms on pseudo-Riemannian manifolds M. Anastasiei, Gabriela Ciobanu and 1. Gottlieb

Finslerian (0, (3)- metrics in weak gravitational models Vladimir Balan and Panayotis C. Stavrinos . . .

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Table of Contents

Applications of adapted frames to the geometry of black holes Liviu Popescu . . . . . . . . . . . . . . . . . . .

Implications of homogeneity in Miron's sense in gauge theories of second order

Adrian Sandovici ................ .

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269

277 The free geodesic connection and applications to physical field theories

Thomas P. Storer . . . . . . . . . . . . . . . . . . . .. 287 The geometry of non-inertial frames

Ion $andru ....... . Self-duality equations for gauge theories

Gheorghe Zet and Vasile Manta

303

313

FOREWORD

In the last decade several international conferences on Finsler, Lagrange and Hamilton geometries were organized in Bra§ov, Romania (1994), Seattle, USA (1995), Edmonton, Canada (1998), besides the Seminars that periodically are held in Japan and Romania. All these meetings produced important progress in the field and brought forth the appearance of some reference volumes. Along this line, a new International Conference on Finsler and Lagrange Geometry took place August 26-31,2001 at the "Al.I.Cuza" University in Ia§i, Romania.

This Conference was organized in the framework of a Memorandum of Un­derstanding (1994-2004) between the "Al.I.Cuza" University in Ia§i, Romania and the University of Alberta in Edmonton, Canada. It was especially dedicated to Prof. Dr. Peter Louis Antonelli, the liaison officer in the Memorandum, an untired promoter of Finsler, Lagrange and Hamilton geometries, very close to the Romanian School of Geometry led by Prof. Dr. Radu Miron. The dedica­tion wished to mark also the 60th birthday of Prof. Dr. Peter Louis Antonelli. With this occasion a Diploma was given to Professor Dr. Peter Louis Antonelli conferring the title of Honorary Professor granted to him by the Senate of the oldest Romanian University (140 years), the "Al.I.Cuza" University, Ia§i, Roma­nia. There were almost fifty participants from Egypt, Greece, Hungary, Japan, Romania, USA.

There were scheduled 45 minutes lectures as well as short communications. All offered opportunities to learn more about the subject and to exchange infor­mation. A social program helpful for cordial personal interactions was organized.

This Volume contains the texts of the short communications and the ab­stracts of reasonable length of the lectures presented at the Conference. The Editors saw the contents of these but the responsibility regarding the ideas and correctness of the results belongs entirely to the authors. The authors provided files upon which this Volume was built.

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The Volume is organized into sections. The first section contains 28 papers on Finsler, Lagrange and Hamilton geometries and the second section includes 7 papers on applications of these geometries to Physics. Each section contains a Preface in which its content is presented. The Volume is opened by an article presenting a short biography and the main scholarly achievements of Professor P.L. Antonelli, as he celebrated his 60th birthday.

June 2001

lalli, Romania Prof. Dr. M. Anastasiei

Acknowledgements

The editors would like to express their sincere thanks to Vivian Spak, who typeset this book, and to Scott Berard, who kept our computers running.

PROFESSOR DR. PETER LOUIS ANTONELLI AT SIXTY

It is the duty of the mathematical community to mark in one way or another the important events from the life of its remarkable gifted members. Such an event is the 60th birthday of Prof. Dr. Peter Louis Antonelli, an outstanding scholar from the University of Alberta, Edmonton, Canada, well-known on all meridians for his contributions in Differential Topology, Differential Geometry and Applications of Mathematics to Biology and Ecology. It is our pleasure to celebrate this event recording the main deeds of his life.

Professor Dr. Peter Louis Antonelli was born on March 5, 1941 in Syracuse, N.Y., United States. He obtained the B. Sc. Degree in 1963 at the Syracuse University, the M. Sc. Degree and then the Ph.D. in 1967 at the same Uni­versity. In the academic year 1967-68 he worked as Assistant Professor at the University of Tennessee, Knoxville. From 1968 to 1970 he was a National Sci­ence Foundation Fellow at the Institute for Advanced Studies, in Princeton, New Jersey, USA. He then moved as Associate Professor to the University of Alberta in Edmonton, Canada where he has been full Professor since 1980.

The Ph.D thesis of Prof. Dr. P. L. Antonelli was titled "Montgomery- Samel­son Fiberings Between Manifolds", his advisor being Prof. Dr. Erik Hemingsen. These fiberings are important in many respects and the young Antonelli solved some difficult problems regarding them. For example, he proposed a structure theory of Montgomery-Samelson fiberings, and studied Montgomery- Samelson fiberings of spheres, and those having finite singular sets. The results he ob­tained were published in some important American Journals as Proc. Amer. Math. Soc. and Bull. Amer. Math. Soc. In this period he met P.J. Kahn, a stu­dent of Fields medalist J. Milnor, and the Romanian, D. Burghelea. Together, they solved some hard problems on the diffeomorphism group of manifolds such as spheres and exotic spheres. Their researches were published in two parts, the first as Lectures Notes in Mathematics 215, Springer- Verlag, 1970 with the ti­tle "The Concordance-Homotopy Group of Geometric Automorphism Groups" and the other, a fifty-page article in the journal Topology entitled "The Non­Finite Homotopy Type of Some Diffeomorphism Groups". The importance of this work was reflected in publication of three BAMS articles summarizing their work.

Since 1970 the interests of Antonelli have been towards the Applications of Mathematics, especially differential geometry, developmental biology, genetics and ecology. He used the academic year 1972-1973 as Visiting Professor at

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the University of Sussex, Biology Department, England and at Galton Genetics Laboratory, London to educate himself in these branches of Applied Mathemat­ics, continuing what he started as a graduate student in mathematical biology at the University of Chicago in 1963-64, where he was a United States Public Health Service Fellow.

On his 14th birthday, he received a calculus text book and, at 17, he won a mathematics contest and acquired a copy of L.P. Eisenhart "Riemannian Ge­ometry", which he has to this day. In these early years, his interests focused in Physics, especially the theory of General Relativity, being a familiar preco­cious regular figure at the Syracuse University Physics department. As a young adult, special and even strange mathematical objects as exotic spheres and spe­cial groups of diffeomorphisms attracted him, which he preferred first to treat in an example and then to look for a general theory. Later on, Antonelli became a practitioner of mathematics as a whole. For him Mathematics appears as a powerful tool for solving problems of the Sciences. He experienced this with a great success. Over a period of thirty years he succeeded to find a common language with famous specialists as N. Kazarinoff, R. Bradbury, P. Sammarco, C. Strobeck, K. Morgan, R. Elliot and R. Seymour. As post-doctoral fellows (37), he had V. Krivan, T. Zastawniak, B. Lackey, 1. Bucataru, D. Hrimiuc and S. F. Rutz, to name a few. They constructed some powerful mathematical models from which were extracted very useful practical results. These models have raised new mathematical problems leading Prof. Antonelli to establish a set of valuable results in several fields of Mathematics, both Pure and Applied.

A glance over the more than 120 research papers published by Prof. Dr. P. L. Antonelli shows how much mathematical knowledge is incorporated. These pa­pers, many of them in collaboration with specialists in certain particular fields, fall in domains as diffusion theory, nonlinear mechanics, geometric probability, stochastic calculus and stochastic geometry, differential game theory, bifurca­tion theory, Hamiltonian systems, geometry of paths, Riemannian, Finslerian and Lagrangian geometries. There are very few mathematicians handling so many different fields and able to mobilize so many different people for a fruitful collaboration.

We would like to say more about the achievements of Prof. Dr. P.L. An­tonelli in Finslerian and Lagrangian geometries. He started by introducing and studying a mathematical model for treating the Volterra-Hamilton equations in Biology in which some special Riemannian and Finslerian metrics were involved. At an international meeting (Debrecen, Hungary, 1992), Prof. Dr. Radu Miron raised the problem of whether the use of a Lagrangian metric could provide a better model for those equations. As a reply Antonelli not only constructed new and better models but also identified and solved new theoretical problems. For example, he dealt with the problem of stability of geodesics for special mth-root metrics (with H. Shimada); he clarified the concept of constant Finslerian con­nection (with M. Matsumoto); he discovered and studied a new class of Lagrange manifold (with M. Anastasiei and D. Hrimiuc); he developed a stochastic cal­culus and a theory of diffusion on Finsler manifold (with T. Zastawniak). This work became the standard text in the field (Kluwer Acad. Press). But, a few

P.L. Antonelli at Sixty xiii

years earlier, Antonelli together with M. Matsumoto and R. lngarden, published a fundamental book on applications of Finsler geometry to Physics and Biology. Prof. Dr. P. L. Antonelli definitely has put his mark upon the geometry of non­Riemannian metrics. Such metric spaces now bear his name. What's more, he always enthusiastically promoted the Lagrange and Hamilton geometries, espe­cially recently, when his interest turned to Seismology and which is reported on in the 3rd article of this Proceedings.

In 2001, upon his 60th birthday, Prof. Antonelli was awarded the degree of Honorary Professor of the "Alexandru loan Cuza" University.

Over the years, Antonelli published alone or in collaboration several books (5) at international publishing houses and acted as editor for a number of volumes published by Kluwer Academic Publishers. He not only frequently publishes important research papers but he is very active and involved in the life of the scientific and academic communities. He actually travelled over the whole world delivering lectures, attending scientific meetings, generally as in­vited speaker or organizing sections at international congresses. Also, he likes to organize scientific meetings and then carefully edits and publishes the pro­ceedings of such meetings (6).

Prof. Peter Antonelli is of an optimistic nature and a very pleasant person. His opinions are always interesting and elegantly expressed. He is a enthusiastic teacher and enjoys introducing the students in the research activities.

We take this opportunity to congratulate him for his outstanding achieve­ments in research and teaching and to wish him power to work and to create in good health and happiness.

June 2001

Ia§i, Romania Prof. Dr. Radu Miron, Member of the Romanian Academy Prof. Dr. Mihai Anastasiei

PREFACE SECTION 1 Lagrange and Hamilton Geometry and Applications in Control and Seismology

Each article in this section has an abstract for more detailed information on a specific article. Here I deal with a more general perspective. Thus, articles fall into several distinct categories. Complex manifolds are of concern in the papers of A. Nicoleta, F. Munteanu, G. Munteanu and o. Muntenau, whereas, Lagrange and Hamilton Geometry are explicitly covered in articles of M. Anas­tasiei, N. Bucataru, M. Cra§mareanu, 1. Mihai and R. Ro§ca, R. Miron and K. Matsumoto, and OJ. ~andru.

There are several articles on Finsler geometry proper, namely, those by S. Bacso, T. Nagano, M. Roman and V.S. Sabau and finally the important works of E. Stoica, L. Tamassy, H. Shimada and V.S. Sabau on Randers Spaces of Constant Curvature.

There are two articles on applications, one in Control Theory, by C. Udri§te and one on Seismology by P.L. Antonelli, S. Rutz and M.A. Slawinski. This article, in particular, displays the power of the software package FINSLER based on MAPLE developed by S.F. Rutz and R. Portugal in Rio de Janeiro.

June 2001

Ia§i, Romania Prof. P.L. Antonelli

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