P.L. Antonelli - Böcker

This volume presents the principles and methods of sprays (path spaces) and Finsler spaces with many applications in the physical and life sciences. Beginning from the classical theory of sprays, the first chapter presents an introduction to modern Finsler differential geometry. The following three chapters can serve as a comprehensive graduate course using the notions of pre-Finsler connections in spray bundles. Topics covered are the calculus of variations and Finsler metric functions, spaces of constant curvature, projective and conformal geometry, two-dimensional Finsler spaces, and Beswald spaces. Chapter 4 deals with the Finslerian view of dissipative mechanics, thermodynamics and information, and geometrical and electron optics. Chapter 5 discusses, from a Finslerian perspective, ecological problems and models, with particular reference to the Great Barrier Reef. Spray connection theory is shown to be indispensable for a logically consistent theory of social interactions. Projective Finsler geometry and Wagner connection theory are used to model time-sequencing changes in growth and development.Some direct applications to fossil measurements in paleontology are also described. For geometers, physicists and theoretical (marine) biologists, the book can also be used as a supplementary graduate text.

Since 1992 Finsler geometry, Lagrange geometry and their applications to physics and biology, have been intensive1y studied in the context of a 5-year program called "Memorandum ofUnderstanding", between the University of Alberta and "AL.1. CUZA" University in lasi, Romania. The conference, whose proceedings appear in this collection, belongs to that program and aims to provide a forum for an exchange of ideas and information on recent advances in this field. Besides the Canadian and Romanian researchers involved, the conference benefited from the participation of many specialists from Greece, Hungary and Japan. This proceedings is the second publication of our study group. The first was Lagrange Geometry. Finsler spaces and Noise Applied in Biology and Physics (1]. Lagrange geometry, which is concerned with regular Lagrangians not necessarily homogeneous with respect to the rate (i.e. velocities or production) variables, naturalIy extends Finsler geometry to alIow the study of, for example, metrical structures (i.e. energies) which are not homogeneous in these rates. Most Lagrangians arising in physics falI into this class, for example.Lagrange geometry and its applications in general relativity, unified field theories and re1ativistic optics has been developed mainly by R. Miron and his students and collaborators in Romania, while P. Antonelli and his associates have developed models in ecology, development and evolution and have rigorously laid the foundations ofFinsler diffusion theory [1] .

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the 16th century. Its phylogeny and history are presented in the Prologue of this volume. The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives. Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenbock formulas, and Finslerian spinors and Dirac operators. The tools developed in this book should find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua.This text should be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.

This is a text on stochastic Finslerian geometry.The theory is rigorously presented and several applications in ecology, evolution and epidemiology are described. Amongst the various topics covered are the role of curvature in Finslerian diffusions, Nelson's stochastic mechanics, non-linear (Finslerian) filtering and entropy production. Two appendices deal with, respectively, the stochastic Hodge theory of Finslerian harmonic forms, and the theory of 2-dimensional Finsler spaces. The latter plays an important role in the applications described in the text. This volume should be of interest to probabilists, applied mathematicians, mathematical biologists and geometers. It can also be recommended as a supplementary graduate text.

This work covers advances in Finslerian geometries, such as anisotropic geometries, and their applications by the Japanese, European and American schools. It contains three introductory articles, one from each of these schools, giving a broad overview of basic ideas. Further papers treat topics from pure mathematics such as complex differential geometry, equivalence methods, Finslerian deformations, constant sprays, homogeneous contact transformations, Douglas spaces, submanifold theory, inverse problems, area theory, and more. This book completes the Kluwer trilogy on Finslerian Geometry by P.L. Antonelli and his associates. It should be of interest to physicists and mathematicians whose work involves quantum field theory, combination theory and relativity, programming and optimization. Mathematical biologists working in ecology and evolution will also find it useful.

This text gives the most recent results in Finsler and related geometries from the Miron school. Both pure and applied topics are covered, including: higher-order geometry, Hamilton and Cartan spaces, Legendre transformations, self-duality in Gauge fields, constant curvature spaces, electromagnetics, gravity theory, black holes, complex Finsler geometry and Finsler-Lagrange-Hamilton structures in control and optimization. There is also an article on Finsler Seismic ray theory which uses the software Finsler based on MAPLE.

It was over three years ago, at the annual meeting of the American Ma- ematical Society in San Diego, California, that Dr. Paul Roos of Kluwer asked me to poll Finsler geometers around the world as to their interest in writing a HANDBOOK OF FINSLER GEOMETRY. The result of that query was a resounding a?rmation, and here at long last, is the ?nal result. You have in your hands,the most complete and authoritative exposition of state-of-the-art Finsler geometry that is possible, today. Each of the eleven partsiscompletely independent of the rest, and each has been written with mathematics and science students in mind. These articles are accessible! P. L. Antonelli Edmonton, Alberta, Canada June, 2003 xv 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. xvii HANDBOOK OF FINSLER GEOMETRY Volume 2 PART 1 Complex Finsler Geometry TadashiAikou Contents 1K...ahler Fibrations 9 1. 1 Fibrations...9 1. 2 LocalTreatments...10 1. 3 Bott Connections...13 1. 3. 1 StructureTensors...14 1. 3. 2 Bott Connections...16 1. 4 K..ahlerFibration ...18 2Complex Finsler Bundles 23 2.1 Vector Bundles Over Complex Projective Space ...23 2. 2 ComplexFinslerMetrics...27 2. 2. 1 Complete Circular Domains and Minkowski Functionals . 27 r r?1 2. 2. 2 Complex Finsler Metrics on C and Kahler ...Metrics on P 30 2. 2. 3 Complex Finsler Metric on Vector Bundles ...

This volume presents the principles and methods of sprays (path spaces) and Finsler spaces with many applications in the physical and life sciences. The book has six chapters and an extensive reference section. Beginning from the classical theory of sprays, Chapter 0 presents an introduction to modern Finsler differential geometry. The following three chapters can serve as a comprehensive graduate course using the notions of pre-Finsler connections in spray bundles. Topics covered are the calculus of variations and Finsler metric functions, spaces of constant curvature, projective and conformal geometry, two-dimensional Finsler spaces, Beswald spaces, (alpha, beta)-metrics, etc. Chapter 4 deals with the Finslerian view of dissipative mechanics, thermodynamics and information, and geometrical and electron optics. Chapter 5 discusses, from a Finslerian perspective, ecological problems and models, with particular reference to the Great Barrier Reef. Spray connection theory is shown to be indispensable for a logically consistent theory of social interactions.Projective Finsler geometry and Wagner connection theory are used to model time-sequencing changes in growth and development. Some direct applications to fossil measurements in paleontology are also described. For geometers, physicists and theoretical (marine) biologists, the book can also be recommended as a supplementary graduate text.

Since 1992 Finsler geometry, Lagrange geometry and their applications to physics and biology, have been intensive1y studied in the context of a 5-year program called "Memorandum ofUnderstanding", between the University of Alberta and "AL.1. CUZA" University in lasi, Romania. The conference, whose proceedings appear in this collection, belongs to that program and aims to provide a forum for an exchange of ideas and information on recent advances in this field. Besides the Canadian and Romanian researchers involved, the conference benefited from the participation of many specialists from Greece, Hungary and Japan. This proceedings is the second publication of our study group. The first was Lagrange Geometry. Finsler spaces and Noise Applied in Biology and Physics (1]. Lagrange geometry, which is concerned with regular Lagrangians not necessarily homogeneous with respect to the rate (i.e. velocities or production) variables, naturalIy extends Finsler geometry to alIow the study of, for example, metrical structures (i.e. energies) which are not homogeneous in these rates. Most Lagrangians arising in physics falI into this class, for example.Lagrange geometry and its applications in general relativity, unified field theories and re1ativistic optics has been developed mainly by R. Miron and his students and collaborators in Romania, while P. Antonelli and his associates have developed models in ecology, development and evolution and have rigorously laid the foundations ofFinsler diffusion theory [1] .

This text gives the most recent results in Finsler and related geometries from the Miron school. Both pure and applied topics are covered. For example Higher-Order geometry, Hamilton and Cartan spaces, Legendre transformations, self-duality in Gauge fields, constant curvature spaces, Electromagnetics, Gravity theory, Black Holes, complex Finsler geometry and Finsler-Lagrange-Hamilton structures in control and optimization. There is also an article on Finsler Seismic ray theory which uses the software FINSLER based on MAPLE.

It was over three years ago, at the annual meeting of the American Ma- ematical Society in San Diego, California, that Dr. Paul Roos of Kluwer asked me to poll Finsler geometers around the world as to their interest in writing a HANDBOOK OF FINSLER GEOMETRY. The result of that query was a resounding a?rmation, and here at long last, is the ?nal result. You have in your hands,the most complete and authoritative exposition of state-of-the-art Finsler geometry that is possible, today. Each of the eleven partsiscompletely independent of the rest, and each has been written with mathematics and science students in mind. These articles are accessible! P. L. Antonelli Edmonton, Alberta, Canada June, 2003 xv 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. xvii HANDBOOK OF FINSLER GEOMETRY Volume 2 PART 1 Complex Finsler Geometry TadashiAikou Contents 1K...ahler Fibrations 9 1. 1 Fibrations...9 1. 2 LocalTreatments...10 1. 3 Bott Connections...13 1. 3. 1 StructureTensors...14 1. 3. 2 Bott Connections...16 1. 4 KahlerFibration ...18 2Complex Finsler Bundles 23 2.1 Vector Bundles Over Complex Projective Space ...23 2. 2 ComplexFinslerMetrics...27 2. 2. 1 Complete Circular Domains and Minkowski Functionals . 27 r r?1 2. 2. 2 Complex Finsler Metrics on C and Kahler ...Metrics on P 30 2. 2. 3 Complex Finsler Metric on Vector Bundles ...

The International Conference on Finsler and Lagrange Geometry and its Applications: A Meeting of Minds, took place August 13-20, 1998 at the University of Alberta in Edmonton, 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 (Romania 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.

The erratic motion of pollen grains and other tiny particles suspended in liquid is known as Brownian motion, after its discoverer, Robert Brown, a botanist who worked in 1828, in London. He turned over the problem of why this motion occurred to physicists who were investigating kinetic theory and thermodynamics; at a time when the existence of molecules had yet to be established. In 1900, Henri Poincare lectured on this topic to the 1900 International Congress of Physicists, in Paris [Wic95]. At this time, Louis Bachelier, a thesis student of Poincare, made a monumental breakthrough with his Theory of Stock Market Fluctuations, which is still studied today, [Co064]. Norbert Wiener (1923), who was first to formulate a rigorous concept of the Brownian path, is most often cited by mathematicians as the father of the subject, while physicists will cite A. Einstein (1905) and M. Smoluchowski. Both considered Markov diffusions and realized that Brownian behaviour nd could be formulated in terms of parabolic 2 order linear p. d. e. 'so Further­ more, from this perspective, the covariance of changes in position could be allowed to depend on the position itself, according to the invariant form of the diffusion introduced by Kolmogorov in 1937, [KoI37]. Thus, any time­ homogeneous Markov diffusion could be written in terms of the Laplacian, intrinsically given by the symbol (covariance) of the p. d. e. , plus a drift vec­ tor. The theory was further advanced in 1949, when K.

Finslerian Laplacians have arisen from the demands of modelling the modern world. However, the roots of the Laplacian concept can be traced back to the sixteenth century. Its phylogeny and history are presented in the Prologue of this volume. The text proper begins with a brief introduction to stochastically derived Finslerian Laplacians, facilitated by applications in ecology, epidemiology and evolutionary biology. The mathematical ideas are then fully presented in section II, with generalizations to Lagrange geometry following in section III. With section IV, the focus abruptly shifts to the local mean-value approach to Finslerian Laplacians and a Hodge-de Rham theory is developed for the representation on real cohomology classes by harmonic forms on the base manifold. Similar results are proved in sections II and IV, each from different perspectives. Modern topics treated include nonlinear Laplacians, Bochner and Lichnerowicz vanishing theorems, Weitzenböck formulas, and Finslerian spinors and Dirac operators. The tools developed in this book will find uses in several areas of physics and engineering, but especially in the mechanics of inhomogeneous media, e.g. Cofferat continua. Audience: This text will be of use to workers in stochastic processes, differential geometry, nonlinear analysis, epidemiology, ecology and evolution, as well as physics of the solid state and continua.