The present book has been written by two mathematicians and one physicist: a pure mathematician specializing in Finsler geometry (Makoto Matsumoto), one working in mathematical biology (Peter Antonelli), and a mathematical physicist specializing in information thermodynamics (Roman Ingarden). The main purpose of this book is to present the principles and methods of sprays (path spaces) and Finsler spaces together with examples of applications to physical and life sciences. It is our aim to write an introductory book on Finsler geometry and its applications at a fairly advanced level. It is intended especially for graduate students in pure mathemat ics, science and applied mathematics, but should be also of interest to those pure "Finslerists" who would like to see their subject applied. After more than 70 years of relatively slow development Finsler geometry is now a modern subject with a large body of theorems and techniques and has math ematical content comparable to any field of modern differential geometry. The time has come to say this in full voice, against those who have thought Finsler geometry, because of its computational complexity, is only of marginal interest and with prac tically no interesting applications. Contrary to these outdated fossilized opinions, we believe "the world is Finslerian" in a true sense and we will try to show this in our application in thermodynamics, optics, ecology, evolution and developmental biology. On the other hand, while the complexity of the subject has not disappeared, the modern bundle theoretic approach has increased greatly its understandability.

Klappentext

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.

Inhalt
Preface.- 0. Introductory Geometrical Background.- 1. Finsler Metrics.- 2. Connections in Finsler Spaces.- 3. Special Finsler Spaces.- 4. Finslerian Physics.- 5. Finslerian Biology.- References.