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 Contrary to these outdated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.
