This book provides a comprehensive introduction to Finsler geometry
in the language of present-day mathematics. Through Finsler
geometry, it also introduces the reader to other structures and
techniques of differential geometry. Prerequisites for reading the
book are minimal: undergraduate linear algebra (over the reals) and
analysis. The necessary concepts and tools of advanced linear
algebra (over modules), point set topology, multivariable calculus
and the rudiments of the theory of differential equations are
integrated in the text. Basic manifold and bundle theories are
treated concisely, carefully and (apart from proofs) in a
self-contained manner. The backbone of the book is the detailed and
original exposition of tangent bundle geometry, Ehresmann
connections and sprays. It turns out that these structures are
important not only in their own right and in the foundation of
Finsler geometry, but they can be also regarded as the cornerstones
of the huge edifice of Differential Geometry. The authors emphasize
the conceptual aspects, but carefully elaborate calculative aspects
as well (tensor derivations, graded derivations and covariant
derivatives). Although they give preference to index-free methods,
they also apply the techniques of traditional tensor calculus. Most
proofs are elaborated in detail, which makes the book suitable for
self-study. Nevertheless, the authors provide for more advanced
readers as well by supplying them with adequate material, and the
book may also serve as a reference.
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