Connections, Sprays and Finsler Structures

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.

Contents:

• Modules, Algebras and Derivations
• Manifolds and Bundles
• Vector Fields, Tensors and Integration
• Structures on Tangent Bundles
• Sprays and Lagrangians
• Covariant Derivatives
• Theory of Ehresmann Connections
• Geometry of Spray Manifolds
• Finsler Norms and Finsler Functions

Readership: Undergraduate and graduate students of mathematics, graduate students of physics, researchers and professionals in differential geometry and mathematical physics.
Key Features:

• This book integrates the theory of Finsler manifolds into the framework of vector bundles, covariant derivatives, Ehresmann connections and sprays. This makes it possible to develop the theory in a rigorous, yet very transparent manner, and to demonstrate how the general apparatus works in a concrete situation
• The authors give a detailed exposition of a major part of the necessary background material, including the indispensable tools of abstract algebra and multivariable calculus. Therefore, the book also applies to theoretical physicists interested in Finsler geometry and its applications. The text is completely self-contained for them
• The book discusses several theorems, together with their complete proofs, which have been available only in journal articles