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This textbook introduces the reader to modern differential geometry through Finsler geometry, without relying on any previous knowledge of differential geometry. In discussing the general theory of manifolds and bundles, the authors give a detailed, clear and precise description of the fundamental concepts and ideas. The discussion of the tangent bundle and some related structures is exhaustive. From that point, several useful subtleties and technical tricks often neglected in other textbooks are presented. In the early chapters, the conventions, notation and terminology are set out. The linear algebra to be used later is expounded, in a general form, on modules. These chapters contain the local calculus that is used in proving many important theorems on existence and uniqueness in the framework of manifolds. This calculus is developed without reference to bases, i.e. free of partial derivatives. The book discusses several theorems, together with their complete proofs, which have been available only in journal articles, thus they have been lacking a uniform and consistent exposition.
List of contents
Sets, Mappings, Operations; Modules, Algebras, Derivations; Calculus in Vector Spaces; Manifolds and Bundles; Tangent Bundle, Vector Fields, and Differential Forms; Covariant Derivatives; Ehresmann Connections and Sprays; Finsler Structures.
