This book provides a systematic presentation of the mathematical foundation of modern physics with applications particularly within classical mechanics and the theory of relativity. Written to be self-contained, this book provides complete and rigorous proofs of all the results presented within. Among the themes illustrated in the book are differentiable manifolds, differential forms, fiber bundles and differential geometry with non-trivial applications especially within the general theory of relativity. The emphasis is upon a systematic and logical construction of the mathematical foundations. It can be used as a textbook for a pure mathematics course in differential geometry, assuming the reader has a good understanding of basic analysis, linear algebra and point set topology. The book will also appeal to students of theoretical physics interested in the mathematical foundation of the theories.
Table of Contents
Introduction. Smooth manifolds and vector bundles. Vector fields and differential equations. Tensors. Differential forms. Integration on manifolds. Metric and symplectic structures. Lie groups. Group actions. Fibre bundles. Isometric immersions and the second fundamental form. Jet bundles. Appendix.
Before his retirement in 2014, Steinar Johannesen was an associate professor of mathematics at Oslo and Akershus University College of Applied Sciences, Norway