ABSTRACT

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.

chapter Chapter 1|8 pages

Introduction

chapter Chapter 2|38 pages

Smooth Manifolds and Vector Bundles

chapter Chapter 3|22 pages

Vector Fields and Differential Equations

chapter Chapter 4|48 pages

Tensors

chapter Chapter 5|46 pages

Differential Forms

chapter Chapter 6|18 pages

Integration on Manifolds

chapter Chapter 7|66 pages

Metric and Symplectic Structures

chapter Chapter 8|46 pages

Lie Groups

chapter Chapter 9|38 pages

Group Actions

chapter Chapter 10|144 pages

Fibre Bundles

chapter Chapter 11|50 pages

Isometric Immersions and The Second Fundamental Form

chapter Chapter 12|66 pages

Jet Bundles