This book presents, in a concise and direct manner, the appropriate mathematical formalism and fundamentals of differential topology and differential geometry, together with essential applications in many branches of physics.

Differential geometry has encountered numerous applications in physics. More and more physical concepts considered as fundamental can be understood as a direct consequence of geometric principles. The mathematical structure of Maxwell's electrodynamics, general theory of relativity, string theory and gauge theories, to name but a few, are of a geometric nature. All of these disciplines require a curved space for the description of a system, and we require a mathematical formalism that can handle the dynamics in such spaces if we wish to go beyond a simple and superficial discussion of physical relationships. This formalism is differential geometry. Even areas like thermodynamics and fluid mechanics greatly benefit from a differential geometric treatment. Not only in physics, but in important branches of mathematics, has differential geometry effected important changes. Aimed at graduate students, and requiring only linear algebra and differential and integral calculus, this book presents, in a concise and direct manner, the appropriate mathematical formalism and fundamentals of differential topology and differential geometry, together with essential applications in many branches of physics.

Preface

Notation

1. Synopsis of General Relativity

2. Curves and Surfaces in E^3

3. Elements of Topology

4. Differentiable Manifolds

5. Tangent Vectors and Tangent Spaces

6. Tensor Algebra

7. Tensor Fields and Commutators

8. Differential Forms and Exterior Calculus

9. Maps Between Manifolds

10. Integration on Manifolds

11. Integral Curves and Lie Derivatives

12. Linear Connections

13. Geodesics

14. Torsion and Curvature

15. Pseudo-Riemannian Metric

16. Newtonian Space-Time and Thermodynamics

17. Special Relativity, Electrodynamics, and the Poincaré Group

18. General Relativity

19. Gravitational Radiation

Bibliography