Introduction to Differential Geometry and General Relativity

Lecture Notes by Stefan Waner,
Department of Mathematics, Hofstra University

These notes are dedicated to the memory of Hanno Rund.

1. Preliminaries: Distance, Open Sets, Parametric Surfaces and Smooth Functions
2. Smooth Manifolds and Scalar Fields
3. Tangent Vectors and the Tangent Space
4. Contravariant and Covariant Vector Fields
5. Tensor Fields
6. Riemannian Manifolds
7. Locally Minkowskian Manifolds: A Little Relativity
8. Covariant Differentiation
9. Geodesics and Local Inertial Frames
10. The Riemann Curvature Tensor
11. A Little More Relativity: Comoving Frames and Proper Time
12. The Stress Tensor and the Relativistic Stress-Energy Tensor
13. Three Basic Premises of General Relativity
14. The Einstein Field Equations and Derivation of Newton's Law
15. The Schwarzschild Metric and Event Horizons
16. White Dwarfs, Neutron Stars and Black Holes by Gregory C. Levine

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References and Suggested Further Reading
(Listed in the rough order reflecting the degree to which they were used)

Bernard F. Schutz, A First Course in General Relativity (Cambridge University Press, 1986)
David Lovelock and Hanno Rund, Tensors, Differential Forms, and Variational Principles (Dover, 1989)
Charles E. Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus (Cambridge University Press, 1963)
Charles W. Misner, Kip S. Thorne and John A. Wheeler, Gravitation (W.H. Freeman, 1973)
Keith R. Symon, Mechanics (3rd. Ed. Addison Wesley)

Further Reading on the Web
For a comprehensive catalog of internet sites on special and general relativity, visit Relativity on the Web.

Last Updated: January, 2002
Copyright © Stefan Waner
Stop by at the Finite Mathematics and Applied Calculus Resource.