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Introduction to Differential Geometry & General Relativity 6th Printing May 2014 Lecture Notes by Stefan Waner with a Special Guest Lecture by Gregory C. Levine Departments of Mathematics and Physics, Hofstra University Introduction to Differential Geometry and General Relativity Lecture Notes by Stefan Waner, with a Special Guest Lecture by Gregory C. Levine Department of Mathematics, Hofstra University These notes are dedicated to the memory of Hanno Rund. TABLE OF CONTENTS 1. Preliminaries ......................................................................................................3 2. Smooth Manifolds and Scalar Fields................................................................8 3. Tangent Vectors and the Tangent Space .......................................................16 4. Contravariant and Covariant Vector Fields .................................................26 5. Tensor Fields ....................................................................................................37 6. Riemannian Manifolds ....................................................................................42 7. Locally Minkowskian Manifolds: An Introduction to Relativity ................52 8. Covariant Differentiation ................................................................................63 9. Geodesics and Local Inertial Frames .............................................................71 10. The Riemann Curvature Tensor ................................ ...
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