Table of Contents

Introduction 1. Gauss's Theorem Egregium 2. Rules of Calculus 3. Connections on the Tangent Bundle 4. Levi-Civita's Theorem 5. Bi-invariant Metrics on a Lie Group 6. Cartan Calculations 7. Gauss's Lemma 8. Variational Formulas 9. The Hopf-Rinow Theorem 10. Curvature, Distance and Volume 11.Review of Special Relativity 12. The Star Operator and Electromagnetism 13. Preliminaries to the Einstein Equation 14. Die Grundlagen der Physik 15. The Frobenius Theorem 16. Connections on Principal Bundles 17. Reduction of Principal Bundles 18. Superconnections 19. Semi-Riemannian Submersions Bibliography Index