| Lecture 1: Manifolds | 1 |
| Lecture 2: Smooth Maps | 9 |
| Lecture 3: Submanifolds | 17 |
| Lecture 4: Tangent Vectors | 23 |
| Lecture 5: Lie Groups | 35 |
| Lecture 6: Differential Equations | 49 |
| Lecture 7: Lie Brackets and Integrability | 59 |
| Lecture 8: Connections | 71 |
| Lecture 9: Riemannian Metrics | 87 |
| Lecture 10: The Levi-Civita Connection | 95 |
| Lecture 11: Geodesics and Completeness | 101 |
| Lecture 12: Tensors | 109 |
| Lecture 13: Differential Forms | 119 |
| Lecture 14: Stokes' Theorem | 129 |
| Lecture 15: de Rham Cohomology | 137 |
| Lecture 16: Curvature | 151 |
| Lecture 17: Extrinsic Curvature | 165 |
| Lecture 18: The Gauss and Codazzi Equations | 171 |
| Lecture 19: Cartan's Moving Frame Method | 177 |
| Lecture 20: The Gauss-Bonnet Theorem | 181 |