These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. They should be more than sufficient for a semester-long course. We discuss smooth curves and surfaces — the main gate to differential geometry. We focus on the techniques that are absolutely essential for further study, keeping it problem-centered, elementary, visual, and virtually rigorous.
Curves
1. Definitions
2. Length
3. Curvature
4. Torsion
5. Signed curvature
6. Supporting curves
Surfaces
7. Definitions
8. First-order structure
9. Curvatures
10. Supporting surfaces
Geodesics
11. Shortest paths
12. Geodesics
13. Parallel transport
14. Gauss–Bonnet formula
15. Semigeodesic charts
16. Comparison theorems
Please email us comments/corrections/suggestions.
Paper version is available at amazon; its exact copy can be downloaded at arXiv.
The source files can be used as you want; see the license below.