What is differential geometry: curves and surfaces

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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.

Table of contents:

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

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Paper version is available at lulu.com; its exact copy can be downloaded at arXiv.

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