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Ryuma ORITA
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Geometry IB

Term 2, Mon. Period 3 & Thu. Period 2 · Faculty of Science, Niigata University

Syllabus

Note

This course is held online via Zoom.

Main references

  1. Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Dover Publications

Recommended reading

  1. S. Kobayashi, “曲線と曲面の微分幾何（改訂版）” (in Japanese), Shokabo
  2. M. Umehara and K. Yamada, “曲線と曲面（改訂版）” (in Japanese), Shokabo

Course contents

  1. Spheres
  2. The implicit function theorem and the inverse function theorem
  3. Definition of regular surfaces
  4. Examples of regular surfaces
  5. The implicit function theorem and regular surfaces, part 1
  6. The implicit function theorem and regular surfaces, part 2; $C^{\infty}$ functions on surfaces, part 1
  7. $C^{\infty}$ functions on surfaces, part 2; $C^{\infty}$ maps between surfaces, part 1
  8. $C^{\infty}$ maps between surfaces, part 2; tangent planes
  9. Differentials
  10. The first fundamental form, part 1 (definition, examples, lengths of curves on surfaces)
  11. The first fundamental form, part 2 (loxodromes, areas of regions on surfaces)
  12. The Gauss map
  13. Self-adjoint maps and quadratic forms; the second fundamental form
  14. The second fundamental form; the various curvatures
  15. Gauss’s Theorema Egregium and the Gauss–Bonnet theorem
  16. End-of-term exam and summary