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Ryuma ORITA
  1. Teaching/

Geometry IA

Term 1, 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. Parametrized differentiable curves; inner products
  2. Exercises
  3. Regular curves; arc length; orientation of vector spaces
  4. Exercises
  5. Cross products; curvature of curves; principal normal vectors
  6. Exercises
  7. Binormal vectors; torsion
  8. Exercises
  9. The Frenet–Serret formulas; the fundamental theorem of space curves
  10. Exercises
  11. Local canonical form; closed curves
  12. Exercises
  13. The Jordan curve theorem; the rotation index theorem; the four-vertex theorem
  14. Exercises
  15. Proof of the four-vertex theorem; the isoperimetric inequality
  16. End-of-term exam and summary