Geometry IB

Term 2, Mon. Period 3 & Thu. Period 4 · Faculty of Science, Niigata University

Syllabus

Note

Held on Mon. Period 3 & Thu. Period 4.

Main references

• M. Umehara and K. Yamada, “曲線と曲面（改訂版）” (in Japanese), Shokabo

Recommended reading

1. S. Kobayashi, “曲線と曲面の微分幾何（改訂版）” (in Japanese), Shokabo
2. Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Dover Publications

Course contents

1. What is a surface?
2. Exercises
3. The first fundamental form, part 1 (definition)
4. Exercises
5. The first fundamental form, part 2 (areas, lengths, and angles)
6. Exercises
7. The second fundamental form, part 1 (definition, curvatures)
8. Exercises
9. The second fundamental form, part 2 (examples, invariance)
10. Exercises
11. Normal curvature; principal directions
12. Exercises
13. Umbilical points; asymptotic directions; geodesics
14. Exercises
15. Theorema Egregium; the Gauss–Bonnet theorem
16. End-of-term exam and summary