Term 1, Mon. Period 3 & Tue. Period 2 · Faculty of Science, Niigata University
Textbooks
- M. Umehara and K. Yamada, “曲線と曲面（改訂版）” (in Japanese), Shokabo
Main references
- S. Kobayashi, “曲線と曲面の微分幾何（改訂版）” (in Japanese), Shokabo
Recommended reading
- Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition, Dover Publications
Course contents
- What is a curve?
- Curvature
- The Frenet formulas
- The fundamental theorem of plane curves; the four-vertex theorem
- Space curves
- Midterm exam
- What is a surface?
- The first fundamental form, part 1 (definition)
- The first fundamental form, part 2 (areas and lengths)
- The first fundamental form, part 3 (angles, maps); the second fundamental form, part 1 (definition)
- The second fundamental form, part 2 (curvatures, invariance)
- The second fundamental form, part 3 (the Weingarten formulas); definition of normal curvature
- Normal and principal curvatures; geodesics
- Theorema Egregium; the Gauss-Bonnet theorem
- End-of-term exam and summary