トピックアウトライン
- 一般
- Week 1. Curves in the Euclidean space (I).
Week 1. Curves in the Euclidean space (I).
Parameterized curves; arc-length parameterization.
Reading [dC] 1-4
Homework [dC] 1-2 #5, 1-3 #5, 6, 10, 1-5 #4, 5. (Due: 9/15 9:50AM)
[dC] do Carmo, Differential geometry of curves and surfaces, 2nd Ed. - Week 2. Curves in the Euclidean space (II).
Week 2. Curves in the Euclidean space (II).
The local theory and local canonical form for space curves.
Reading Review differential calculus on multivariate functions; for instance
[Adams, Chapter 13]: the definitions of partial derivatives;
[Rudin, Chapter 9]: derivatives of a multivariate function (Definition 9.11 and Theorem 9.17).Homework [dC] 1-5 #1, 7, 12, 13, 15, 17. (Due 9/22 9:50AM)
- Week 3. Basics about surfaces in the Euclidean spaces (I).
Week 3. Basics about surfaces in the Euclidean spaces (I).
Regular surfaces; change of parameters; differentiable functions on surfaces.
Reading [dC] Examples 4 and 5 in 2-3 (surfaces of revolution and tangent surfaces).
Homework [dC] 2-2 #11, 16, 17, 2-3 #5, 8, 13. (Due: 9/30 9:50AM)
- Week 4. Basics about surfaces in the Euclidean spaces (II).
Week 4. Basics about surfaces in the Euclidean spaces (II).
Tangent planes; the differential of a map; the first fundamental form.
Homework [dC] 2-4 #4, 10, 12, 13, 17, 21
- Week 5. Basics about surfaces in the Euclidean spaces (III).
Week 5. Basics about surfaces in the Euclidean spaces (III).
The first fundamental form; areas; orientation of surfaces.
Homework [dC] 2-5 #3, 9, 10, 13, 14.
- Week 6. Basics about surfaces in the Euclidean spaces (IV).
Week 6. Basics about surfaces in the Euclidean spaces (IV).
The definition of the Gauss map and its fundamental properties.
- Week 7. Basics about surfaces in the Euclidean spaces (V).
Week 7. Basics about surfaces in the Euclidean spaces (V).
The Gauss map in local coordinates (the 2nd fundamental form).
- Week 8. Basics about surfaces in the Euclidean spaces (VI).
Week 8. Basics about surfaces in the Euclidean spaces (VI).
Vector fields and the stronger form of the ODE theorem. Midterm.
- Week 9. Basics about surfaces in the Euclidean spaces (VII).
Week 9. Basics about surfaces in the Euclidean spaces (VII).
Vector fields and the stronger form of the ODE theorem.
- Week 10. Intrinsic geometry of surfaces (I).
- Week 11. Intrinsic geometry of surfaces (II).
- Week 12. Intrinsic geometry of surfaces (III).
- Week 13. Intrinsic geometry of surfaces (IV).
- Week 14. Intrinsic geometry of surfaces (V).
- Week 15. Global differential geometry (I).
Week 15. Global differential geometry (I).
First and second variation of arc-length; Bonnet's theorem.
- Week 16. Global differential geometry (II).
- Week 17.