Geometry I (2025 Fall)

Course Description

The material that will be covered in the Fall semester includes (but not restricted to) the following:

這是一門兩學期的課(雖然對有些同學來說第二學期不是必修)，所以上面所列的內容不一定會在第一學期全部上完。不論上到哪裡，下學期就會從那裏開始接下去上。另外，這門課會使用多變數微積分，線性代數以及一些高微的知識，這些算是這門課的pre-requirement。

黃武雄,初等微分幾何講稿(第一卷，曲線篇，曲面篇)
A.N. Pressley, Elementary Differential Geometry. SUMS (這本書非常適合自己念，淺顯易懂)
B. O'Neill, Elementary Differential Geometry.
L.Tu, An Introduction to Manifolds, UTX.
M. do Carmo, Differential forms and applications, UTX.
S. Morita, Geometry of Differential forms.
I.M.Singer and J.A.Thorpe, Lecture notes on Elementary Topology and Geometry, UTM.
M.Spivak, A Comprehensive Introduction to Differential Geometry I.

9/2(二): Course outline (時間是第8堂)
9/3(三): (Tutorial: 第一週不上習題課)
9/4(四): 2.2: definition of surfaces, examples, and regular value theorem
9/9(二): 2.3: change of parametrizations, 2.4: tangent planes (時間注意是89堂)
9/10(三): (Tutorial: 2.2, 2.3)
9/11(四): 2.4: differential of a map between surfaces, 2.5: first fundamental form
9/16(二): 2.5: surface area, part of 2.8, and 2.6: orientability (時間注意是89堂)
9/17(三): (Tutorial: 2.3, 2.4)
9/18(四): 2.6: orientability; Review of change of parametrizations, some notes
9/23(二): 3.2: Gauss map, differential of Gauss map (時間注意是89堂)
9/24(三): (Tutorial: 2.4, 2.5 )
9/25(四): 時間移到 9/9 以及 9/16
9/30(二): 時間移到 9/23
10/1(三): (Tutorial: 2.5, 2.6)
10/2(四): 3.2: Second fundamental form
10/7(二): 3.2: Gaussian curvature, mean curvature, Dupin indicatrix (時間注意是89堂)
10/8(三): (Tutorial: 2.6, 3.2)
10/9(四)*: Finishing 3.2, start 3.3
10/14(二)*: 3.3: Gauss map in local coordinate (時間注意是89堂)
10/15(三): (Tutorial: 3.2)
10/16(四)*: 3.3: Gauss map in local coordinate
10/21(二)*: 3.3: calculating the principal direction and the asymptotic direction (時間注意是89堂)
10/22(三): (Tutorial: 3.2, 3.3)
10/23(四)*: Midterm: chapter 2.1-2.6, 3.2-3.3

10/28(二)*: 時間移到 10/7

10/29(三): (Tutorial: 3.3)

10/30(四)*: 時間移到 10/14 以及 10/21

11/4(二)*: 3.3: Geometric meaning of the Gauss curvature (時間注意是89堂)

11/5(三): (Tutorial: 檢討考卷 )

11/6(四)*: 3.3: 3rd fundamental form

11/11(二)*: 時間移到 11/4

11/12(三): (Tutorial: University Holiday (運動會))

11/13(四)*: 時間移到 11/18 以及 11/25

11/18(二)*: 3.5: minimal surface (時間注意是89堂)

11/19(三): (Tutorial: 3.5 )

11/20(四)*: 3.5: ruled surface, 3.4: vector fields

11/25(二)*: 3.4: special parametrizations 4.2: isometry (時間注意是89堂)

11/26(三): (Tutorial: 3.4, 4.3)

11/27(四)*: 4.3: intrinsic geometry

12/2(二)*: 4.4: parallel transport (時間注意是89堂)

12/3(三): (Tutorial: 4.2)

12/4(四)*: 4.4: geodesic

12/9(二)*: 4.5: Gauss-Bonnet Theorem, 2.6: orientation(時間注意是89堂)

12/10(三): (Tutorial: 4.3)

12/11(四)*: 時間移到 12/2 以及 12/9

12/16(二)*: Final exam

12/??: 看考卷

Last Updated: October 2, 2025
URL: http://www.math.nthu.edu.tw/~nankuo/G2025F.html