Geometry I (2021 Fall)

Textbook: M. do Carmo, Differential Geometry of curves and surfaces.
Room: 綜三203
Time: T89R78
因應疫情發展，第一週及第二週改為純線上上課，直播連結是這裡。
分組: (A組) 學號小於等於108021171 (27人) (B組) otherwise (28人)。
Tutorial: Wed 7-9 pm , 綜三203, 助教: 徐靖家。習題課前幾堂助教做了錄影。

期中考及期末考除了公假、病假、喪假外，其餘事假均不接受。公假、喪假要事先有假單，生病可事後請假但需付考試當時就診證明(包含被要求居家檢疫)。

由於疫情不穩定，下面的進度只是預估。考試日期也是預估，很可能會有改變，請同學注意網頁的UPDATE

Course Description

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

1. Regular surfaces
2. Tangent spaces, Differential of a Map
3. First fundamental form
4. Gauss Map, Gauss Theorem
5. Second fundamental form
6. Parallel transport, geodesics
7. Gauss-Bonnet Theorem

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，並且下學期還會有其他教材。另外，如同我第一堂課說的，這門課會使用很多線性代數以及一些高微，這些是這門課的pre-requirement。

References

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.

Syllabus (目前暫時維持正常課表時間T8R78，如果之後要改成各一半會再公布)

9/14(二): outline
9/15(三): (Tutorial: 第一週不上習題課)
9/16(四): 2.2: definition of surfaces, examples
9/21(二): National Holiday
9/22(三): (Tutorial: 2.2)
9/23(四): 2.3: regular value theorem , definitions (such as regular surfaces, differentiable functions, etc.) don't depend on the coordinate system chosen
9/28(二): University Holiday
9/29(三): (Tutorial: 2.3)
9/30(四): 2.4: tangent planes, differential of a map between surfaces
10/5(二): 2.4: tangent planes, differential of a map between surfaces
10/6(三): (Tutorial: 2.4)
10/7(四): 2.5: first fundamental form, surface area
10/12(二): 2.5: first fundamental form, surface area
10/13(三): (Tutorial: 2.5)
10/14(四): 2.6: orientation
10/19(二): 3.2: Gauss map
10/20(三): (Tutorial: 2.6)
10/21(四): 3.2: differential of Gauss map, second fundamental form
10/26(二): 3.2: differential of Gauss map, second fundamental form
10/27(三): (Tutorial: 2.6, 3.2)
10/28(四): 3.2: Gaussian curvature, mean curvature, Dupin indicatrix
11/2(二): 開會停課，時間補在 11/16 期中考)
11/3(三): (Tutorial: 3.2)
11/4(四): 3.3: Gauss map in local coordinate

11/9(二): 3.3: Gauss map in local coordinate

11/10(三): (Tutorial: 部分 3.3 )

11/11(四): 3.3: calculating the principal direction and the asymptotic direction

11/16(二): Midterm: chapter 2.1-2.6, 3.2 (時間注意是89堂)

11/17(三): (Tutorial: University Holiday)

11/18(四): 3.3: Geometric meaning of the Gauss curvature

11/23(二): 3rd fundamental form

11/24(三): (Tutorial: 檢討考卷)

11/25(四): 3.5: minimal surface

11/30(二): 3.5: ruled surface

12/1(三): (Tutorial: 3.3 & 部分 3.5)

12/2(四): 3.5: ruled surface, 3.4: vector fields

12/7(二): 3.4: vector fields, special parametrizations (上兩堂，i.e.是89堂)

12/8(三): (Tutorial: 部分 3.5 & 3.4)

12/9(四): 4.2: isometry, 4.3: intrinsic geometry

12/14(二): 4.4: parallel transport

12/15(三): (Tutorial: 4.2)

12/16(四): 4.4: parallel transport

12/21(二): 4.4: geodesic

12/22(三): (Tutorial: 4.3)

12/23(四): 4.4: geodesic

12/28(二): 4.4: geodesic

12/29(三): (Tutorial: 4.4)

12/30(四): 4.4: geodesic curvature

1/4(二): 課移到 12/7

1/5(三): (Tutorial: 4.4)

1/6(四): Final exam

1/13(四): (在教室)看考卷 3:30 - 4:30 pm

Homework

2.2: 1,5,6,7,8,13,16,19
2.3: 1,3,7,8,14,16
2.4: 11,12,13,15,17,19,20,21
2.5: 1,3,5,6,9(related to HW3),11,14
2.6: 1(also assume that both V_1&V_2 are connected),2,4,5
2.8: read the proposition, this is HW4
3.2: 2,5,6,8,9,14,15,16,17,18
3.3: 2,5,15,16,20,21
3.4: 5,6,7,8,10
3.5: 11,12,13,14
4.2: 3,4,7,8,11,12,13,18
4.3: 1,2,3,4,5,6,8
4.4: 1,2,5,6,7,8,9,10,12,13,15

Evaluation (期末成績完全依計算公式得到)

Homework 20%
Midterm exam 40%
Final exam 40%

Last Updated: January 7, 2022
URL: http://www.math.nthu.edu.tw/~nankuo/G2021F.html