課程資訊
課程名稱
幾何學導論
Introduction to Geometry 
開課學期
109-1 
授課對象
理學院  數學系  
授課教師
張瑞恩 
課號
MATH5356 
課程識別碼
221 U6580 
班次
 
學分
4.0 
全/半年
半年 
必/選修
必帶 
上課時間
星期三3,4(10:20~12:10)星期五3,4(10:20~12:10) 
上課地點
普503普503 
備註
此課程研究生選修不算學分。
限學士班學生
總人數上限:70人
外系人數限制:30人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1091MATH5356_ 
課程簡介影片
 
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課程概述

Applying basic calculus and linear algebra to study the surface inside 3 dimensional space gives fruitful results and forms the basis of modern differential geometry, which, in turns, provide the framework of general relativity and quantum field theory.

Our course would concentrate on the concept of "curvature", "surface" and the interplay between them. We shall conduct a course from the local behavior of a surface to the global property of it. 

課程目標
1. Curvature of Curves.
2. Curvature of Surfaces.
3. Internal Curvature of Surfaces.
4. Curvature and Global Property of Surfaces 
課程要求
 
預期每週課後學習時數
 
Office Hours
 
參考書目
 
指定閱讀
M. do Carmo, Differential Geometry of curves & surfaces
A. Pressley, Elementary Differential Geometry
The second book is available from Springer link (it can be downloaded from NTU IP)
https://link.springer.com/book/10.1007%2F978-1-84882-891-9 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
期中考 
35% 
 
2. 
期末考 
35% 
 
3. 
作業 
20% 
 
4. 
小考 
10% 
 
 
課程進度
週次
日期
單元主題
第1週
9/16, 9/18  9/16 開學,微分幾何簡介,先備知識,Parametric curves[AP Chap1, DC 1.1-4]
9/18 Plane curves[AP 2.2] 
第2週
9/23, 9/25, 9/26  9/23 Space curves[AP 2.1,2.3, DC 1.5-6]
9/25 Some other topics related to curves, calculus of variation
9/26 補10/2的課,助教課 
第3週
9/30  9/30 Quiz 1. Regular surfaces. [AP 4.1-2, DC 2.1-2]
10/02 中秋節放假 
第4週
10/07  10/07 The tangent space[AP 4.4, DC 2.3-4]
10/09 國慶日放假 
第5週
10/14, 10/16  10/14 The first fundamental form[AP 6.1, DC 2.5]
10/16 Mapping between surfaces[AP 4.3, DC 4.2] 
第6週
10/21, 10/23  10/21 Quiz 2. Orientation of surfaces, Gauss map[AP 4.5, DC 2.6]
10/23 The second fundamental form[AP 7.1-3, DC 3.2] 
第7週
10/28, 10/30  10/28 Mean curvature, Gauss curvature[AP 7.1-3, DC 3.2]
10/30 Curvature in local coordinates[AP 8.1-2, DC 3.3] 
第8週
11/04, 11/06  11/04 Topics related to mean curvature, minimal surface[AP 8.5, DC 3.5]
11/06 Ruled surface, Flat surface[DC 3.5, AP 8.4] 
第9週
11/11, 11/13  11/11 Quiz 3, Review
11/13 Midterm 
第10週
  11/18 自主學習週放假
11/20 校慶放假 
第11週
11/25, 11/27  11/25 Christoffel symbol, Compatible euqations[AP 10.1-2, DC 4.3]
11/27 Compatible equations[AP 10.1-2, DC 4.3] 
第12週
12/02, 12/04  12/02 Covariant derivative, Parallel transport[AP 7.4, DC 3.4, 4.4]
12/04 Geodesics[AP 7.3, 9.1-3, DC 4.4] 
第13週
12/09, 12/11  12/02 Gauss-Bonnet theorem[AP 13.1-4, 13.7,DC 4.5]
12/04 Gauss-Bonnet theorem[AP 13.1-4, 13.7,DC 4.5] 
第14週
12/16, 12/18  12/16 Quiz 4. Gauss-Bonnet theorem[AP 13.1-4, 13.7,DC 4.5]
12/18 Hyperbolic geometry[AP Chap 11] 
第15週
12/23, 12/25  12/23 Hyperbolic geometry[AP Chap 11]
12/25 Hyperbolic geometry, Geodesic normal coordinates[AP Chap 11, DC 4.6] 
第16週
12/30  12/30 Relativity. Special relativity and Minkowski space
(Reference: Semi-Riemannian Geometry, O' Neill)
1/01 元旦放假 
第17週
1/06, 1/08  1/06 Quiz 5. Relativity: Spacetime with constant curvature, de Sitter spacetime and Anti-de Sitter spacetime
1/08 Relativity: Schwarzschild spacetime and Kruskal spacetime for blackholes

(Reference: Semi-Riemannian Geometry, O' Neill) 
第18週
1/13, 1/15  期末考