課程資訊
課程名稱
微分幾何一
Differential Geometry (Ⅰ) 
開課學期
111-1 
授課對象
理學院  數學研究所  
授課教師
蔡忠潤 
課號
MATH7301 
課程識別碼
221 U2930 
班次
 
學分
3.0 
全/半年
半年 
必/選修
必修 
上課時間
星期三9(16:30~17:20)星期五3,4(10:20~12:10) 
上課地點
天數101天數101 
備註
總人數上限:30人 
 
課程簡介影片
 
核心能力關聯
核心能力與課程規劃關聯圖
課程大綱
為確保您我的權利,請尊重智慧財產權及不得非法影印
課程概述

Modern differential geometry encompasses a wide array of techniques and results. Beginning with an overview of smooth differential manifolds (including coordinates, vector fields, tangent bundles, differential forms, tensors) we will then discuss Riemannian manifolds (those for which metric notions such as length, volume, etc. are defined), connections (leading to Hessian and Laplacian), exponential map, geodesics, submanifolds, and curvature. A major goal of the first semester is the Hodge theorem, which combines geometry, topology and analysis.

The tentative plan can be found below.  

課程目標
Provide an essential foundation in differential geometry for students aiming at using it in various kind of further applications in mathematics or modern sciences, and open a way to pursue work or research in modern geometry. 
課程要求
Undergraduate required courses: Linear algebra, advanced calculus, algebra, geometry, complex analysis, introduction to ODE, introduction to PDE.
Optional (not absolutely required): Topology, algebraic topology. 
預期每週課後學習時數
 
Office Hours
每週三 15:30~16:20 
參考書目
[CE] Jeff Cheeger and David Ebin, Comparison theorems in Riemannian Geometry.
[W'] Chin-Lung Wang, Differential Geometry (in http://www.math.ntu.edu.tw/~dragon/courses.html)
[dC] Manfredo do Carmo, Riemannian Geometry.
[T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature. 
指定閱讀
[W] Warner: Foundation of Differentiable Manifolds. 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
homework 
30% 
You have two jokers: the lowest two grades will be discarded. 
2. 
midterm 
35% 
10/28 
3. 
final 
35% 
12/23 
 
課程進度
週次
日期
單元主題
第1-4週
  [W] ch.1: smooth manifolds 
第5-7週
  [W] ch.2 & 4: tensors, differential forms, integrations 
第8-11週
  [CE] ch.1: basic Riemannian geometry 
第12-15週
  [W] ch.6: Hodge theory