課程資訊
課程名稱
微分幾何一
Differential Geometry (Ⅰ) 
開課學期
109-1 
授課對象
理學院  數學研究所  
授課教師
王金龍 
課號
MATH7301 
課程識別碼
221 U2930 
班次
 
學分
3.0 
全/半年
半年 
必/選修
選修 
上課時間
星期三1,2(8:10~10:00)星期五3(10:20~11:10) 
上課地點
天數102天數102 
備註
總人數上限:80人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1091MATH7301_DG1 
課程簡介影片
 
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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. Examples on space forms, Lie groups and symmetric spaces will be emphasized.
A significant part of the remainder of the course will study the effects curvature has on geometry and topology. In particular, this includes the linear theory of de Rham theorem and Hodge theory of harmonic forms, Bochner principles, and the non-linear theory on applications of second variational formula for geodesics and minimal sub-manifolds. 

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, analysis, algebra, geometry, complex analysis, introduction to ODE, introduction to PDE.
Optional (not absolutely required): Topology, algebraic topology.

提醒: 本課程是數學研究所的必修課, 習題與考試均有嚴格要求. 因此, 大學部必修課中, 線性代數, 分析(導論), 幾何學(導論), 複變(導論) 四科若有任何一科未曾修過或尚未通過而想修本課程者, 必須先通過第一週的期初測驗. 無法投入大量精力而又希望對微分幾何有初步了解的同學, 建議旁聽就好.  
預期每週課後學習時數
 
Office Hours
每週一 13:10~14:00 
參考書目
Kobayashi and Nomizu, Foundations of Differential Geometry, I and II.
Schoen and Yau: Lectures on Differential Geometry.
Spivak : A Comprehensive Introduction to Differential Geometry, I-II. 
指定閱讀
Chin-Lung Wang: Differential Geometry
Warner: Foundation of Differentiable Manifolds.
Do Carmo: Riemannian Geometry 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Hoework 
30% 
 
2. 
MIdterm Exam 
35% 
 
3. 
Final Exam 
35% 
 
 
課程進度
週次
日期
單元主題
第1週
9/16,9/18  Manifolds and tangent spaces 
第2週
9/23,9/25  Submanifolds, Whitney embeddings and Sard's theorem 
第3週
9/30,10/02  Vector fields, flows and Lie derivatives 
第4週
10/07,10/09  Tensors, differential forms 
第5週
10/14,10/16  Lie derivatives on tensors, Cartan's theory 
第6週
10/21,10/23  Stokes's theorem, de Rham theory 
第7週
10/28,10/30  Riemannian structure, covariant derivatives 
第8週
11/04,11/06  Geodesics, Exp, and curvature tensor 
第9週
11/11,11/13  Hilbert--Einstein action, Midterm Exam 
第10週
11/18,11/20  Variations of geodesics, Jacobi fields 
第11週
11/25,11/27  Second fundamental forms and HD variations 
第12週
12/02,12/04  Harmonic forms and elliptic operators 
第13週
12/09,12/11  Fourier transform and Sobolev spaces 
第14週
12/16,12/18  Proof of Hodge theorem, Bochner principle 
第15週
12/23,12/25  Category of Lie groups and Lie algebras 
第16週
12/30,1/01  Geometry on Lie groups and homogeneous spaces 
第17週
1/06,1/08  Introduction to symmetric spaces