課程資訊
課程名稱
微分幾何二
Differential Geometry (Ⅱ) 
開課學期
99-2 
授課對象
理學院  數學研究所  
授課教師
李瑩英 
課號
MATH7302 
課程識別碼
221 U2940 
班次
 
學分
全/半年
半年 
必/選修
選修 
上課時間
星期三5(12:20~13:10)星期五3,4(10:20~12:10) 
上課地點
天數304天數304 
備註
總人數上限:40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
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課程概述

With background from last semester, we will explore different aspects of Geometry and introduce basic notions and ideas to various directions. The class will emphasize on students’ independent studies and presentations. We will discuss vector bundles, connections in vector bundles, the moving frame method, De Rham Cohomology and Harmonic Differential Forms, Yang-Mill Functional and Yang-Mill equations, Chern Classes, the covariant derivatives of tensors and the rules on exchanging order of derivatives, the Laplacian of 2nd fundamental form (in general co-dimension and manifolds), Simon’s identity and its applications, the 1st and 2nd variation formula of area (also the restricted case that encloses fixed volume), minimal surfaces, some applications of the 2nd variation formula of area, spin and spin^c structures, Dirac operator and Weitzenbock formulas, the proof of Toponogov Theorem, properties of Killing vector fields, the Bochner method, Symmetric Spaces, various notions of convergence for Riemannian manifolds. At suitable stages of the class, we will also briefly introduce some important developments in Geometry such as Donaldson theory, Seiberg-Witten equations, positive mass theorems, Chern-Simon forms, Morse Theory and Floer Homology, Geometric Analysis, Curvature flow and the resolve of Poincare conjecture. 

課程目標
◎ Set up the foundation for students to get into the field of geometry.
◎ Introduce the basics and essentials in Differential and Riemannian Geometry to students in all fields.
 
課程要求
Differential Geometry (I) 
預期每週課後學習時數
 
Office Hours
 
參考書目
1) Riemannian Geometry and Geometric Analysis, Jurgen Jost,
Fifth edition. Universitext. Springer-Verlag, Berlin, 2008.
2) Riemann Geometry, Peter Petersen
Springer Science, Graduate Texts in Mathematics,(Spring e-books)
3) Riemannian Geometry, Do Carmo
Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1992.
 
指定閱讀
 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
Homework and in class performance 
20% 
 
2. 
Oral 
20% 
 
3. 
Presentation  
60% 
 
 
課程進度
週次
日期
單元主題