課程資訊
課程名稱
微分幾何二
Differential Geometry (Ⅱ) 
開課學期
108-2 
授課對象
理學院  數學研究所  
授課教師
蔡宜洵 
課號
MATH7302 
課程識別碼
221 U2940 
班次
 
學分
3.0 
全/半年
半年 
必/選修
必修 
上課時間
星期三2(9:10~10:00)星期五3,4(10:20~12:10) 
上課地點
天數305天數305 
備註
研究所數學組基礎課。
總人數上限:40人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1082MATH7302_ 
課程簡介影片
 
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課程概述

This class will go through the following topics :

Outline :
1. Jacobian Field and Index Theorem
2. Toponogov's Theorem and Applications
3. Topics on Differential Sphere Theorem
4. Kaehler Manifolds and Its Related Topics 

課程目標
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 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
 
參考書目
These lectures will be based on my own notes along with
1. F.W. Warner, Foundations of Differential Manifolds and Lie Groups, 1971, 1983.
2. M. Do Carmo, Riemannian Geometry, 1992.
3. P. Petersen, Riemannian Geometry, 2nd edition, 2006.
4. I. Chavel, Riemannian Geometry, 2nd edition, 2006.
5. J. Cheeger and D.G. Ebin, Comparison Theorems in Riemannian Geometry,
1975.

指定閱讀
1. Taubes, C., “Differential geometry”: discusses bundle theory in great detail
2. Kobayashi-Nomizu: “Foundations of Differential geometry”: bundle, principle bundles and related topics.
3. Berline, N., Getzler, E., Vergne, M., “Heat kernels and Dirac operators”: a more specialized book on bundle theory and used it for various index theorems
 
評量方式
(僅供參考)
   
課程進度
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