課程資訊
課程名稱
微分幾何一
Differential Geometry (Ⅰ) 
開課學期
104-1 
授課對象
理學院  數學研究所  
授課教師
蔡宜洵 
課號
MATH7301 
課程識別碼
221 U2930 
班次
 
學分
全/半年
半年 
必/選修
選修 
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102天數102 
備註
總人數上限:30人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1041MATH7301_ 
課程簡介影片
 
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課程大綱
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課程概述

1.Jacobi fields, 2nd variation, Jacobi equation, conjugate points,minimizing property of geodesics, Index Lemma, Jacobi’s theorem, two proofs
2. Myers-Bonnet theorem, Cartan-Hadamard theorem, Rauch comparison theorem with applications to injectivity radius
3. Space of constant curvature, group theory viewpoints, geodesics, Jacobi fields
4. Cartan-Ambrose-Hicks Theorem
5, 6, 7. Miscellaneous
a) flows and transformations
b) Killing vector fields
c) volume element and divergence
d) Ricci curvature and volume growth
e) 2nd Bianchi identity applied to Einstein manifolds
f) Cut locus, injectivity radius, Klingenberg’s lemma
8. vector bundles, bundle maps, pull-back bundles, complex vector bundles
9. connection, curvature form, Bianchi identity
10. Chern classes, invariant polynomials, Chern character, unitary connection
11. Examples and application of Chern classes, immersions and embeddings in complex projective spaces
12. Pontrjagin classes, Euler class, relation with Chern classes,Todd class, A-hat genus
13. star operator, Hodge decomposition theorem, Poincare duality,
14. Kunneth formula, Bochner-Weitzenbock formula, proof
15. divergence, application of B-W formula to topology of manifolds, index of de Rham complex, remark on Index theorem
16. Gauge theory, Erlanger Program, historical remarks, principle bundles
17. Examples of Lie groups, SU(2)-bundles, Yang-Mills equation, self-duality equation  

課程目標
Provide an essential foundation in differential geometry, and the idea about how to use calculus/analysis to study geometry 
課程要求
 
預期每週課後學習時數
 
Office Hours
 
參考書目
[dC] do Carmo, Riemannian geometry.
[GHL] Gallot-Hulin-Lafontaines, Riemannian geometry.
[H] Helgason, Differential geometry, Lie groups and symmetric spaces.
[N] Noel J.Hicks, Notes on Differential geometry.
[CE] Cheeger and Ebin, Comparison theorems in Riemannian geometry.
[W] Warner, Foundations of differentiable manifolds and Lie groups.
[BT] Bott and Tu, Differential forms in algebraic topology. 
指定閱讀
 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
習題 
30% 
 
2. 
期中考 
35% 
 
3. 
期末考 
35% 
 
 
課程進度
週次
日期
單元主題