課程資訊
課程名稱
 微分幾何一
Differential Geometry (Ⅰ) 
開課學期
 103-1 
授課對象
 理學院  數學系  
授課教師
 蔡忠潤 
課號
 MATH7301 
課程識別碼
 221 U2930 
班次
  
學分
 3 
全/半年
 半年 
必/選修
 選修 
上課時間
 星期三8(15:30~16:20)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/1031dg 
課程簡介影片
 
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課程概述

1. smooth manifolds (coordinates, vector fields, Lie derivatives, tangent bundles, vector bundles, differential forms, tensors, etc.)
2. basic Riemannian geometry (geodesics, exponential map, curvature, etc.)
3. Hessian and Laplacian, Hodge theory.






 

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
 1. point-set topology
2. familiar with linear algebra
3. advanced calculus, in particular, Taylor theorem and inverse/implicit function theorem
4. fundamental theory of ordinary differential equation: existence and uniqueness, smoothness of solutions in initial conditions 
預期每週課後學習時數
  
Office Hours
 每週二 16:00~17:00
每週二 11:00~11:50 
指定閱讀
 1. Taubes, Differential geometry. Bundles, connections, metrics and curvature. 
參考書目
 2. do Carmo, Riemannian geometry.
3. Cheeger and Ebin, Comparison theorems in Riemannian geometry.
4. Warner, Foundations of differentiable manifolds and Lie groups. 
評量方式
(僅供參考)
  
No.
項目
百分比
說明
1. 
 作業 
30% 
  
2. 
 期中考 
30% 
  
3. 
 期末考 
35% 
  
4. 
 上課情況 
5% 
  
 
課程進度
週次
日期
單元主題
Week 4
 10/08  vector field and derivation, maps between vector bundles. Reference: [T, §3] 
Week 11
 11/26,11/28  fall break 
Week 16
 12/31  holonomy and curvature, construction of first Chern class from curvature. Reference: [T, §13~14] 
Week 1-1
 9/17  topological manifolds and smooth manifolds. Reference: [T, §1] 
Week 1-2
 9/19  submanifolds, immersion and embedding, projective space. Reference: [T, §1] 
Week 2-1
 9/24  partition of unity and exhaustion function. Reference: [T, §1] 
Week 2-2
 9/26  embedding into Euclidean spaces, Lie group, matrix groups. Reference: [T, §2] 
Week 3-1
 10/01  vector bundles, tautological bundle. Reference: [T, §3] 
Week 3-2
 10/03  tangnet bundle and cotangent bundle. Reference: [T, §3] 
Week 5-1
 10/15  subbundle and quotient bundle. Reference: [T, §4] 
Week 5-2
 10/17  pull-back bundle, symmetric square and exterior powers of cotangent bundle, push-forward and pull-back, exterior derivative. Reference: [T, §4, §5 and §12.1] 
Week 6-1
 10/22  exterior derivative, orientation and integration, de Rham cohomologies. Reference: [T, §12.1 and §12.2] and [BT, §3] 
Week 6-2
 10/24  manifold with boundary, Stokes theorem, the notion of left/right invariant for Lie groups, exponential map for matrix groups. Reference: [BT, §3] and [T, §5.4 and §5.5] 
Week 7-1
 10/29  exponential map (continued), almost complex structure. Reference: [T, §2.4 and §6] 
Week 7-2
 10/31  complex vector bundle, orientation for vector bundle, metric on vector bundle. Reference: [T, §6 and §7] 
Week 8-1
 11/05  Riemannian manifold, geodesics. Reference: [T, §8] 
Week 8-2
 11/07  examples of geodesics: hypersurface, special orthogonal group. Reference: [T, §8] 
Week 9-1
 11/12  (geodesic) exponential map. Reference: [T, §9] 
Week 9-2
 11/14  exponential map (continued), Gaussian coordinate and Gauss lemma, properties of geodesics. Reference: [T, §9] and [CE, §1.2 ~ §1.3] 
Week 10-1
 11/19  properties of geodesics (continued), spherical geometry and hyperbolic geometry. Reference: [T, §9] 
Week 10-2
 11/21  Midterm 
Week 12-1
 12/03  principal G-bundle. Reference: [T, §10] 
Week 12-2
 12/05  group action, associated vector bundles. Reference: [T, §10] 
Week 13-1
 12/10  covariant derivative. Reference: [T, §11] 
Week 13-2
 12/12  covariant derivative (continued), connection on principal G-bundle, corresponding covariant derivative on associated bundle. Reference: [T, §11] 
Week 14-1
 12/17  local expression of connection on principal G-bundle. Reference: [T, §11] 
Week 14-2
 12/19  Lie derivative, Cartan formula, curvature of covariant derivative. Reference: [T, §12] 
Week 15-1
 12/24  curvature of connection on principal G-bundle, curvature and commuting derivatives, Frobenius theorem. Reference: [T, §12~13] 
Week 15-2
 12/26  Frobenius theorem and flat connection, flat connection over the circle, holonomy map. Reference: [T, §13] 
Week 17-1
 1/07  example for first Chern class, total Chern class. Reference: [T, §14] 
Week 17-2
 1/09  example for second Chern class, Pontryagin class. Reference: [T, §14]