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 |
|
核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
|
為確保您我的權利,請尊重智慧財產權及不得非法影印
|
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% |
|
|
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] |
|