課程大綱

課程資訊

課程名稱	大域微分幾何 Introduction to Global Differential Geometry
開課學期	99-2
授課對象	理學院數學系
授課教師	黃武雄
課號	MATH5313
課程識別碼	221 U3760
班次
學分	3
全/半年	半年
必/選修	選修
上課時間	星期二3,4,@(10:20~)
上課地點	天數102
備註	總人數上限：50人
Ceiba 課程網頁	http://ceiba.ntu.edu.tw/992Global_Diff_Geom
課程簡介影片
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課程概述	(1) Holonomy and Gauss curvature, Hopf-Poincare, Gauss-Bonnet, Moving Frames and Applications: Clifford Tori, Lorentz Model, Degree theorems and vector fields. (2) Select topics from the following: Isoperimetric Inequalities, Jacobi fields and Synge theorem, Space forms. The course basically treats the two dimensional surfaces, but many concepts may be easily extended to higher dimensional Riemannian manifolds.
課程目標	希望這門課程，成為跨入幾何專業的橋樑，更重要的是培養幾何直觀。
課程要求	大三上學期幾何課、線性代數及高等微積分有關Stokes定理與反函數定理之部分。＊Grading scheme：依上課參加討論的表現，及期末報告而定。這是選修課，我希望修課者基於自己的興趣來學東西。在分數的壓力下，我不相信學到的東西會有多大的價值。
預期每週課後學習時數
Office Hours
參考書目	Huang,W.: A Rapid Course on Global Surface theory, 收錄於微分幾何與活動標架法前卷。 John Milnor: Topology from the Differentiable Viewpoint. Robert Ossermann: The Isoperimetric Inequality, Bulletin of the Amer.Math.Soc Some more related papers
指定閱讀
評量方式 (僅供參考)

課程進度

週次	日期	單元主題
第1週	2/22	Review of Surface Theory
第2週	3/01	Review of Surface theory(II);Hopf-Poincare Thm in high dimensions
第3週	3/08	(i)Lecture Part A, $1-9: holomomy and Euler number; (ii)Hopf-Poincare in High dimensions,including a sketch of Morse Index theorem and a topological proof of the fundamental thm of algebra-following John Milnor.
第4週	3/15	$9與 $10:Gauss equation and Codazzi eq.. ; Topics:Combinatorial version on Gauss- Bonnet, Morse Index and Gauss Theorema Egregium.
第5週	3/22	先用組合觀點重看 Gauss-Bonnet Thm, Morse Index Thm 及 Gauss Theorema Egregium; 再討論 moving frames in Euclidean spaces.
第6週	3/29	Structure Equations of Surfaces in R^3. Idea of Chern's intrinsic proof of Gauss-Bonnet Theorem on Surfaces. Sphere Bundles and connection forms.
第7週	4/05	春假
第8週	4/12	開放討論兼複習
第9週	4/19	Using moving frames to compute Riemann curvature tensor on Clifford torus and hyperbolic model ;The converse of homotopy degree theorem and counter examples.
第10週	4/26	Differentiable manifolds;tangent vectors as operators; framed Cobordism, Pontryagin manifolds.
第11週	5/03	From ODE to PDE; Lie bracket and Lie derivatives;Various integrability conditions(I).
第12週	5/10	Various integrabilty conditions(II); Frobenious integration theorem.
第13週	5/17	The birth of Riemannian geometry--by finding integrability condition for flat metrics and thus introducing Riemann curvature tensor; Fundamental theorem of surface theory and it's extention to higher dimensions(I).
第14週	5/24	Affine manifold,covariant derivatives verse Lie derivatives; Equations of parallelism and of geodesics,Equivalence of parallelism and affine connection; Tensors and tensor fields, torsion tensor.
第15週	5/31	Connection and metrics, Curvature tensor in modern form; Second Variation of geodesics, Synge theorem, Hopf- Rinow theorem, Frankel theorem.
第16週	6/07	Conjugate locus and Cut locus. Jacobi fields and its construction.
第17週	6/14	Minimization of Energy functional by Jacobi fields. Global theory of length minimization. Applications to global geometry of Riemannian manifolds: Theorems of Bonnet-Myer and Hadamard.