Fall 2019 MATH 6006
Differential Geometry 1 微分幾何(一)
Syllabus 課程大綱
宣導事項
教學進度
9/10/2019(星期二): Welcome to the class.
9/12/2019(星期四): Preliminaries on point set topology.
9/17/2019(星期二): Preliminaries on point set topology.
9/19/2019(星期四): Definitions and examples of topological manifolds.
9/24/2019(星期二): Definitions and examples of smooth manifolds.
9/26/2019(星期四): Manifolds with boundaries.
10/1/2019(星期二): Smooth maps and diffeomorphisms.
10/3/2019(星期四): Tangent spaces and differentials.
10/8/2019(星期二): Abstract definition of tangent spaces.
10/10/2019(星期四): No class (台灣國 慶)
10/15/2019(星期二): Differentials of smooth maps.
10/17/2019(星期四): Computations in coordinates, change of coordinates.
10/22/2019(星期二): Introduction to tangent bundles.
10/24/20191(星期四): Tangent bundle to circle and spheres, velocity vectors.
10/29/2019(星期二): Submersions, immersions, and embeddings.
10/31/2019(星期四): Maps of constant ranks.
11/5/2019(星期二): Embeddings.
11/7/2019(星期四): Immersions vs. embeddings, submersions.
11/12/2019(星期二): Presentation of problems.
11/14/2019(星期四): Introduction on Grassmannians.
11/19/2019(星期二): Embeddings and regular level sets.
11/21/2019(星期四): Regular level sets, immersed submanifolds, tangent space to submanifolds.
11/26/2019(星期二): Sard's Theorem.
11/28/2019(星期四): Proof of Sard's Theorem, Introduction of Whitney embedding theorem.
12/3/2019(星期二): Proof of Whitney embedding theorem.
12/5/2019(星期四): Finish the proof of Whitney embedding theorem.
12/10/2019(星期二): Introduction to tensor products.
12/12/2019(星期四): Symmetric and alternating tensors, wedge products.
12/17/2019(星期二): Introduction and properties of differential forms.
作業
Homework 1 (截止日 9/26/2019 星期四):
Chapter 1 Problems: 3, 6, 7, 10, 11.
Homework 2 (截止日 10/24/2019 星期四):
Chapter 2 Problems: 3, 5, 8 (do only the real case), 11.
Chapter 3 Problems: 1, 2, 3.
Homework 3 (截止日 11/7/2019 星期四):
Chapter 3 Problem: 6.
Chapter 4 Problems: 2, 4, 6, 8.
Homework 4 (截止日 12/5/2019 星期四):
Chapter 5 Problems: 1, 5, 6, 10, 20.
Chapter 6 Problems: 1.
Homework 5 (截止日 1/9/2020 星期四)
Chapter 11 Problems: 5, 7, 9, 11.
Chapter 12 Problems: 10.
Chapter 14 Problems: 1, 5, 6.
Chapter 15 Problems: 1, 2, 5, 7, 13.
Chapter 16 Problems: 2, 6, 9, 10, 12.
考試相關
講義
Overview 9/9/2019
9/12/2019: Preliminaries on Topology
9/12/2019: Definitions and Examples of Smooth Manifols (updated)
9/30/2019: Smooth Maps
9/30/2019: Tangent Spaces and Differentials
11/21/2019: Submersions, Immersions, and Embeddings
11/21/2019: Submanifolds and Related Topics