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