基本素養 Basic Literacy

核心能力 Competence

掌握連續與逼進的能力
Ability to handle continuity phenomena and approximations.
處理數字及符號運算的能力
Ability to operate with symbols and digits.
透視形像與空間的能力
Ability to visualize space, shape, and images.
利用計算機處理數學的能力
Ability of computer to do mathematics.
應用隨機理論的能力
Ability to apply stochastic mathematics.
處理大量數據的能力
Ability to handle large amount of data.
處理離散型數學的能力
Ability to do mathematics relating to discrete models.
獨立研究與撰寫論文的能力
Ability to do independent researches and to write thesis.
使用計算機的能力
Abilities to use computer as a tool.

課程概述 Course Description

這是微分幾何導論的後續課程。課程會講授微分幾何的一些更進階和深入的內容。其中涉及的主題可以包括流形上的微積分、黎曼流形、複流形、辛幾何、纖維叢、李群跟微分拓撲。這個課程的目標是讓學生對現代微分幾何及相關領域的研究有一個較全面的理解,從而幫助發展研究興趣。
This is a sequel to the course "Introduction to differential geometry". We will cover more advanced and specific topics in various aspects of differential geometry. The list of topics may include calculus on manifolds, Riemannian manifolds, complex manifolds, symplectic geometry, fiber bundles, Lie groups, and differential topology. It will enable students to have a comprehensive view on the modern development of differential geometry and its related branches.

課程學習目標 Course Objectives

  • Introduction to tensors and Riemannian metrics
  • Concepts and computations of connections, geodesics, curvatures and submanifolds
  • Relations between curvature and topology
  • 課程進度 Course Outline

    週次 Week進度說明 Progress Description
    1Introduction
    2Tensors, Manifolds, and Vector Bundles
    3Definitions and Examples of Riemannian Metrics
    4Connections
    5Geodesics
    6Geodesics
    7Geodesics and Distance
    8Geodesics and Distance
    9Curvature
    10Curvature
    11Riemannian Submanifolds
    12Riemannian Submanifolds
    13The Gauss–Bonnet Theorem
    14Jacobi Fields
    15Jacobi Fields
    16Curvature and Topology
    17Curvature and Topology
    18Final Exam
     以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

    有關課程其他調查 Other Surveys of Courses

    1.本課程是否規劃業界教師參與教學或演講? 否
    Is there any industry specialist invited in this course? How many times? No
    2.本課程是否規劃含校外實習(並非參訪)? 否
    Are there any internships involved in the course? How many hours? No
    3.本課程是否可歸認為學術倫理課程?
    Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics?
    4.本課程是否屬進入社區實踐課程?
    Is this course recognized as a Community engagement and Service learning course? Which community will be engaged?