課程資訊

Differential Geometry (Ⅰ)

109-1

MATH7301

221 U2930

3.0

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1091MATH7301_DG1

Modern differential geometry encompasses a wide array of techniques and results. Beginning with an overview of smooth differential manifolds (including coordinates, vector ﬁelds, tangent bundles, differential forms, tensors) we will then discuss Riemannian manifolds (those for which metric notions such as length, volume, etc. are deﬁned), connections (leading to Hessian and Laplacian), exponential map, geodesics, submanifolds, and curvature. Examples on space forms, Lie groups and symmetric spaces will be emphasized.
A significant part of the remainder of the course will study the effects curvature has on geometry and topology. In particular, this includes the linear theory of de Rham theorem and Hodge theory of harmonic forms, Bochner principles, and the non-linear theory on applications of second variational formula for geodesics and minimal sub-manifolds.

Provide an essential foundation in differential geometry for students aiming at using it in various kind of further applications in mathematics or modern sciences, and open a way to pursue work or research in modern geometry.

Undergraduate required courses: Linear algebra, analysis, algebra, geometry, complex analysis, introduction to ODE, introduction to PDE.
Optional (not absolutely required): Topology, algebraic topology.

Office Hours

Kobayashi and Nomizu, Foundations of Differential Geometry, I and II.
Schoen and Yau: Lectures on Differential Geometry.
Spivak : A Comprehensive Introduction to Differential Geometry, I-II.

Chin-Lung Wang: Differential Geometry
Warner: Foundation of Differentiable Manifolds.
Do Carmo: Riemannian Geometry

(僅供參考)

 No. 項目 百分比 說明 1. Hoework 30% 2. MIdterm Exam 35% 3. Final Exam 35%

 課程進度
 週次 日期 單元主題 第1週 9/16,9/18 Manifolds and tangent spaces 第2週 9/23,9/25 Submanifolds, Whitney embeddings and Sard's theorem 第3週 9/30,10/02 Vector fields, flows and Lie derivatives 第4週 10/07,10/09 Tensors, differential forms 第5週 10/14,10/16 Lie derivatives on tensors, Cartan's theory 第6週 10/21,10/23 Stokes's theorem, de Rham theory 第7週 10/28,10/30 Riemannian structure, covariant derivatives 第8週 11/04,11/06 Geodesics, Exp, and curvature tensor 第9週 11/11,11/13 Hilbert--Einstein action, Midterm Exam 第10週 11/18,11/20 Variations of geodesics, Jacobi fields 第11週 11/25,11/27 Second fundamental forms and HD variations 第12週 12/02,12/04 Harmonic forms and elliptic operators 第13週 12/09,12/11 Fourier transform and Sobolev spaces 第14週 12/16,12/18 Proof of Hodge theorem, Bochner principle 第15週 12/23,12/25 Category of Lie groups and Lie algebras 第16週 12/30,1/01 Geometry on Lie groups and homogeneous spaces 第17週 1/06,1/08 Introduction to symmetric spaces