課程資訊

Differential Geometry (Ⅰ)

101-1

MATH7301

221 U2930

Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1011DG1

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

Office Hours

Hirsch: Differential Topology.
Schoen and Yau: Lectures on Differential Geometry.
Milnor: Morse Theory.

M. Spivak: A Comprehensive Introduction to Differential Geometry, vol. 1, 2.
Warner: Foundation of Differentiable Manifolds and Lie Groups.
Cheeger-Ebin: Comparison Theorems in Riemannian Geometry.
Do Carmo: Riemannian Geometry.

(僅供參考)

 No. 項目 百分比 說明 1. midterm exam 35% 2. final exam 35% 3. homework 30%

 課程進度
 週次 日期 單元主題 第1週 9/12,9/14 Manifolds and tangent bundles. 第2週 9/19,9/21 Submanifolds and Whitney embedding. 第3週 9/26,9/28 Sard's theorem, flows and Lie derivatives. 第4週 10/03,10/05 Frobenius theorem, smoothness of ODE. 第5週 10/10,10/12 Tensors and differential forms. 第6週 10/17,10/19 Cartan's formula and de Rham complex. 第7週 10/24,10/26 Manifolds with boundary, Stokes theorem. 第8週 10/31,11/02 De Rham theorem. 第9週 11/07,11/09 Midterm Exam on 11/07. Riemannian structure, Levi-Civita connection. 第10週 11/14,11/16 Geodesics, curvature tensor. 第11週 11/21,11/23 Variations of geodesics in complete manifolds. 第12週 11/28,11/30 Jacobi fields, Cartan-Hadamard, Cartan-Ambrose-Hicks. 第13週 12/05,12/07 Variations of Riemannian submanifolds, Gauss equations. 第14週 12/12,12/14 Hodge theorem for harmonic forms. 第15週 12/19,12/21 Sobolev spaces and elliptic regularity. 第16週 12/26,12/28 Lie groups. 第17週 1/02,1/04 Review and Final Exam.