課程資訊

Differential Geometry (Ⅰ)

111-1

MATH7301

221 U2930

3.0

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 major goal of the first semester is the Hodge theorem, which combines geometry, topology and analysis.

The tentative plan can be found below.

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

[CE] Jeff Cheeger and David Ebin, Comparison theorems in Riemannian Geometry.
[W'] Chin-Lung Wang, Differential Geometry (in http://www.math.ntu.edu.tw/~dragon/courses.html)
[dC] Manfredo do Carmo, Riemannian Geometry.
[T] Clifford Taubes, Differential geometry. Bundles, connections, metrics and curvature.

[W] Warner: Foundation of Differentiable Manifolds.

(僅供參考)

 No. 項目 百分比 說明 1. homework 30% You have two jokers: the lowest two grades will be discarded. 2. midterm 35% 10/28 3. final 35% 12/23

 課程進度
 週次 日期 單元主題 第1-4週 [W] ch.1: smooth manifolds 第5-7週 [W] ch.2 & 4: tensors, differential forms, integrations 第8-11週 [CE] ch.1: basic Riemannian geometry 第12-15週 [W] ch.6: Hodge theory