课程大纲
COURSE SYLLABUS
1.
课程代码/名称
Course Code/Title
MAT8024 微分流形 Differentiable manifolds
2.
课程性质
Compulsory/Elective
Compulsory
3.
课程学分/学时
Course Credit/Hours
3/48
4.
授课语言
Teaching Language
English
5.
授课教师
Instructor(s)
Stavros Garoufalidis
6.
是否面向本科生开放
Open to undergraduates
or not
Yes
7.
先修要求
Pre-requisites
（ 如 面 向 本 科 生 开 放 ， 请 注明 区 分 内 容 。 If the course is open to
undergraduates, please indicate the difference.）
Some knowledge of basic topology
MA323 拓扑学
8.
教学目标
Course Objectives
（ 如面向本科生开放， 请注明区分内容。 If the course is open to undergraduates, please indicate the
difference.）
This is a first graduate course on smooth manifolds, introducing various aspects of their topology,
geometry, and analysis. We will start at the beginning with the definition of a smooth manifold, look
at some examples, and then explore the basic associated objects, including submanifolds, tangent
vectors, bundles, and derivatives. We will apply the inverse function theorem to geometric issues like
transversality, and then look at vector fields, associated flows, and the Lie derivative. Differential
forms on manifolds will also be a focus, including how to differentiate and integrate them. Time
permitting, we might look at the very basics of Lie groups, foliations (the Frobenius theorem), Morse
theory, or de Rham cohomology. In addition to treating the foundations of the subject carefully, this
course aims to emphasize examples and geometric intuition throughout.
9.
教学方法
Teaching Methods
（ 如 面 向 本 科 生 开 放 ， 请 注 明 区 分 内 容 。 If the course is open to undergraduates, please indicate the
difference.）
Guides and facilitates student learning and overall understanding of the material. Student learning is
measured through formal and informal forms of assessment, including group projects, student portfolios,
and classroom participation. Teaching methods may include classroom participation, demonstration, rote
memorization, memorization, or a combination of these.
10.
教学内容
Course Contents
（如面向本科生开放，请注明区分内容。 If the course is open to undergraduates, please indicate the
difference.）
Section 1
smooth manifolds
Section 2
Smooth maps
Section 3
Tangent Vectors
Section 4
Submersions, Immersions, and Embeddings
Section 5
Submanifolds
Section 6
Sard’s Theorem
Section 7
Lie Groups
Section 8
Vector Fields
Section 9
Intergral Curves and Flows
Section 10
Vector Bundles
…………
11.
12.