# Fall 2020, MATH 538: Differential Geometry and
Manifolds

**Instructor: Thomas Yu, **email:** yut@drexel.edu
, **Office hours (on zoom):** M-W-F 1:00-3:00 p.m**

**Course Info.
and Syllabus**

**Lecture notes: **

**(I) local theory of curves
**

**(II) Geometry of linear maps
**

** (III) Regular surfaces: global
and local issues **

**(IV) Higher dimensional regular
surfaces **

**(V) Differentiable manifolds
**

** (VI) Tangent spaces and
derivatives **

**(VII) Submanifolds **

**(VIII) SO(3) **

**(IX) ****Gauss'
Theorema Egregium **

**(X) Riemannian
manifolds (lecture notes by Greg Naber)**

** **

**Computer Demos**
**Assigned Textbook****:
None, but it is helpful to have the following two references.**

**References: (i) ****John M.
Lee, Introduction to Smooth Manifolds (2nd edition)****,
(ii) M. Do Carmo, Differential Geometry of Curves and Surfaces**

**Pre-requisite:** Linear Algebra, Multivariate
Calculus, Vector Calculus

**Grading policy:** 90% HW, 10% class participation

**Assignments: **HW#0, HW#1, HW#2, HW#3

**Background references: (i) advanced calculus (e.g. Folland,
little
Spivak), (ii) vector calculus (e.g. Marsden
& Tromba), (iii) linear algebra (e.g. Hoffman
& Kunze, Strang)
(iv) point-set topology (E.g. Lectures 1-5 of Greg
Naber's notes)**

**Other references: **

**(i) ****John
Lee's four books**

**(ii) Spivak,
A Comprehensive Introduction to Differential Geometry (Not only
comprehensive, but also not terse. He explains the motivations behind
the abstract stuff, at the expense of writing a lot. He actually found
his own publishing company in order to publish these 5 volumes. In
Volume 2, he translates the original papers of Gauss and Riemann and
tells their stories.)**

**(iii) Boothby,
An Introduction to differentiable manifolds and Riemannian geometry**

**(iv) Nigel
Hitchin's notes (I find it rewarding to read these notes. But
they maybe a little terse for a first-timer, unless you have the
luxury to sit into to his lectures.)**

**(v) Greg
Naber's notes (you may like the thoughtful expositions, the
choice of examples, as well as the neat handwriting & drawings.)**

### Policies:

#### Lateness and Absence

Midterm and final exams makeups will not be allowed, except for the REAL
emergencies. Those should be communicated to (and agreed to by) me ahead of
time whenever possible. If not possible it will have to be supported by a
solid evidence.
#### Withdrawing from the course

You may withdraw from the course up to the last day of the sixth week of
class. If you do not withdraw by then, you will receive a grade for the
course.
#### Academic Honesty

The university's Academic Honesty policy is in full effect for this course.
Please read Section 11: "Academic Honesty" in the Drexel University Student
Handbook to make sure you are familiar with this policy.