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.