Math 435: Differential Geometry, Spring 2024

Announcements

All announcements will be posted on Blackboard.

Lecture notes

You can find a brief outline of the lectures here.
Here are the notes for just Lecture 42.

Homework

Homework 13 has been posted.
It is due on Wednesday, April 24.

See the Homework page for older homework assignments.

Overview

This course will apply advanced topics in calculus to differential geometry, which is the study of curves in the plane and of curves and surfaces in space, as well as of manifolds, which are higher-dimensional analogues of curves and surfaces.

In the first half of the course, we will be going through more or less all of Michael Spivak’s classic book Calculus on Manifolds. This will in part be a review of the main concepts from multivariable calculus: partial derivatives, the Jacobian matrix, and multi-dimensional integration. We will then introduce the conept of differential form, which provide an extremely elegant geometric reformulation of integration theory.

We will then introduce manifolds and differential forms on manifolds. The culmination of this part of the course is the differential forms version of Stokes’ Theorem, which includes as special cases the classical Stokes’ Theorem, as well as Green’s Theorem and the Divergence Theorem, and generalizes these to manifolds of all dimensions.

In the second half of the course, we will turn from the general theory of manifolds, to the study of curves and surfaces, following (parts of) Kristopher Tapp’s book Differential Geometry of Curves and Surfaces.

The main emphasis of this part of the course will be the notion of curvature, and it will culminate with the Gauss-Bonnet Theorem, which relates the curvature of a surface to its topology, that is, to its overall shape.

For a more detailed (preliminary) week-by-week list of topics to be covered, please see the lecture plan here.

Textbook

In the first part of the course, we will be following

After that, we will be following

Prerequisites

Multivariable calculus (including multivariable integration) and linear algebra.
The official prerequisites are: at least one of Math 226, Math 227, or Math 229, and at least one of Math 225 or Math 245.

This page, Blackboard, Gradescope

Most information and material related to the course will be posted on this page.

The exceptions are:

• Gradescope will be used to submit homework assignments, and to see your graded homework.
• Blackboard will be used to make announcements.
• Email should be used to email to contact the instructor or TA.

Grading scheme

• 40% Homework
• 30% Midterm Exam
• 30% Final Exam

Homework

Homework assignments will be assigned every week.

Homework is to be submitted on Gradescope (you should see the course there if you are enrolled; if you have any problems, please inform the instructor or TA). When you submit on Gradescope, please don’t forget to match your scanned pages with the problems.

Homework must be submitted by the posted due dates. If you expect to have issues submitting the homework on time, or if you are having difficulties with gradescope, please write to the instructor or TA as soon as possible, and attach a scanned copy of your submission (this is a wise practice for any class).

You are encouraged though not required to type up your homework using LaTeX. If you do not have experience with LaTeX and would like to learn how to use it, please ask the instructor or TA.

Exams

• The midterm will be on Monday, February 26, in class (e.g., at the usual time 11:00am-11:50am, in the usual place KAP 137).
  
• The final exam will be held on Wednesday, May 1 from 11 am to 1 pm, in KAP 137, as per the USC exam schedule.
  

Both exams are closed book, closed notes exams, with no calculators or other electronic aids permitted. The final exam will cover all topics from the semester, but will have greater emphasis on topics developed after the midterm.

Office hours

The TA’s and instructor’s office hours are TBA.

Statement for students with disabilities

Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. DSP is located in GFS 120 and is open 8:30 a.m.–5:00 p.m., Monday through Friday. Website for DSP (https://dsp.usc.edu/) and contact information: (213) 740-0776 (Phone), (213) 740-6948 (TDD only), (213) 740-8216 (FAX) dspfrontdesk@usc.edu.

Statement on academic integrity

USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one’s own academic work from misuse by others as well as to avoid using another’s work as one’s own. All students are expected to understand and abide by these principles. SCampus, the Student Guidebook, contains the University Student Conduct Code (see University Governance, Section 11.00), while the recommended sanctions are located in Appendix A.

Statement on academic integrity

USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one’s own academic work from misuse by others as well as to avoid using another’s work as one’s own. All students are expected to understand and abide by these principles. SCampus, the Student Guidebook, contains the University Student Conduct Code (see University Governance, Section 11.00), while the recommended sanctions are located in Appendix A.

Emergency preparedness/course continuity in a crisis

In case of a declared emergency if travel to campus is not feasible, USC executive leadership will announce an electronic way for instructors to teach students in their residence halls or homes using a combination of Blackboard, teleconferencing, and other technologies. See the university’s site on Campus Safety and Emergency Preparedness.

Disclaimer

This syllabus is not a contract, and the Instructor reserves the right to make some changes during the semester.