MATH 433 Differential Geometry Spring 2025

Instructor: Dr. Micah Warren, Fenton 318, 541-346-5618

Syllabus subject to change

Time and Location:

M,W, F. 1100-1150, 103 PETR

Office Hours:

Final:

Friday, June 13, 2025, 10:15 a.m. (Verify this for yourself before booking flights)

Standard Course description:

Plane and space curves, Frenet-Serret formula surfaces. Local differential geometry, Gauss-Bonnet formula, introduction to manifolds.

Textbook:

Classic and inexpensive textbook - Do Carmo Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition

Material:

Time permitting - we hope to get through Chapter 4 of Do Carmo.

Prerequisites:

Prereq: MATH 282, 342; one from MATH 232, 262, 307.

Learning Outcomes:

The aim of this course is to introduce Riemannian geometry, from the starting point of submanifold of Euclidean space. Submanifolds of Euclidean space are, for example, curves and surfaces. Students should be able to compute curvature and other invariants of submanifolds on Euclidean space, based on parametric or nonparametric descriptions of the manifold.

Grading:

There will be weekly assignments, two midterms and a final. They will count toward the grade as follows:

Homework:

Homework will be assigned on Canvas. It will be due on Wednesday. First assignment due April 12. Lowest homework score will be dropped. Not all selected problems will be scored. You are encouraged to work with each other on the homework. Each assignment should be written up on your own. As a general rule, you are responsible for (that is, may be expected to know on an exam) any material that was covered by a homework problem, whether or not a similar example was covered in lecture.

Exams:

No calculators/smart watches, etc are available for the exams. Obviously, no collaboration on the exams. I expect the exams to happen on the fourth and eighth weeks.

AI/LLM/Wolfram Alpha/Claude etc policy:

Obviously none of this stuff will be allowed to solve math problems on exams. It's fine, if you want to get yourself unstuck doing homework problems to consult these as resources, I can't stop you. A generally good practice is to do the problem yourself and then check your work. It's very bad practice to become reliant on these tools to hold your hands through the tough parts.

Some of these LLMs are OK at talking you through misunderstandings. It's totally fine to tell an LLM "I'm struggling to understand the difference between extrinsic and intrinsic geometry." The LLM might be able to give you examples or explanations that work for you. The important thing is that you check to make sure you can do the problem and it actually works. LLMs are just making things up, most of the time for basic stuff they get it right. For 400 level classes, they do OK. The important thing is that you verify yourself that your understanding actually works out.

Final:

The final exam is cumulative and scheduled for Friday, June 13, 2025, 10:15 a.m. The same rules apply as for quizzes. Faculty legislation prohibits final exams from being administered early.

Makeup policy:

Attendance and Classroom environment:

I won't be taking attendance or prioritizing your life for you, but in class, you shouldn't be texting or doing coursework for another class. If you have a paper for another course due at 3:00, that's your own problem - you should leave the class and finish the paper. However, I have loads of anecdotal evidence that attending class regularly is highly correlated with scoring well in a class.

If you have a question, please ask me. At any one point in time during the class, there may be a large portion of class who is not quite with me, for whatever reason - oftentimes a stupid question is just what other students need to refocus their attention. In fact, there is well-documented evidence that attention spans during a lecture will almost always fail every fifteen minutes. I keep this in mind and welcome opportunities to reset. If you ever want me to take a two minute break, just stop me and tell me! If it's good for you it's probably good for the rest of us.

Academic dishonesty:

Any type of academic dishonesty will not be tolerated. In the event of academic dishonesty, the offense will be reported to the Office of Student Conduct and Community Standards and the student will be sanctioned up to receiving a failing grade in the course.

Other general university policies:

Mercifully, I no longer have to include 8 pages of policies, as these are linked in Canvas!