## Differential Geometry

### MATH 423, Fall 2022

University of Illinois at Urbana-Champaign, Department of Mathematics

**What this course is about:** This is a first introduction to the use of calculus to understand and characterize curves and surfaces. Perhaps you have seen a bit of this in Calculus III when you studied parametric curves. This course goes well beyond that, and you will learn about Gaussian curvature, Gauss' Theorema Egregium, Gauss-Bonnet theorem and many other topics about surfaces and curves. You will also learn the machinery of *differential forms*, which are a useful set of tools, not taught in calculus classes. Learning about a concrete set of surfaces and learning to study these using differential forms will help you in future courses on differentiable manifolds (Math 481, 518), general relativity, and higher level courses on differential geometry (Math 519).

**Instructor:** Anil Hirani

**Office Hour:** TBA

**TA:** TA details and office hours will be on Moodle (eventually)

**Lectures:** TR 9:30 -- 10:50, Altgeld Hall, Room 341

**Course LMS and communication:** There will be a Moodle page for the course. We may also have a Campuswire class feed (details will be on Moodle). For communicating with the instructor you are encouraged to use the advanced, newly re-discovered communication protocol called H2HIP (human to human, in person). This option is only available if you come to class or my office.

**Homeworks:** There will be a graded homework most weeks assigned from the textbook and posted on Moodle. Homework will be due *in class on paper*. A selection of problems will be graded by the grader. You are encouraged to discuss the homeworks with others and me, but you should write up solution on your own. Late homeworks will only be accepted in case of medical / family emergency. Your lowest homework score will be dropped.

**Required Textbook** Barrett O'Neill,
*Elementary
Differential Geometry*, Revised Second Edition, 2006. Available free from on-campus, thanks to our library.

**Suggested book** Tristan Needham, Visual Differential Geometry and Forms, 2021.

**Exams:** There will be two midterm exams and a final exam. Conflict exams will be available if arrangements have been made with me at least one week in advance of the scheduled exam.

**Grading:** Homeworks 20%, Two Midterm Exams: your lower score worth 20%, higher score worth 25%,
Final Exam 35%.

Page maintained by Anil Hirani.
Last modified: Mon Aug 22 12:01:46 CDT 2022