Differential Geometry Spring 2021
Lectures: MWF 9:10 am – 10:00am, HayesHealey 129. You may attend lectures either remotely or in person. I will send out an email with the lecture zoom link. If you can’t find it send me an email.
Office hours: Thursdays 4pm6pm. Office hours will be over zoom. I will send out the zoom link.
Syllabus: We will loosely follow do Carmo’s book. We will cover material up through chapter 10, and maybe beyond if we have time/inclination. We will cover: Riemannian metrics, connections, geodesics, curvature, submanifolds, GaussBonnet theroem, Jacobi fields, curvature comparison theorems. My lecture notes, which will be updated as we go, are available here.
Book: do Carmo, “Riemannian geometry.” Another good reference is Lee’s book “Introduction to Riemannian manifolds.”
Grade: There will be one takehome midterm, and one final. Your grade will be weighted by: 50% homework, 25% midterm, 25% final.
Midterm: The midterm will take place the week of March 22. I haven’t decided exactly how it will work yet, but you won’t have homework for that week.
Homework: There will be regularity homework assignments, typically due the Friday after it is handed out. I will post the homeworks and solutions here as they appear.
Course Notes Part 1 2018 Edition Course Notes Part 2 2018 Edition
Course synopsis:
This course is a graduatelevel introduction to foundationalmaterial in differential geometry. Differential geometry underliesmodern treatments of many areas of mathematics and physics, includinggeometric analysis, topology, gauge theory, general relativity, andstring theory. The main topics of study will be organized into twooverall sections, differential topology and Riemanniangeometry . Additional advanced topics will beconsidered if time permits.
The textbook for this course is Riemannian Geometry byManfredoPerdigao do Carmo.As a supplementary source, some of the material covered in the classcan be found in Riemannian Geometry by Gallot, Hulin, andLafontaine, and Smooth Manifoldsby Lee.
The grading for this class is based on weekly homework sets and atakehome final exam.
The formal prerequisite for this course is Math 532, which we willmainly use for the Inverse and Implicit Function Theorems. If youhaven’t taken Math 532, please consult with me to see if Math 621 isappropriate for you. Our course is fastpaced and is likely to beextremely difficult if you haven’t at least taken 500level courses inthe past. For some indication of the level of our course, you may wantto take a look at the textbook and at my lecture notes from last year’scourse .
Mathematics 621 Spring 2018
Wednesdays and Fridays, 3:054:20, Physics 205
Office hours: Mondays 2:003:00, Thursdays 11:0012:00
I will need to be away several times during the semester. To make upfor missed classes, we will be holding class on occasional Mondays3:054:20. Here’s the current list of canceled and makeup classes:
 Wednesday January 17: no class
 Monday January 22:makeup class, 3:054:20 in Physics205
 Wednesday January 28:no class
 Monday February 5:makeupclass, 3:054:20 in Physics 205
 Monday February 12:makeup class, 3:054:20 in Physics205
 Wednesday February 14:no class
 Friday February 16: noclass
 Monday February 19: makeup class, 3:054:20 inPhysics205
 Wednesday February 28: no class
 Wednesday April 11: noclass
 Monday April 23: makeup class, 3:054:20 in Physics 205
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Coursework Exam And Grades
Coursework will consist of weekly homework assigments and a take home final exam. The exam will be posted sometime late December and will be due January 4.
The final will be worth twice as much as a single homework assignment. So each of the 10 assignments will be worth 100/12% and the final 100/6% of the final score.
In grading the assignments and final exam, I will consider not only correctness, but how well they are written. You may lose points if your writing is careless, disorganized, or difficult to read.
I hope and expect that cumulative scores will be such that letter grades will be assigned according to the following scale:

A : cumulative score in

B : cumulative score in [80%, 90%)

C : cumulative score in [65%, 80%)

D: cumulative score in [50%, 65%)

F: cumulative score less than 50%.
The cutoffs for the letter grade sign will be set at the very end of the course, when all the scores have been computed.
I may change these cutoff scores if I find it necessary, although no changes will be made that would result in a tougher scale than the above.