# 21-470 Topics in Analysis: Differential Geometry

### Description

Differential Geometry stands at a crossroads in mathematics. It is the point where topology, geometry and analysis combine. It is the language for many fields of modern mathematics, and finds applications in physics, robotics and other fields.

In this course, we will introduce the basic concepts, and work to develop an intuitive understanding of them. We begin with the basics. Chapters 1-3 review some of the background material and develop the definition of a differentiable manifold. Chapters 3 and 4 introduce the idea of a tangent vectors on a manifold, and go on to discuss the tangent bundle and other types of tensor. Chapters 6 and 7 discuss calculus on manifolds, the ideas of multivariable calculus and analysis in this new setting. Chapter 8 discusses curvature, hilighting some classical results along the way.

While it will not be possible to cover the entire textbook, it is hoped that a sufficient amount of each topic can be covered to give a good feeling for the flavor of the subject.

### News

Friday 7 May: The Final Exams are available for pickup. They have been for several days, but I have not heard from a few of you. I thought I would post the news here.

Tuesday 27 April: The reading assignments and homework assignments for Week #15 have been posted. Follow the link from the Schedule page.
Friday 16 April: The reading assignments and homework assignments for Week #14 have been posted. Follow the link from the Schedule page.
Friday 9 April: The reading assignments for Week #13 have been posted, along with the exercises due on Friday. Follow the link from the Schedule page.

Tuesday 6 April: The reading assignments for Week #12 have been posted, along with the exercises due on Friday. Follow the link from the Schedule page.
Tuesday 23 March: The reading assignments for Week #10 have been posted, along with the exercises due on Friday. Follow the link from the Schedule page.
Thursday 18 March: The recent "show and tell" items from class can be found on the Web. Here are the Enneper's Surface snow sculpture, and the Costa surface, both the Gauss map version, and the LEGO version. Click around on theses pages, there is lots of interesting stuff. Some of it is related to mathematics. You also might find this gallary of surfaces interesting. It 23 surfaces in different categories that can be rotated freely.
Thursday 18 March: The reading and homework for Week #9 have been posted. Follow the link from the Schedule page.
Monday 1 March: The reading assignment for Week #8 has been posted. There is no assignment due this week, aside from the postponed assignment from Friday.
Tuesday 24 February: The reading and homework for Week #7 have been posted. We have gotten a bit out of sync with the Schedule. I will try to revise that before too much longer.
Monday 16 February: Solutions to a selection of problems from Chapter 1 are available outside my office door, including most of the latest homework assignment.
Monday 16 February: Solutions to problems 1.2.1 and 1.3.13 are available outside my office door. More to come.
Monday 16 February: Now the graded homeworks really are outside my office door.
Sunday 15 February: Your graded HW#5 (and any previous assignments) can be picked up from my office. They are in an envelope on my door, so come by whenever is convenient.
Sunday 15 February: I will be holding office hours from 2:00-4:00 on Tuesday before the exam.
Sunday 15 February: Exam #1 will be held in class on Wednesday 18 February. You may bring a half-page of notes with you to the exam. (One half of an 8.5x11 sheet of paper, written on one side.) The exam will cover Sections 1.1-1.5 and 2.1-2.4 of Oprea. Check back here often for exam related updates.
Sunday 15 February: The reading assignment for Week #6 has been posted. As Exam #1 is being held in class on Wednesday, there will be no homework assignment.
Monday 9 Febraury: Solutions to the first and third homework assignments are available outside my office door. The second homework assignment should make an appearance soon.
Monday 9 Febraury: The reading and homework for Week #5 have been posted. There is an assignment due on Friday 13 February.
Monday 2 February: The reading and homework for Week #4 have been posted. There is an assignment due on Friday 6 February.
Tuesday 27 January: CORRECTION TO 1.3.13: There is an error in the definition of linking number. After summing the +1's and -1's you must divide by 2. I've noted this on the homework assignment page as well.
Sunday 25 January: I also changed the reading assignment, as it referred to a section that is not in the first edition of the textbook. I guess I'll have to go through the schedule to see if there are other changes that need to be made. I'll have to do that anyway if we decide to move some of the material relating to higher dimensional manifolds to earlier in the semester.
Sunday 25 January: I've added a comment about problem 1.3.9 that may help with your understanding of the situation.
Friday 23 January: The homework for Week #3 has been posted. I'll add some comments about a few of the problems soon. Follow the link from the Schedule Page.
Thursday 22 January: The reading for Week #3 has been posted.
Monday 21 January: I have made a correction to Wednesday's homework assignment. Problem 1.2.12 should have been 1.2.1. I have made this change on the assignment page.
Thursday 15 January: I've updated (and extended) the homework assignment for Week #1, which is now due on Wednesday 21 January.
Thursday 15 January: Since the bookstore has stocked the first edition of the textbook, we will work from this one rather than the second edition. I'll have to update the homework assignmnet.
Tuesday 13 January: The reading and homework assignment for Week #1 have been posted. There is a short assignment due this Friday, January 16.
Monday 12 January: Welcome to 21-470 Topics in Analysis: Differential Geometry.