(Credit: Paul Newlander)

MAT367S Differential Geometry

Classes: MWF1; Room: SS2106.

Course description

Course syllabus

Prerequisites:

MAT257Y1/(MAT224H1, MAT237Y1,MAT246H1,and permission of instructor)

The course is designed as a continuation of the geometric concepts encountered in MAT257Y. Consequently, some familiarity with manifolds inside Euclidean space (as discussed e.g. in MAT257) will be assumed, as well as some familiarity with differential forms. You should be very comfortable with the implicit and inverse function theorems from multivariable calculus. You'll also need a good background knowledge of linear algebra (ideally, on the level of MAT247). While MAT267/MAT244 is not technically a prerequisite, it would be helpful to be familiar with the existence and uniqueness theorem for systems of first order ODE's.

Lecture Notes

Suggestions are welcome; please let me know in case you find errors, typos, etc.

Lecture Notes

(I'm using the Springer fonts to make it look like a book.) Not all topics in the notes will be covered in class, and you're not required to know topics that were not covered in class or as homework.

Problem sets

Here I will post the problem sets for the course. (They will be posted approximately one week before the due date.) The solutions should be handed back in class, on the given due date. (If you cannot make it to class, I will also accept pdf files (scanned or typed) sent by email, but this would have to reach me before class time.)

Our late policy is as follows: No late assignments will be accepted.

You have to write up the solutions yourself, in your own words. If you find the solutions in books or on the internet, you must quote your source (and still write it up in your own words!)

Problem set #1 Due: January 25, 2016
Problem set #2 Due: February 8, 2016
Problem set #3 Due: February 24, 2016
Problem set #4 Due: March 11, 2016
Problem set #5 Due: March 28, 2016
Problem set #6 Due: April 8, 2016

Links

An animated Klein bottle
A large Klein bottle
A picture of Boy's surface
A video of Boy's surface
An animation of Boy's surface
Turning the sphere inside out (part I)
Turning the sphere inside out (part II)

References

Of the following references, Boothby's book (first half) is perhaps closest to our approach. The book by Gadea et al contains many worked exercises for manifolds; note that UofT library has an electronic copy.

  • William Boothby: An Introduction to Differentiable Manifolds and Riemannian Geometry. Academic Press.
  • Pedro M. Gadea, Jaime Munoz Masque, Ihor V. Mykytyuk: Analysis and algebra on differentiable manifolds: a workbook for students and teachers Springer, 2013
  • John Lee: Introduction to Smooth Manifolds. Springer.
  • Loring Tu: An introduction to manifolds. Springer.