(Credit: Paul Newlander)

## MAT367S Differential Geometry

**Classes:** MWF1;
** Room:** SS2106.

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.

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.

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

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)

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.