(Credit: Paul Newlander)

## MAT367S Differential Geometry

Classes: MWF1; Room: MP137

 Course description

Course syllabus

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

I will update and polish the notes during the semester. Suggestions are welcome; please let me know in case you find errors, typos, etc.

Lecture Notes

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

The required homework problems for this course will be sent online, via Crowdmark. You will be asked to submit the solutions electronically, via Crowdmark.

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!)

 Calendar

##### Week 1: January - January 13
• Material covered: Introduction, informal discussion of surfaces. Definition of manifolds. Examples: 2-spheres, affine lines in $\mathbb{R}^2$.

Of course, the best option (for us) is if you type the solutions using LaTeX .

If you encounter any problem with the Crowdmark system, please let me know quickly via email. Hopefully, once you get used to the uploading process, you'll get to like the system. The marked assignments will be returned online; if there are any questions about the marking you can send an email to Mykola (or, if necessary, me), including the link to your problem set.

• ##### Week 2: January 16 - January 20
• Material covered: Compatibility of charts, atlases, maximal atlases. Review of equivalence relations'. The Hausdorff condition; example of a non-Hausdorff manifold. Rigorous definition of a manifold. Examples: Spheres, projective spaces.
• My office hours are set for Fridays, 3:30-4:30. Mykola's hours will be T 10-11 and W 3-4 in BA 6135 (or appointment, if neither time works). I noticed a bad typo on the syllabus: Term test 1 is on Monday, February 27 (not the 28th).
• Homework #1 is due today at 10 pm. Again: please don't leave it to the last minute to submit your work! And if you encounter problems with the uploading, please send me an email right away. Normally it should go smoothly, though.
• ##### Week 3: January 23 - January 27
• Material covered: More examples: Real and complex Grassmannians. Open subsets of manifolds, compactness. Orientated manifolds.
• ##### Week 4: January 30 - February 3
• Material covered: More examples: Smooth functions on manifolds, smooth maps between manifolds.
• ##### Week 5: February 6 - February 10
• Material covered: Examples of smooth maps. Maps to and from projective spaces. The Hopf fibration. Submanifolds. (Examples; basic results: submanifolds inherit a manifold structure such that the inclusion map is smooth, their manifold topology coincides with their subspace topology.)
• ##### Week 6: February 13 - February 17
• Material covered: Submanifolds given as level sets. The rank of a smooth map between manifolds; maps of maximal rank. Submersions, immersions, local diffeomorphisms.
• This coming week is Reading week' -- no classes. Office hours will be held, though. Please remember that TERM TEST 1 is on Monday, February 27 (right after reading week). The test will take place during class time, in the usual classroom. I will announce more about the test via Blackboard; I'll also post a practice test on Blackboard, under Course Materials'. Please note, however, that we will use Crowdmark for the exam; hence the format will be a bit different.
• ##### Week 7: February 27 - March 3

• Monday, February 27, was the date of term test #1.
• Material covered: Tangent vectors as directional derivatives. Three equivalent definition of the tangent space, using curves, charts, or the product rule.
• ##### Week 8: March 6 - March 10

• Material covered: The tangent map $T_pF$ of a smooth map $F$. Special case: If $F$ maps between open subsets of $\mathbb{R}^m,\mathbb{R}^n$ then $T_pF$ is the Jacobian matrix. Tangent spaces to submanifolds $S\subseteq M$, tangent spaces to regular level sets $S=F^{-1}(q)$, calculation of critical points of $h|_S$ when $S\subseteq \mathbb{R}^m$ is a submanifold. Matrix Lie groups $G$ and their Lie algebras' $\mathfrak{g}$.
• ##### Week 9: March 13 - March 17

• Material covered: Vector fields and Lie brackets. Related vector fields. Vector fields tangent to a submanifold.
• ##### Week 10: March 20 - March 24

• Material covered: Differential forms: Review, coordinate-free interpretation.
• ##### Week 11: March 23 - March 29

• Wednesday, March 25, was the date of term test #2.
• Material covered: Exterior differential, Cartan calculus. Integration of differential forms.
• ##### Week 12: March 30 - April 5

• Material covered: Stokes' theorem, volume forms, and applications: linking numbers, winding numbers.

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.