Manifolds (Differentieerbare varieteiten)

Blok 1, 2021/2022

After exam info:

Here you can download the exam and some solutions , written in a rush so that I could send them to you right after the exam ... but unfortunately the "blackboard" does not seem to work properly today (I tried to send you an email a couple of times, but the email function seems to be dead). Hopefully you will see this here in time (while the exam is still fresh in your mind).
This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2022. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc. I will try to make use also of the blackboard.
THE LECTURES:

LECTURER: Marius Crainic.

THE WERKCOLLEGES (devoted to solving exercises from the lecture notes, getting feedback for the homeworks, and/or asking for help/discussing aspects of the course that you did not understand):

TEACHING ASSISTANTS: Aaron Gootjes-Dreesbach, Bouke Jansen, Ruben de Vries

HOMEWORKS: Each Wednesday, in the afternoon, you will receive a homework that you have to "hand in" by the next Wednesday, 9AM. If you are not present at the wekcollege on Wednesday please discuss with the TAs in advance about the way you can hand in your homework. Please do not send your homework to the lecturer (then your homework may get lost ...).

Each homework will be corrected taking into account also the mathematical quality of the writing, which will count for 25 percent of the mark. If you score low on that, you have the chance to improve your writing (hence also your mark) by submitting your homework a second time (but please do not abuse, otherwise we have to put an upper limit on how many homeworks each one of you can resubmit).

Passing the course: that will be subject to some minimal requirements, and the final marks will depend on several items:

LECTURE NOTES: We will be using an updated version of the lecture notes from the last year (but please be aware that they may still contain some typo); comparing with the previous year, besides correcting some typos, we are also trying to add a lot more pictures to support the theory (thanks to Aaron!). The new version of the notes will be made available as we go. We start with the updated version of the reminder chapter .

September 13: Chapter 1 and 2 together .

October 3: Chapter 1,2 and 3 together (includes the previous version).

October 17: Chapter 1,2, 3 and 4 together (includes the previous version).

October 26: The complete sets of notes. Please be aware that the Chapter on integration and Stokes has been modified quite a bit comparing with the previous years (it is more streamlined now, hopefully easier to grasp).

THE SCHEDULE WEEK BY WEEK

(I will try to add, after each lecture, a description of what was discussed in the lectures + the exercises from the lecture notes that you are supposed to do during the werkcollege)

♣ WEEK 37/Lecture 1 (September 12): Reminder on smooth maps, differentials, inverse functions theorem, immersions/submersions, charts, representing functions in charts.

Here is a copy of what was on the blackboard (more or less) during this lecture.

Werkcollege (September 12 and September 14): read the lecture notes containing the reminders on Topology and Analysis. Recommended exercises for Wednesday (September 14): 1.54, 1.60, 1.44, 1.45, 1.48.

♣ WEEK 37/Lecture 2 (September 14): End the reminder on smooth submanifolds in $R^{L}$, then start with abstract manifolds (up to smooth structures).

Here is a copy of what was on the blackboard (more or less) during this lecture.

Homework (for the next week on Wednesday): homework 1.

Suggested problems for the werkcollege held today (regarding the 2nd lecture): 2.5, 2.9, Check the properties of A^max mentioned right above Def. 2.10, 2.11.

♣ WEEK 38/Lecture 3 (September 19): Recap on smooth structures; then (smooth manifolds) and smooth maps, with some examples and particular classes (e.g. diffeomorphisms, submersions, etc). In the last 10 minutes: recalled the regular value theorem (the version for functions between (opens in) Euclidean spaces).

Here is a copy of lecture 3 on the blackboard.

Suggested problems for the werkcollege held today (regarding the 3th lecture): 2.21, 2.22, 2.31, 2.41, 2.52.

♣WEEK 38/Lecture 4 (September 21): Examples: spheres (via RFT or via various atlases), torus, real/complex projective spaces, matrices, invertible matrices, started $O(n)$.

Here is blackboard-version of lecture 4 (more or less).

Homework (for the next week on Wednesday): homework 2.

♣ WEEK 39/Lecture 5 (September 26): continued with examples/constructions: finished O(n), then products, Lie groups, examples of those (the other Lie groups U(n) etc), embedded submanifolds, embeddings.

Here is blackboard of lecture 4 (more or less).

♣WEEK 39/Lecture 6. (September 28): Finished with examples, submanifolds, embeddings, immersions. Started a bit with tangent spaces.

Here is lecture 6 - the blackboard version (more or less).

Homework (for the next week on Wednesday): homework 3.

♣ WEEK 40/Lecture 7 (October 3): Tangent spaces.

Please note: the lecture notes have been updated (I included also the chapter on tangent vectors/fields): Chapter 1,2 and 3 together .

Here is lecture 7 - the blackboard version.

Suggested problems for the werkcollege held today (regarding the 6th lecture): 2.75, Check that the map f_0 from Ex. 2.68 with codomain R^3 is an embedding, 2.84, 2.82.

♣WEEK 40/Lecture 8 (October 5): F

Here is the blackboard for lecture 8.

Homework (for the next week on Wednesday): homework 4.

♣ WEEK 41/Lecture 9 (October 10): Vector fields.

Here is the blackboard for lecture 9.

♣WEEK 41/Lecture 10 (October 12): Integral curves, flows of vector fields (assuming $X$ is complete; the general case will be done at the beginning of the next lecture).

Here are blackboard snapshots for lecture 10.

And here is homework 5 (for the next week on Wednesday), which starts with a couple of pages of bla-bla (hopefully of some use to you) and then an exercise which is perhaps long, but more generous in providing points (in particular, you do not have to answer all the questions to get the normal maximum number of points).

♣ WEEK 42/Lecture 11 (October 17): Flows in the general case, proof that on compact manifolds all vector fields are complete, then the principle of ``Lie derivatives along vector fields'' for various ``objects'', with the following particular cases of "objects": functions and vector fields. In those case one recover the old $L_X$ on functions, and the Lie bracket $[X, -]$ on vector fields. With the consequence on relating $[X, Y]$ to the commutativity of the flows of $X$ and $Y$. Then we moved to the next chapter, a bit: recalled the notion of the dual of a vector space, dual basis, then the definition of the cotangent spaces $T_{p}^{*}M$ with examples of cotangent vectors given by differentials $(df)_p\in T_{p}^{*}M$, and then the canonical basis $(d\chi_1)_p, \ldots, (d\chi_m)_p$ (basis of $T_{p}^{*}M$) associated to a/any chart $\chi$ around $p$.

Here are blackboards for lecture 11 (except for blackboard number 6, which seems to have been missed. But that contained very little- just the slogan: for a vector field $X$ on $M$, and any ``natural object $\xi$ on $M$'', the Lie derivative of $\xi$ along $X$ (measuring ``the variation of $\xi$ along the flow of $X$'') is: and I just copied formula (3.6.3) from the lecture notes.

Note: I have added the chapter on differential forms. In case you missed it, here are chapters 1,2, 3 and 4 together .

♣WEEK 42/Lecture 12 (October 19): 1-forms, then did the linear algebra for $k$-form: $T^kV^*$, $\Lambda^kV^*$, $Alt$, defined the wedge product motivated by the desire to produce forms of high-degree from ones of lower degree (and, ultimately, be able to exhibit bases for $\Lambda^kV^*$), discussed the main properties of the wedge product and discussed how wedging 1-forms one can produce bases for $\Lambda^kV^*$.

Homework (for the next week on Wednesday): homework 6.

Here are the blackboard version of lecture 12.

♣ WEEK 43/Lecture 13 (October 24): Differential forms of arbitrary degrees, DeRham differential $d$, the Lie derivatives $L_V$ and the interior products $i_V$ (to be continued).

Here is the blackboard version of lecture 13.

♣WEEK 43/Lecture 14 (October 26): Looked again at $k$-forms, DeRham differential, Lie derivatives and interior products (and Cartan's magic formula): the main properties that determines the uniquely (recap), explicit computations, and then conceptual proofs of most of the properties. Then, the last 10 min, the def of volume forms, with a look in coordinates, and the examples of the canonical volume forms on $R^m$ and on spheres.

Here is blackboard-version of lecture 14 .

Comment 1: we also discussed about the exam a bit. If you missed the class, please talk to your colleagues who were present. In principal, the most important piece of information is that some of you (about 3) opted for having an open book exam, i.e. they can use the lecture notes for the course (but with my strong advise of not spending all three hours going through the notes), while the rest followed my advise of preparing ``cheat sheets'' (2 sheets, i.e. 4 pages in total).

Homework 7.

Comment 2: please try to hand in Homework within the usual limit of one week. That is particularly important if you want to discuss with the teaching assistants specific points from your homework. However, we will accept the homeworks that are handed in later, but no later than the exam.

Comment 3: Homework is the last homework that will count for now. I will try to give another homework, next week, to be handed in by the retake exam. That will have to be done only by the people that will do the retake.

Comment 4: The complete sets of notes. Please be aware that the Chapter on integration and Stokes has been modified quite a bit comparing with the previous years (it is more streamlined now, hopefully easier to grasp).

♣ WEEK 44/Lecture 15 (October 31): Volume forms, orientations, integration on manifolds without boundary, Stokes.

Blackboard-lecture 15 .

♣WEEK 44/Lecture 16 (November 2): DeRham cohomology, application to the hairy ball theorem (arbitrary dimension): a sphere has a nowhere vanishing vector field if and only if the dimension of the sphere is odd.

Homework (to be handed in before the retake): compute the integral from exercise 6.10 (page 172 of the lecture notes).

Blackboard-lecture 16 (but, unfortunately, the picture with blackboard no 5 is missing. But that is not needed for the rest: it is the giving two examples of homotopy equivalences (to get a feeling about that notion): between $R^m$ and a point (contracting the Euclidean space into a point), and between $R^{m}-\{0\}$ and $S^{m-1}$ (retracting the Euclidean space with origin removed into the sphere inside it).

After exam info:

Here you can download the exam and some solutions , written in a rush so that I could send them to you right after the exam ... but unfortunately the "blackboard" does not seem to work properly today (I tried to send you an email a couple of times, but the email function seems to be dead). Hopefully you will see this here in time (while the exam is still fresh in your mind).

♣End exam: Tuesday, November 8, 17:00, Ruppert Witt.

IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. MORE PRECISELY, ALTHOUGH I DO TRY TO KEEP IT UPDATED (E.G. BY SPECIFYING WHICH IS THE HOMEWORK), SOMETIMES I AM NOT ABLE TO DO IT. IN SUCH CASES, PLEASE CONTACT YOUR COLLEAGUES THAT WERE PRESENT AT THE LECTURE OR WERKCOLLEGE.