# Manifolds (Differentieerbare varieteiten) 2018 (blok 1)

This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2018. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc.

# Results of the exam:

(a couple of the last homeworks do not have the last exercise separately corrected, hence the final mark of that homework is not added to the file but, in all such cases, it seems that it does not make any difference to the very final mark).

----> For those who want to see their exam: you can drop by our offices on any of the next two Mondays (November 26, December 3). You find us on the 8th floor of the math (Freudenthal) building- both me (M. Crainic) as well as Maarten Mol.

----> For those who plan to do the retake:

please make sure that you know how to compute the flow of a vector field, and how to use the regular value theorem to show that something is a manifold and to compute its tangent space. I think all that is a MUST for passing this course!

When, where, etc:

THE LECTURES:

------ Mondays: 11:00-12:45 in 611AB (Math Building/HFG)

------ Wednesdays: 11:00-12:45 in C101, ANDRO.

LECTURER: Marius Crainic.

TEACHING ASSISTANTS: Maarten Mol and Thomas Blom.

THE WERKCOLLEGES:

------ Mondays: 09.00 - 10.45: 611AB (Math Building/HFG).

------ Tuesdays: 17:00- 16:45: 611AB (Math Building/HFG).

HOMEWORKS:

There will be homeworks during the course, that will count for the final mark. Probably around 4-5 homeworks. You will receive them at the end of the last class of the week (Wednesdays) and you have to hand them in one week later. Note: please do not send your homework by email, unless you aggree with the TA. In any case: please never send the homework to me (the lecturer) by email!!! The danger is that it will get lost.

Here I will soon add a link where you can check the marks for the homeworks.

Marks: there will be:

- hand in exercises- see above. The everage of the marks for all the homeworks will give one mark HW (maximum 10).

- final exam, for which you will receive a mark E (maximum 10).

Note: you are allowed to bring with you, and use during the exam, three sheets of A4 papers (= six pages) containing definitions, theorems, etc from the course.

Coordinates of the exam: TBA.

Final mark: The final mark will be obtained by combining E (exam mark) and HW (homework mark), by the formula:

max{(7 E+ 3 HW)/10, (17E+ 3H)/20}

Lecture notes: Please be aware that the lecture notes may still contain typos. So, if there is something that you do not understand, please ask it at the werkcollege; also, if you find typos, please communicate them to the teaching assistants. This will be of great help to improve the lecture notes and make them into a regular "dictaat" that can be printed for students.

In principle, I will be using the file that I made last year, but which I will correct/improve as we go along with the course. But, as last year, once I post some material, later I will no longer make any big change to what was already posted (i.e. I will make sure that the numbering of the theorems etc will not change). So, in principle, once you printed a part, you do not have to print it again.

While the various parts of the lecture notes will be made available as we go, just for your curiosity, here is the link to the lecture notes of the last year.

The schedule week by week (here we will 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 10): Reminder on Topology.

Exercises for the werkcollege: read the lecture notes (see above) and try to redo the exam from the last year's course on Topology.

WEEK 37/Lecture 2 (September 12): Reminder on Analysis, up to (but excluding) section 5 (submanifolds of \$R^m\$).

Exercises for the werkcollege: see the lecture notes.

WEEK 38/Lecture 3 (September 17): Reminder on Analysis: submanifolds of \$R^m\$; tangent spaces.

Exercises for the werkcollege: see the lecture notes.

WEEK 38/Lecture 4 (September 19): Abstract manifolds. Here are the part of the notes that contain the chapter on manifolds (Chapter 3)

Exercises for the werkcollege: 3.1, 3.2, 3.3, 3.4, 3.6, 3.11, 3.12.

WEEK 39/Lecture 5 (September 24): The projective space again, submanifolds of \$R^n\$, submanifolds of a general manifold \$M\$, concentrating on embedded ones, then on immersed ones, and paying special attention to the regular value theorem.

Exercises for the werkcollege: see the notes.

WEEK 39/Lecture 6 (September 26): Lie groups (examples and def) and started talking about tangent vectors.

Exercises for the werkcollege: try 2 from Lie groups, and then as many as you can from tangent vectors. But, at tangent vectors, please carefully look at the theory, stare at it, ask yourself questions (don't be afraid of asking questions that are too simple!), try to get a feeling of what it really is about.

WEEK 40/Lecture 7 (October 1): Tangent vectors again (all points of view).

Exercises for the werkcollege: Make sure you did Exercises 4.1, 4.3, 4.5. Then go on to the other viewpoint on tangent vectors and do 4.9, 4.8, 4.12. If you have time and you think you really understood what tangent vectors are, then have a look at 4.6 as well.

WEEK 40/Lecture 8 (October 3): Finished tangent vectors and then started vector fields, up to (and including) the Lie bracket of vector fields.

Homework: Homework for Tuesday, October 9.

WEEK 41/Lecture 9 (October 8): Vector fields up to (but excluding) integral curves/flows.

Exercises for the werkcollege:

WEEK 41/Lecture 10 (October 10): Integral curves/flows.

Homework: Homework for Wednesday, October 17..

WEEK 42/Lecture 11 (October 15): 1-forms

Exercises for the werkcollege:

WEEK 42/Lecture 12 (October 17): Differential forms

Homework: Exercise 5.10.

WEEK 43/Lecture 13 (October 22): De Rham differential.

Exercises for the werkcollege:

WEEK 43/Lecture 14 (October 24): End the discussion on DeRham differential, interior products and Lie derivatives, Cartan's magic formula, orientation on vector spaces.

Exercises for the werkcollege: Homework for the next week..

WEEK 44/Lecture 15 (October 29): Stokes on manifolds without boundary, consequence on volume forms not being exact, and start of DeRham cohomology and the cohomological formulation of Stokes.

WEEK 44/Lecture 16 (October 31):

Exercises for the werkcollege:

Homework:

WEEK 46: EXAM (November 7): 09.00 - 12.00. Gebouw: RUPPERT, Zaal: Blauw.

WEEK 1/2018: RETAKE EXAM (January 2nd), 09.00 - 12.00, Gebouw: BBG, Zaal: 385.

# 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.

Enjoy the sphere (and not only).