This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2019. Here you will find all the practical informations about the course, changes that take place during the year, lecture notes, etc.

A few comments:

- I have made a few comments on some of the exam sheets (most notably recommendations for the retake). It may be useful/interesting for you to have a brief look.

- if you want to discuss or just see your exam: Monday, December 2nd, at 17:15, in the Atlas room in the KBG (Koningsbergergebouw). The plan is: I will briefly discuss the solutions of the exam exercises (on the blackboard) and then you can have a look at the exams (and eventually ask questions).

- as predicated, I think the marks are below your real potential- just that you simply had not enough time to properly revise the material (and "let it sink in"). This is also clearly shown by the fact that many of you solved the more difficult exercises (on stuff related to the last part of the course), but did not do well on the more basic/simpler stuff from the earlier parts of the course. Therefore, I am repeating the recommendation to all of you: spend a couple of more days with the material somewhere before the holidays, then go over the material once more in the day before the retake, then come to the retake to improve your mark!

- for those doing the retake: you cannot get a lower mark! (i.e., if it is lower, I will keep the previous one). But please do not abuse this: i.e. please do spend a bit of time revisiting the material- do not come just without preparing a bit, just because "there is nothing to loose".

THE LECTURES:

------ Mondays: 11:00-12:45 in BBG 061 in. the weeks 37-41, and Ruppert C in the weeks 42, 43 and 44.

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

LECTURER: Marius Crainic.

TEACHING ASSISTANTS: Maarten Mol.

THE WERKCOLLEGES:

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

------ Wednesdays: 09.00 - 10.45: 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, at the start of the werkcollege on Wednesday.
**Note: please do not send your homework by email, unless you agree 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.

- hand in exercises- see above. The average 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:November 6th 2019, 9:00-12:00 in Educ Beta.
** 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}

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 chapters with the reminders on Topology and Analysis.

The chapter on manifolds (Chapter 3).

The chapter on tangent vectors and vector fields (Chapter 4).

The chapter on differential forms and Stokes (Chapter 5).

♣ **WEEK 37/Lecture 1 (September 9):** Reminder on Topology and Analysis.

Werkcollege (September 9 and September 11): read the lecture notes containing the reminders on Topology and Analysis.

♦ **WEEK 37/Lecture 2 (September 11):** Reminder on Topology and Analysis .

Exercises for the werkcollege: see the lecture notes.

Homework: see here.

♦ **WEEK 38/Lecture 3 (September 16):** Definition of (smooth) manifolds, smooth maps and first examples (opens inside Euclidean spaces, spheres, embedded submanifolds of the Euclidean spaces).

Exercises for the werkcollege: lecture notes.

♦**WEEK 38/Lecture 4 (September 18):** Recap of the basic notions (smooth manifolds, smooth maps, immersion/submersion) then more examples (most notably the real and the complex projective spaces, the orthogonal group O(n)), embedded submanifolds and the general regular value theorem.

Exercises for the werkcollege: lecture notes.

Homework: Exercise 3.21.

♦ **WEEK 39/Lecture 5 (September 23): ** more on submanifolds (of general manifolds) and Lie groups.

Exercises for the werkcollege:

♦**WEEK 39/Lecture 6. (October 25): ** Immersed submanifolds; motivation for tangent spaces; the first definition of tangent spaces.

Exercises for the werkcollege:

Homework: With the definition of the tangent spaces $T_pM$ given in the lecture (Def. 4.1 in the lecture notes), define yourself the differentials $(dF)_p: T_p M \rightarrow T_{F(p)} N$ (for smooth functions $F: M\rightarrow N$ between two abstract manifolds $M$ and $N$, and $p\in M$), prove that $(dF)_p$ is always a linear map, and show that the chain rule still holds in this generality.

♦ **WEEK 39/Lecture 7 (September 30): ** Tangent Vectors.

Exercises for the werkcollege:

♦**WEEK 39/Lecture 8 (October 2): ** Tangent vectors.

Exercises for the werkcollege:

Homework: no homework.

♦ **WEEK 40/Lecture 9 (October 7): ** Started with vector fields, up (and including) to the interpretation the interpretation as derivation- with application to the construction of the Lie bracket of vector fields.

Exercises for the werkcollege:

Homework (for October 16): : see here. This is longer than previous times (except for the one of last week ...). However, I hope it is more doable (just that it will take a bit more time); on the other hand, I will count it as two homeworks (hence, if you do it well, it has a larger weight in the final mark).

♦**WEEK 40/Lecture 10 (October 9):** Integral curves, flows of vector fields (discussed the complete case, started the general- to be recalled and continued next time).

Exercises for the werkcollege:

Homework: see above.

♦ **WEEK 41/Lecture 11 (October 14):** Finished integral curves. Then moved on to 1-forms.

Exercises for the werkcollege:

♦**WEEK 41/Lecture 12 (October 16):** 1-forms; then one final look back at vector fields versus 1-forms, in the following table . Then moved to k-forms and discussed the linear algebra (on a vector space $V$), with wedge product, up to exhibiting a basis for the exterior powers.

Exercises for the werkcollege:

Homework: Do exercise 5.1 from the new set of lecture notes (be aware: unlike the notes of last year, the exercise has no 4 questions!). .

♦ **WEEK 42/Lecture 13 (October 21):** Differential forms; DeRham differential.

Exercises for the werkcollege:

♦**WEEK 42/Lecture 14 (October 23):**

Exercises for the werkcollege:

Homework: this is a homework that counts more, to make up for the points lost in the first homework.

♦ **WEEK 43/Lecture 15 (October 28):**

Exercises for the werkcollege:

♦**WEEK 43/Lecture 16 (October 30):**

Exercises for the werkcollege:

Homework: TBA.

**Enjoy the sphere ** (and not only).