2026/2027, blok 1
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: Thijmen Kruithof and Yoram Veltmaat.
HOMEWORKS: Each week there will be a homework. The homework exercise will be announced on Monday or Wednesday afternoon. You can work on it till the next Monday morning, but you are NOT required to write a complete solution at home and then hand it in. Instead, on each Monday, at the werkcollege, in the last 30-40 minutes of the werkcollege, you are supposed to write down the solution to the homework exercise (without access to other material) and hand it in to the TAs at the end.
Passing the course: that will be subject to some minimal requirements, and the final marks will depend on several items:
F= min{10, max{(6 E+ 4 HW)/10+ B} }
rounded off to the closest integer if F< 6, and to a half integer otherwise. The requirements for passing the exam are: H and E have to be at least 5 (before rounding) and F has to be at least 6.
LECTURE NOTES: Lecture notes 2026/2027 .
♣ Monday, September 7: Lecture 1. An introduction: some of the keywords of this course; Theorem 1.50; the notion of topological chart and smooth compatibility of charts. Plan for the next time: the definition of manifolds (smooth atlases, smooth structure= maximal smooth atlases; first examples).
Blackboard photos of lecture 1..
Exercises for the next two werkcolleges (Wednesday and next Monday morning): 1.24, 1.38, 1.53, Examples 2.3 and 2.4, 2.11, 2.26 then, if you want to avoid topology, look at 2.34 and 2.179.
♣ Wednesday, September 9: Lecture 2. In the first part we discussed the concept of "(topological) chart on a space $M$" pointing out that, while when $M$ is inside some $R^n$ one can make sense of the concept of (classically-)SMOOTH chart (by making use of classical analysis on $R^n$), in general we cannot talk about "smooth chart on a topological space". In the second part we moved to the concept of smooth structure/manifolds (with the intermediate concepts of: change of coordinates between charts, smooth compatibility of charts, smooth atlases, and smooth structures). While "smooth charts of a topological space" is a concept that is not defined/does not make sense, "smooth charts of a manifold" makes sense basically by the definition of smooth structure (a collection of (compatible) charts which, ultimately, are called the "smooth charts of the manifold"). Take your time to think about what we are doing here ... what we want (be able to talk about smooth charts) is realised by brute force ... kind of tautologically ... it looks a bit cheap, doesn't it?
Blackboard photos of lecture 2..
♣ Monday, September 14: Lecture 3. Revisited the concept of smooth (abstract) manifold, and smooth maps between them, with the particular cases of observable, smooth curves, and diffeomorphisms, Then we looked as some examples like Euclidean spaces and opens inside them, copies of them in disguise (like space of matrices), a weird one on $R$ for which $t\mapsto t^3$ is used as chart, then we moved the spheres and projective spaces as smooth manifolds. .
Blackboard photos of lecture 3.
Exercises for the next two werkcolleges (Wednesday and next Monday morning): go over the blackboards from Monday's lecture and make sure that the examples there are clear, and read Remark 2.13 from the notes. Also, do not forget to solve the exercise left on the bottom of blackboard no 2 (under Remark 3), saying that for $f: M\to N$, if the smooth structures of $M$ and $N$ are induced by two smooth atlases $A$ and $B$ (not necessarily maximal) then for $f$ to be smooth it suffices to check the definition using only charts from $A$ and $B$, respectively. Also, do not forget to go again over the weird structure on $R$. i.e., do Exercise 2.55. Then 2.38. Then do Exercise 2.40 and the part of Exercise 2.11 from which you removed the first two lines. Then the weird "non-Hausdorff manifold" from Exercise 2.179. Then Exercise 2.183.
♣ Wednesday, September 16: Lecture 4. Revisited the basic concepts and examples, adding the first version of the RVT (regular value theorem), products of manifolds, submanifolds. Then we looked a bit at Lie groups, also explaining hot the RVT can be used to prove that $O(n)$ is a smooth manifold and, then, a Lie group.
Blackboard photos of lecture 4..
♣ Monday, September 21: Lecture 5. Tangent spaces (geometric and abstract)- see the blackboard fotos. .
Blackboard photos of lecture 5..
Exercises for the werkcollege: 2.110, 2.112, then think about items (i) and (ii) of the homework, then 2.111, then move to item (iv) of the homework. Once all of these are clear, show that you master everything by doing Exercise 2.122 (probably the right moment for that exercise would be the werkcollege from Wednesday September 30).
♣ Wednesday, September 23: Lecture 6. Recap on tangent spaces/tangent vectors plus the fact that any tangent vector $v$ induces an important operation on $C^{\infty}(M)$, then we discussed the differential of a smooth map (and introduced the concepts of submersions/immersions between general manifolds) and we explained that, in the case of (submanifolds of) $\mathbb{R}^n$, via the standard identifications, the differential becomes the usual differential from calculus.
Blackboard photos of lecture 6..
Exercises for the werkcollege: next to the ones from last time (see above): have a look at the one at the bottom of blakboard 1', then move to the bottom of blackboard 2 and make sure you understand the two descriptions of $\partial_v$ and show that they are the same.
♣ Monday, September 28: Lecture 7. Tangent vectors as derivations; the definition of vector fields. More on vector fields next time.
Blackboard photos of lecture 7..
Exercises for the next two werkcollege: 3.22, 3.24, 3.25, 3.28, 3.34.
♣ Wednesday, September 30: Lecture 8. Vector fields as derivations; the Lie bracket of vector fields.
Blackboard photos of lecture 8..
♣ Monday, October 5: Lecture 9. Integral curves, and the start of the discussion of the flow.
Blackboard photos of lecture 9..
Exercises for the next two werkcolleges: 3.64, 3.68, 3.65, 3.80, 3.75 and, if you have time try one of the more elaborate ones from the end of the chapter, such as 3.102, 3.104 or 3.105.
♣ Wednesday, October 7: Lecture 10. The flow of vector fields and basic properties, including the completeness of all vector fields on compact manifolds (with proof). Then the general principle of "Lie derivatives of objects on $M$ along vector fields $X$", as a measure of the change of the object along the flow of $X$, with the example where the "objects" were functions $f$ (when we recover the old $L_X(f)$_ and the example where the "objects" were vector fields $Y$ (when we recovered the Lie bracket $[X, V]$).
Blackboard photos of lecture 10..
♣ Monday, October 12: Lecture 11. First a reminder on vector fields plus the fact that the flows of two vector fields $X$ and $Y$ commute if and only if their Lie bracket vanishes: $[X, Y]= 0$ (or: " $X$ and $Y$ commute"). Then, making use of the construction $V\mapsto V^*$ that associated to any vector space its dual (another vector space!) and the fact that any basis of $V$ induces one of $V^*$ (the dual basis), we went through the basics on $TM$ and vector fields and replaced $V$ by $V^*$ for each $V= T_pM$, obtaining the cotangent spaces $T_p^*M$ and, instead of vector fields $X$, which go $p\mapsto X_p\in T_pM$, 1-forms $\omega$, which go $p\mapasto \omega_p\in T_p^*M$. We adapted the list of basics on vector fields to a similar list for 1-forms; in particular, we explained that 1-forms can also be interpreted as $C^{\infty}(M)$-linear maps from $\mathfrak{X}(M)$ to $C^{\infty}(M)$. At the end I also pointed out one great advantage of 1-forms over vector fields: they can be moved along maps- more precisely, for any smooth $F: M\to N$, one can pull-backl 1-forms on $N$ to 1-forms on $M$, i.e. we have an induced map $F^*: \Omega^1(N)\to \Omega^1(M)$. Next time we will proceed similarly but, instead of using the operation $V\mapsto V^*$, we will use $V\mapsto \Lambda^kV^*$, where $\Lambda^kV^*$ consists of k-multilinear maps on $V$ that are antisymmetric (what we discussed today will be recovered when $k= 1$).
Blackboard photos of lecture 11..
Exercises for the werkcollege: 4.4, 4.7, 4.11, 4.12, (a)-(d) of 5.33, (a)-(e) of 4.72, 4.8. If you have more time, start looking at the questions from the previous exams that belong to the part that we have already covered.
♣ Wednesday, October 14: Lecture 12. With the aim of discussing differential forms in arbitrary degrees, or $k$-form for arbitrary natural number $k$, most of the lecture was linear algebra: how to construct and deal with the space $\Lambda^kV^*$ associated to an arbitrary vector space $V$ (consisting of $k$-multilinear maps on $V$ which are also skew-symmetric), including the operation of wedge-product and the basis induced by a basis of $V$. Once that was done, we went through the basics on 1-forms and just replaced $1$ by $k$, and one argument $X$ by $k$-arguments $X_1, \ldots, X_k$. What was extra was the operation of wedging two forms: for $\omega\in \Omega^p(M)$ and $\eta\in \Omega^q(M)$ we can form $\omega\wedge \eta\in \Omega^{p+q}(M)$. And what was truly new was the DeRham differential $d: \omega^k(M)\to \Omega^{k+1}(M)$. We discussed the DeRham differential only rather briefly: I just mentioned its basic properties and explained in one example how those basic properties are actually enough for doing computations.
The content of this lecture was rather dense, as I was trying to gain some time for you for getting used with such computations. Regarding the theory, we will spend more time with this material, including also some more motivation and intuitive explanations (which I skipped for now to get a bit further). Regarding the computations, since the example at the end was rather fast, I wrote down some similar examples of computations of $d(\omega)$ in which I go slower (the first three pages of the notes below), followed by a similar computation of pull-backs $F^*(\omega)$ (last page of the notes). Here are the handwritten notes with those computations.
Blackboard photos of lecture 12..
♣ Monday, October 19: Lecture 13. Recap on differential forms plus some more explanations/insight/strategy on how to compute things. Then we discussed in more detail DeRham differential, interior products and Lie derivatives (all acting on forms) and the Cartan magic formula.
The blackboard from the werkcollege..
Blackboard photos of lecture 13..
Exercises for the next two werkcolleges: 4.54, 4.56, 4.66, 4.71, and then look at the exercises from the previous exams that you did not look at yet. .
♣ Wednesday, October 21: Lecture 14. Volume forms and orientability; differential forms of top-degree (i.e. $m$-forms, where $m= \dim M$) and the fact that, locally, they are encoded by functions and, when changing coordinates, those functions change by (multiplying with) the Jacobian function of the change of coordinates. Given the change of coordinates formula for the standard integration on $\mathbb{R}^n$, it follows that, on manifolds, it is not the functions that can be integrated (without making extra-choices), but top-degree forms. We stated the existence and uniqueness property of the integration operation and very briefly discussed the proof (that will be discussed next time). Making use of the integration, we made sense of the volume of a manifold w.r.t. a fixed volume form. Then we discussed Stokes' theorem and then, making use of the volume, we deduced that, on compact manifolds, volume forms are never exact. That will be used in the last lecture to provide a proof of the hairy ball theorem, as a beautiful application of differential forms.
Blackboard photos of lecture 14..
Exercises for the werkcollege: on Monday focus on computing some integrals. You need the help of the TAs for a couple of (simple) tricks.
♣ Monday, October 26: Lecture 15. Recap again on integration on Stokes + partitions of unity and the proof of the existence of the integration operation. Then variations on the notion of manifold (analytic manifolds, complex manifolds, manifolds with boundary). Then briefly explained how an orientation on a manifold with boundary induces an orientation on the boundary plus stokes on manifolds with boundary. Then, starting from the theorem that any compact topological manifold can be topologically embedded in some Euclidean space, as an application of partitions of unity, we discussed the smooth version of the theorem and, more generally, the concept of smooth embedding + the fact that that is equivalent to being an immersion and a topological embedding, while the very last condition follows from continuity and injectivity of the domain is compact (Inleiding Topologie stuff).
Blackboard photos from the werkcollege preceeding the lecture...
Blackboard photos of lecture 15..
♣ Wednesday, October 28: Lecture 16. Recap on forms and DeRham differential, then DeRham cohomology as a way to encode the distinction between closed and exact forms, then a couple of comments on how to associate numerical invariants to manifolds, including the Euler characteristic (related to the Euler formula $E-V+F= 2$). The homotopy invariance of cohomology, with some simple examples of homotopies and homotopy equivalences. And then a proof of the hairy ball theorem, saying that on even dimensional sphere does not have a nowhere vanishing vector field. Along the way I may have given some hints for the exam.
Blackboard photos of lecture 16..