Manifolds (Differentieerbare varieteiten)

Blok 1, 2023/2024

This is the web-site for the course "Differentieerbare varieteiten" given in blok 1, Fall 2023.

We will try to keep here all the practical informations about the course, updates/changes that take place during the year, lecture notes, etc. The most recent additions are:

  • The exam
  • Sept 10: the lectures notes with Chapter 1 (reminders) and Chapter 2. See below.
  • Sept 14: the lectures notes with Chapter 1 (reminders) and Chapter 2 were replaced with the correct ones. See below.
  • Sept 27: homework 3 and the lectures notes to which Chapter 3 was added. See below.
  • Sept Oct 18: the notes with Chapter 4 included.
  • Sept Oct 30: Full Lecture Notes included.

    • 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: Anna Fokma, Esther Steenkamer

    HOMEWORKS: (subject to changes during the first week) 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). The new version of the notes will be made available as we go. We start with Chapter 1 (reminder on Topology and on Analysis) and Chapter 2 (definition of manifolds) (originally I made a mistake and I posted some older version, I changed those to the intended ones on September 14). And here is notes with Chapter 3 (tangent vectors/vector fields) .

    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 11): An introduction of the keywords. Notion of chart. Started looking at Theorem 1.38 and 1.37. To be continued.

    Blackboard photos for this lecture.

    Werkcollege (September 11 and September 13): read the lecture notes containing the reminders on Topology and Analysis and start having a first look at the definition of manifold.

    ♣ WEEK 37/Lecture 2 (September 13): finished discussing Theorem 1.38 and 1.37, including those two examples on the unit circle (see foto of the hand-written version ) and immersions/submersions. Then we started chapter II: talked about smooth compatibility of topological charts, smooth atlases, smooth structures (= maximal smooth atlases), smoothness w.r.t. and atlas (of an R-valued function), first examples of atlases, the maximal atlas (i.e. the smooth structure) induced by an atlas. TBC

    Blackboard photos of Lecture 2 .

    The exercises for today's tutorial were: 1.52,54,58(,extra: 59),60.

    Homework(to be handed in by next Wednesday, by the start of the werkcollege): Regarding the main theorem discussed in the lecture ("Theorem 1.38 and 1.37"): we gave two examples/illustration of the Theorem, both on the circle $S^1$- as in this foto. The homework kindly asks you to:

  • a). Do something similar for the unit 2-sphere S^2=\{(x, y, z)\in R^3: x^2+ y^2+ z^2= 1\}.
  • b). For each of the resulting charts $\chi$ and $\chi'$ that you find, prove that they are smooth (in the sense of Definition 1.34), and actually diffeomorphisms.
  • c). Then give another similar example on S^2, but where $\chi$ comes from the stereographic projection.
    • Explanation: for functions on opens inside Euclidean spaces, which are standard/elementary (polynomials, trygonometric, exponentials, well-defined fractions, etc, and composition of such) you are allowed to use what you know from Analysis, in particular that they are smooth. For functions defined on other subsets of Euclidean spaces (which show up in item b) please use Definition 1.34 and provide the arguments.

    ♣ WEEK 38/Lecture 3 (September 18): recap and end the discussion on atlases and smooth structures, define smooth manifolds, smooth maps, went through all kind of remarks on the definition, up to "variations", and started looking at some examples (from the sections with examples we only looked at the spheres so far).

    Blackboard photos of Lecture 3 .

    The exercises for today's tutorial were: 1.56,1.57 and 2.5,9,11.

    An attempt to record this lecture, I hope it is useful at least for the physics students that could not be present: first 10 min, followed by the rest of the first part and then the second part of the lecture .

    ♣WEEK 38/Lecture 4 (September 20): Reminder on the notion of smooth structures and manifolds, smooth maps, the classical smooth structures on embedded manifolds in Euclidean spaces, the regular value theorem as an extremely valuable tool, then looked at S^2 again from 3 different points of view, illustrated in two examples how to write maps between manifolds w.r.t. charts ("in local coordinates"), mentioned some similar examples (torus, complex space $C^m$, n x n matrices, invertible matrices, orthogonal matrices), and general constructions (smooth structures induced on opens, and products of smooth structures) and then, finally, the projective spaces $P^m$. Then, at the very end, mentioned some special/distinguished types of smooth maps: diffeomorphisms, submersion from an m-dimensional manifold to an n-dimensional one (as the ones that, in appropriate coordinates, can always be described as "forgetting the last n-m coordinates", hence which only make sense when $n>= m$), immersions from an m-dimensional manifold to an n-dimensional one (as the ones that, in appropriate coordinates, can always be described as "adding m-n zeros to complete the vector in $R^m$ to one in $R^n$", hence which only make sense when $n<= m$), local diffeomorphisms (with the example/exercise of the map $R\to S^1$ sending $t$ to $(cos(2\pi t), sin(2\pi t))$).

    The exercises for today's tutorial were: 2.19, 2.20 (hint: section 2.1.4), 2.21, 2.30

    Blackboard photos of Lecture 4 .

    Homework: Homework 2 (note: a small typo in item 4, namely "that" instead of "hat", has been corrected on Sept 25), to be handed in before the start of the lecture of September 27th.

    ♣ WEEK 39/Lecture 5 (September 25): Embedded submanifolds, immersions, smooth embeddings.

    Blackboard photos of some parts of Lecture 4 . Something went wrong with taking pictures today (also, the blackboard were used non-linearly, e.g. blackboard 1 was used later on to make blackboard 4 ... hence a bit difficult to show this in pictures ...). Unfortunately, blackboard -3- does not have a picture on its own (bits appear on some of the other pictures), as the quality of that photo makes it unusable, while the last two blackboards have no picture at all (I have no idea what happened, since I am pretty sure I did (attempt to) take pictures). Those two blackboard contains the family of maps $f_{\lambda}$ from $\R$ to the torus, all immersions, but which for $\lambda$ rational were giving rise to embeddings of the circle on the torus (winding around the torus a couple of times), whille for $\lambda$ irrational were immersions of $\R$ which are not embeddings (the image being a line that winds around the torus infinitely times, and densely). The other thing on those the blackboards was the theorem that a smooth map is a smooth embedding if and only if it is an immersion and a topological embedding; with the corollary that, if the domain $M$ of the map is compact, then $f: M\to N$ is a smooth embedding if and only if it is an injective immersion.

    The videos (I hope they are of some use, doing it myself, without any assistance ... I may forget turning the camera, etc. Maybe in the future ask a colleague to help me). Part 1 and Part 2.

    The exercises for today's tutorial were: 2.34, 2.48, 2.57, 2.71, 2.73.

    Plan for the next time: wrap up the stuff on embeddings and immersions, then start with tangent vectors.

    ♣WEEK 39/Lecture 6 (September 27): A generous recap on embedded submanifolds, smooth embeddings with various nice examples, then briefly discussed the notion of immersed submanifold of a given manifold $N$, more informative in terms of what can happen (various different ways to make a picture into an actually immersed submanifold), advising you to think about them as pairs $(M_0, f)$ consisting of a smooth manifold $M_0$ and an injective immersion $f: M\to N$. Then recalled a bit what the classical tangent spaces are, for embedded submanifolds $M\subset \R^n$ (in the way that you have seen in analysis in more variables, or perhaps in other courses).

    The aim for next week: MAKE SENSE OF THE TANGENT SPACES $T_pM$ FOR ANY (ABSTRACT) SMOOTH MANIFOLD $M$ AND ANY POINT $p\in M$. There is a more precise list of wishes:

    - TW1: for any manifold $M$ and any $p\in M$ have a vector space $T_pM$

    - TW2: smooth maps $F: M\to N$ should induce linear maps $(dF)_p: T_pM\to T_{F(p)}$ in such a way that the chain rule is still respected

    - TW3: nothing new in the classical case: i.e., for embedded submanifolds $M\subset \R^n$ (with the induced smooth structure), $T_pM$ should be canonically isomorphic to the classical tangent space $T^{class}_{p}M$.

    - TW4: the leading slogan should be implemented: $T_pM$ is made of ``speeds of smooth curves in $M$ through $p$''.

    Homework: Homework 3 .

    The exercises for the tutorial were: 2.61,62,74a (use 62!), 76,77.

    Blackboard photos of Lecture 6 . Please be aware that, as some of you have pointed out, there is something not quite right in the way the colors were used in the picture with the line winding around the torus (densely). We started with red, rotate now to the left, then after a full rotation (the line does not close and keeps on going) we started drawing it in blue, then after another rotation we started coloring it in green, etc etc, and we used those colors in the arguments (e.g. also to describe the immersion from the line to the torus). The point is that, starting from the original line with red, since when rotating towards the left (clockwise) we see blue, then green, after a turn when approaching the start from the other direction, it should be red, blue, green again- put in the picture the last two colors are switched.

    Note: you can now look at the notes with Chapter 3 (tangent vectors/vector fields) included.

    ♣ WEEK 40/Lecture 7 (October 2): We recalled the classical (red) tangent space for embedded submanifolds of Euclidean spaces, then stated the ;list of "tangent wishes" describing what we expect/want from the general tangent spaces for general manifolds. Then I explained some intuition/motivation for two possible defintions: the white $T_pM$ as functions defined on the collection of charts through $p$, and the green T_pM defines as derivations (satisfying the derivation identity) on the space of smooth real-valued functions on $M$. Then started looking at TW1, speeds TW2, and some remarks in $R^m$, both ``the white version'' as well as ``the green version''. At the end I was encouraging you to try to define a natural map from the white $T_pM$ to the green one (not to read it in the notes, but try to do it yourself, for up to 30-60 min!)

    The exercises for the tutorial were: 1.44,45,46,48.

    Blackboard photos of Lecture 7 .

    ♣WEEK 40/Lecture 8 (October 4): Finished with tangent spaces, except for the proof that any element in the green tangent space of $R^m$ at $p= 0$, i.e., any derivation for $R^n$ at $0$, is a linear combination of the partial derivative operators. I will take 5 min of the next lecture to finish that.

    Plan for the next lecture: a recap of tangent spaces and how to work with them, and then move on to vector fields.

    Blackboard photos of Lecture 8 .

    Homework: Homework 4 . NOTE: YOU CAN ASK QUESTIONS RELATED TO THIS HOMWEWORK ALREADY DURING THE WERKCOLLEGE ON MONDAY (AND YOU CAN ALSO DISCUSS WITH COLLEAGUES; but make sure that you write down the solutions/explain it yourself in the best way you can). If it is too much in too little time, you can hand it later, by Friday afternoon (by email), at the expense of loosing 2 points. It is important to invest some time into this, and if people have trouble with it even in one week time from now, then we have to look at it, and we will, once more, in a werkcollege.

    The exercises for the tutorial were: 13.1,6,7.

    A rough schematic summary on tangent spaces that you may find useful.

    And here summary of tangent spaces inside $\R^n$ that you may find useful, in an animated (old style ...) version.

    ♣ WEEK 41/Lecture 9 (October 9):

    Vector fields, operations with vector fields, vector fields as derivations, Lie bracket of vector fields.

    Blackboard photos of Lecture 9 .

    The exercises for the tutorial were: 3.10, 3.14(1+4), 3.17, 3.24 3rd bullet.

    ♣WEEK 41/Lecture 10 (October 11): Integral curves.

    Blackboard photos of Lecture 10 (but "blackboard 8" is missing, somehow I did not get a picture of that).

    The exercises for the tutorial were: 3.35, 38, 40 (m=2), 43, 47 (2nd bullet), 48.

    Homework: prove that $[X, fY]= f[X, Y]+ L_{X}(f) Y$ for all vector fields $X$ and $Y$ and all smooth functions $f$.

    Homework:

    ♣ WEEK 42/Lecture 11 (October 16): Flows of vector fields, including the proof that, on compact manifolds, all the vector fields are complete.

    Blackboard photos of Lecture 11

    The exercises for the tutorial were: 3.75,78,79.

    ♣WEEK 42/Lecture 12 (October 18):

    Blackboard photos of Lecture 12

    The exercises for the tutorial were: 3.84, 3.85, 3.88, 3.104.

    Homework: Homework 5 (exam questions from last year ... 😜). Notice that the last question is a bonus one, in case you want to get more than the usual maximum of 10pts.

    NEW NOTES: you can now look at the notes with Chapter 4 (differential forms) included.

    ♣ WEEK 43/Lecture 13 (October 23): Differential forms of arbitrary degrees.

    Blackboard photos of Lecture 13

    The exercises for the tutorial were: 4.4,5,7,14,18,25.

    ♣WEEK 43/Lecture 14 (October 25): $d$, $\Lie_V$, $i_V$, Cartan's magic formula.

    Blackboard photos of Lecture 14

    The exercises for the tutorial were: 4.38,39, 52, 60(1,2, second part of 3). If you want more practice with forms you can look at 61(1-3) and/or 63(a,b).

    Homework: Homework 5

    Homework: Previous exams. More to be added this evening: the new homework, plus last part of the notes (including exams from previous years).

    ♣ WEEK 44/Lecture 15 (October 30): Integration and Stokes.

    A little recap on the blackboard during werkcollege.

    The entire set of lecture notes.

    Blackboard photos of Lecture 15 .

    The exercises for the tutorial were: 4.59,4.60c,4.63c, more practice: 3.93.

    ♣WEEK 44/Lecture 16 (November 1): DeRham cohomology and a very cool application: the hairy ball theorem (more precisely, we proved that an m-sphere admits a nowhere vanishing vector field if and only if $m$ is odd).

    Blackboard photos of Lecture 16 .

    The exercises for the tutorial were: 5.5, 5.8, 4.9, 5.28.

    ♣End exam: Tuesday, November 7, 17:00, BEATRIX - 7E ETAGE (BBG 001 for the extra-time).

    IMPORTANT: PLEASE DO NOT RELY 100% ON THIS WEB-PAGE. I.E., IF YOU MISS A CLASS, PLEASE KEEP IN TOUCH WITH YOUR COLLEAGUES THAT WERE PRESENT, JUST TO MAKE SURE YOU STAY INFORMED.