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Manifolds (Summer 2020)




Contents

Online teaching information

Due to the exceptional situation at TU Darmstadt and elsewhere caused by the Covid-19 epidemic, all classes and exercise sessions will be taught entirely online, at least until June 1st, 2020 until the end of the semester.

The online classes will be held via Zoom. You can register on Zoom for free. You can use Zoom directly from a browser, but it is probably better to install the Zoom app designed for your device (Windows, Mac OS, Linux, Android, or iOS).

All online lectures will occur in the Zoom meeting "Manifolds". Here is the information to join this meeting:

  1. Meeting ID: 894 3330 2319
  2. Password: X that solves 3X = 9339

I also have a "Personal Meeting Room" that you can join to ask me questions, in the style of "Office Hours". The Meeting ID of that room is 356 234 4610 and the password is the same.

Information for the exercise sessions: see Exercises and homework section below.

I will video record the lectures and make the videos available online on this website, as well as the lecture notes in PDF. See the Material section below.

Exercise sessions, homework, Studienleistung

Philipp Käse will be in charge of the exercise sessions and the homework. Please refer to this information sheet (updated on 04.05.2020!) for information and details.

Course description, prerequisites, references

Contents

This is an introductory course on differential manifolds. Contents will include:

  1. Topological manifolds, differential manifolds, the tangent bundle, Whitney's embedding theorem.
  2. Vector fields, Lie bracket, flow and integral curves, the Frobenius theorem.
  3. Differential forms, de Rham cohomology, integration, Stokes's theorem.

Prerequisites

Prerequisites for this course include multivariable calculus and other topics that are part of a typical undergraduate mathematics curriculum (such as: general topology, linear algebra, differential equations).

References

The main references I will be using for this course are:

  1. John Lee's book Introduction to smooth manifolds (second edition). To a lesser extent, I will also use his book Introduction to topological manifolds (second edition).
  2. Prof. Karsten Große-Brauckmann's lecture notes Manifolds (2018). Click here to download (PDF).
  3. Jacques Lafontaine's book An introduction to differential manifolds. (English edition came out in 2015, original French text re-edited in 2010.)

I will give more references in the lectures.

Class material

Videos of the lectures

Click on each lecture below to view/download the video:

  1. Lecture 1 (93.9 MB)
  2. Lecture 2 (80.1 MB)
  3. Lecture 2 Complement (44.5 MB)
  4. Lecture 3 (200.7 MB)
  5. Lecture 4 (223.0 MB)
  6. Lecture 5 (244.8 MB)
  7. Lecture 6 (248.0 MB)

Lecture notes

Below are the handwritten notes (on a tablet) or slides for each lecture:

  1. Lecture 1 (5.3 MB)
  2. Lecture 2 (5.9 MB)
  3. Lecture 2 Complement (0.2 MB)
  4. Lecture 3 (0.3 MB)
  5. Lecture 4 (0.5 MB)
  6. Lecture 5 (0.2 MB)
  7. Lecture 6 (0.8 MB)

Exercise sheets


Course schedule

  1. Lectures: Thursdays 17:10-18:50 (refer to table below for details and location)
  2. Exercise sessions: Fridays 09:50-11:30 (refer to table below for details and location)
  3. Office hours: By appointment

NB: The course schedule below is subject to updates!


Date Subject Location Information
Thu. 23.04.2020 Lecture #1 Online Please connect to Zoom 5 minutes early.
Fri. 24.04.2020 Exercise session #1 Online
Thu. 30.04.2020 Lecture #2 Online
Thu. 07.05.2020 Lecture #3 Online
Fri. 08.05.2020 Exercise session #2 Online
Thu. 14.05.2020 Lecture #4 Online
Thu. 21.05.2020 Lecture #5 Online 21.05 is a Holiday. The class will nevertheless happen at the usual time online.
Fri. 22.05.2020 Exercise session #3 Online
Thu. 28.05.2020 Lecture #6 Online
Thu. 04.06.2020 Lecture #7 Online
Fri. 05.06.2020 Exercise session #4 Online
Thu. 11.06.2020 Lecture #8 Online 11.06 is a Holiday. The class will nevertheless happen at the usual time online.
Thu. 18.06.2020 Lecture #9 Online
Fri. 19.06.2020 Exercise session #5 Online
Thu. 25.06.2020 Lecture #10 Online
Thu. 02.07.2020 Lecture #11 Online
Fri. 03.07.2020 Exercise session #6 Online
Thu. 09.07.2020 Lecture #12 Online
Thu. 16.07.2020 Lecture #13 Online
Fri. 17.07.2020 Exercise session #7 Online


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