Differential Geometry I Autumn 2021

Joaquim Serra
Tommaso Goldhirsch
Time and Location:
Monday, 14:15 - 16:00 in ML H 44
Wednesday, 14:15 - 16:00 in HG E 5


Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in \(\mathbb{R}^n\), geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.

The course will follows the Differential Geometry I course taught by Prof. Urs Lang in 2019 (see literature below).

Lecture and Video Recordings

Lectures will take place in-person and will be live streamed. According to ETH Guidelines, a COVID certificate together with an ID card or Legi should be presented to enter the room for in-person teaching.

The lecture will be streamed live on the ETH Video Portal.
Recordings will be available and should be published here on the morning of the following day.

Time Room Livestream
Monday 14:15-16:00 ML H 44 Livestream ML H 44
Wednesday 14:15-16:00 HG E 5 Livestream HG E 5

See the official documentation for more tecnical informations.

Missing video: October 6th and October 13th

Due to some technical issues, the recordings of the lectures of Wednesday October 6th and of Wednesday October 13th are compromised. Here you can find some notes supplementing the audio: October 6th and October 13th .

Written notes: differential forms and Stokes' Theorem

Here you can find some handwritten notes covering the last part of the course: Differential forms and Stokes' Theorem .


The new exercise sheet will be uploaded on this page on Monday. You are supposed to have a look at it before the exercise class, so that you can ask questions if you need to. You have time until the following Monday at 12:15 to upload your solutions.

Please, upload your solution via the SAM upload tool.

In order to access the website you will need a NETHZ-account and you will have to be connected to the ETH-network. From outside the ETH network you can connect to the ETH network via VPN. Here are instructions on how to do that.

Make sure that your solution is one PDF file and that its file name is formatted in the following way:

solution_<number of exercise sheet>_<your last name>_<your first name>.pdf

Example: solution_2_Surname_Name.pdf.

Exercise Sheet Due By Upload Link Solutions
Exercise Sheet 1 October 4, 12:15 Upload Sheet 1 Solution 1
Exercise Sheet 2 October 11, 12:15 Upload Sheet 2 Solution 2
Exercise Sheet 3 October 18, 12:15 Upload Sheet 3 Solution 3
Exercise Sheet 4 October 25, 12:15 Upload Sheet 4 Solution 4
Exercise Sheet 5 November 1, 12:15 Upload Sheet 5 Solution 5
Exercise Sheet 6 November 8, 12:15 Upload Sheet 6 Solution 6
Exercise Sheet 7 November 15, 12:15 Upload Sheet 7 Solution 7
Exercise Sheet 8 November 22, 12:15 Upload Sheet 8 Solution 8
Exercise Sheet 9 November 29, 12:15 Upload Sheet 9 Solution 9
Exercise Sheet 10 December 6, 12:15 Upload Sheet 10 Solution 10
Exercise Sheet 11 December 13, 12:15 Upload Sheet 11 Solution 11
Exercise Sheet 12 December 20, 12:15 Upload Sheet 12 Solution 12

Exercise Classes

The exercise classes will start the second week of the semester. You can enrol in one exercise class on MyStudies

Thursday 13-14HG E 22 Alessio Pellegrini en
Thursday 16-17IFW C 31 Elia Mazzucchelli en
Friday 13-14HG F 3 Giovanni Ambrosioni en


Date: Saturday, February 5th (05.02.22).
Time: 9:00-12:00. The doors will open at 8:30.
Location: By LAST name:
From B to H in room F3 of the main building (HG)
From K to Z in room F1 of the main building (HG)

Special Rules: Wear a surgical or FFP2 mask. Please bring your student card (legi) and your valid Covid certificate (3G), PRINTED.

The exam consists in 3 parts. You can decide in which order to solve them and how to manage your time.
Part 1. Multiple Choice, in the style of Exercise Sheet 12.
Part 2. Two exercises, choose one. You will be presented with two exercises similar to those in the Exercise Sheets, or to examples seen in class. You have to choose one, and solve it. If you hand in both solutions, we will grade the first one that we find and ignore the second!
Part 3. A slightly more challenging exercise, based on the material covered during the semester.

No written aids are allowed. Please bring your own paper and write your solution in either black or blue (no green, no red, no pencil). Bring your own Tipp-Ex to use during the Multiple Choice part of the exam. Don't use a Tipp-Ex for the open exercises (Part 2 and Part 3).


The exam took place on Saturday, February 5th 2022. Here you can find the Multiple Choice exercises as well as the Open Questions.


Partial lecture notes from the course taught in Fall 2019 are available from Prof. Lang's website.