Welcome to the course SF2722 Differential Geometry!

News

If you are not registered until Jan 20, you may submit your solutions to the first exercise sheet via email to the TA Rasmus Johansen Jouttijärvi.

Course content

In this lecture we will deal with the most important fundamentals of the geometry of curved spaces, so-called manifolds. First we will introduce an intrinsic notion for manifolds, i.e. one that does not require a surrounding space. On these we introduce (semi-)Riemannian metrics that allow us to measure lengths and angles in these spaces. As a further consequence, we can also introduce locally shortest connecting lines, so-called geodesics, and various concepts of curvature.

Exercise sheets

The course will be complemented by an exercise class, taking place approximately every second week. For each class, you will get a sheet of homework problems in advance that you have to submit until a particular date. You can submit as a group of two people.

On each sheet, there will be four problems and you are able to get at maximum eight points. The solutions will be discussed with the teaching assistant in the exercise class. You get bonus points if you are presenting your solution in the class.

In order to do the exam, you have to get at least 50% of the points on all of the exercise sheets. The points do not contribute to the final grade.

Literature

The main source I use for preparation are handwritten lecture notes from a previous course I taught at the University of Hamburg. The notes are avaiable here. We do not have time to cover all of it. The current plan is to skip Section 3 and Section 5.2 of these notes as well as some details in other sections. Depending on the progress we may or may not cover Chapter 8.

I decided also to write a TeX-Version of the notes which contains all material of the handwritten notes we are able to cover. I also add the exercises you are supposed to solve plus some additional exercises you can have a look at if you are interested. A preliminary version, covering the content up to Section 3.3 is avaiable here. Thanks to Alex Nash, we have another set of lecture notes, avaiable here, which already contains the content up to Section 8.

Here is a list of further literature you can consult:

Lecture notes I used for the preparation of my own ones are

Differential geometry, lecture notes by Christian Bär, Summer term 2013
Differential geometry, lecture notes by David Lindemann, Summer term 2020
Riemannian geometry, lecture notes by Michael Kunzinger and Roland Steinbauer, Winter term 2016/17

Some books you might be interested in:

John M. Lee: Introduction to Smooth Manifolds, Graduate Texts, 2013 in Mathematics
Barrett O'Neill: Semi-Riemannian geometry with applications to relativity, Academic press, 1983
Peter Petersen: Riemannian geometry, Graduate Texts in Mathematics, 2006
Frank W. Warner: Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, 1983

Course Logbook

Date	Content	Handwritten lecture notes.	TeX notes KK / TeX notes AN
Jan 17	Topology, topological and smooth manifolds	until Example 1.2.8 (b)	until Example 1.2.4 (b)
Jan 20	Examples of smooth manifolds, smooth maps, tangent space	Example 1.2.8 (c) to Definition 1.4.3	Example 1.2.4 (c) to Definition 1.4.3
Jan 27	Tangent space as a vector space, tangent maps	Proposition 1.4.4 to Remark 1.4.11	Proposition 1.4.4 to Remark 1.4.11
Jan 31	Submanifolds	Section 1.5	Section 1.5
Feb 10	Vector fields and and 1-forms	Definition 2.1.1 to Definition 2.2.5	Definition 2.1.1 to Definition 2.2.5
Feb 14	One-forms and tensor fields.	Remark 2.2.6 to Definition 2.3.8	Lemma 2.2.6 to Definition 2.3.9
Feb 24	Tensor fields and scalar producs	Theorem 2.3.9 to Example 4.1.9	Theorem 2.3.10 to Example 3.1.9
Feb 28	Semi-Riemannian metrics	Definition 4.1.10 to Example 4.4.5	Definition 3.1.10 to Example 3.3.4
Mar 21	Riemannian & Lorentzian metrics, covariant derivative	Definition 4.3.1 to Definition 5.2.1	Definition 3.4.1 to Definition 4.2.1
Mar 24	The covariant derivative of submanifolds	Lemma 5.2.2 to Example 5.2.12	Definition 4.2.2 to Example 4.2.16
Mar 31	Geodesics	Chapter 6 up to Lemma 6.2.8	Chapter 5 up to Lemma 5.2.8
Apr 4	The exponential map	Lemma 6.2.9 to Definition 7.1.1	Lemma 5.2.8 to Definition 6.1.4
Apr 21	Riemannian curvature tensor and sectional curvature	Lemma 7.1.2 to Proposition 7.3.1	Lemma 6.1.5 to Proposition 6.3.1
Apr 25	Ricci and scalar curvature	Corollary 7.3.2 to Lemma 8.1.2	Corollary 6.3.2 to Lemma 7.1.2
May 2	Differential forms and the exterior derivative	Remark 8.1.3 to Lemma 8.2.8	Remark 7.1.3 to Lemma 7.2.11
May 5	Integration of manifolds and manifolds with boundary	Example 8.2.9 to Lemma 8.4.4	Remark 7.2.12 to Lemma 7.4.3
May 12	Stokes' theorem, Gauss' divergence theorem, outlook	From Remark 8.4.5	From Remark 7.4.4

I skip some parts of the handwritten notes. Up to some minor comments and examples, the content of the TeX-Version is a subset of the one of the handwritten notes. The most important changes are the following:

I skip all notions of pullbacks and pushforwards which appear in Section 1.3 and Chapter 2.
I skip the discussion on Derivations in Section 1.4 and Section 2.1.
In the TeX-Version, I introduce the Lie Bracket by a local formula which is Corollary 2.1.13 in the handwritten notes. In the exercises you will prove that it is a coordinate-invariant expression. In the TeX-Version I prove (2.1.3) using this different definition, whereas in the handwritten notes, I use (2.1.3) as a definition of the Lie-Bracket.
I completely skip Section 3 in the handwritten notes on flows and the Lie-Derivative
I skip Definitions 4.4.6 and 4.4.7
I restructure Sections 4.3, 4.4 and 5.2
I skip Section 5.3 (parts may appear in other sections).

Additional exercises

If anyone is looking for additional relevant exercises to do, I have created a document with some of my personal recommendations:

Recommended-2.pdf

I will add to it, during the course.

Since we didn't have time to discuss all the exercises for the first TA session, I have collected my notes for the last two exercises in the following document:

Exercise1.pdf

Problem 17 of Homework Assignment #5:

Problem17.pdf

Problem 28 of Homework Assignment #7:

Problem28.pdf

Homework Assignment #8:

HW8.pdf

Homework Assignment #9:

HW9.pdf

Examination

There will be an oral exam after the course. You need 50% on the points of the exercises in order to attend.

The oral exams will take place at the following dates: May 30, June 2, June 9 with the following timeslots: 9:20, 10:00, 10:40, 11:20, 13:00, 13:40, 14:20. Registration is via email and slots will be provided on a first come, first serve basis. If all slots are booked, other slots will be opened. If you are not able to make it on one of these dates, other times can be arranged on an individual basis.

Contact information

Klaus Kröncke (lecturer)

Rasmus Johansen Jouttijärvi (teaching assisstant)