Welcome to the course SF2722 Differential Geometry!

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. Per problem, you get up to two points, which means a maximum of eight points per sheet. The solutions will be discussed with the teaching assistant in the exercise class. You get up to two bonus points per problem you are presenting in the exercise class.

There will be eight exercise sessions and eight corresponding problem sheets. The first problem sheet is not part of the homework.

In order to do the exam, you have to get at least 50% of the points on all of the exercise sheets. That means, at least 28 of the 56 points you can achieve on the seven homework problem sheets. The points do not contribute to the final grade. If your are not able to submit enough exercises in time, let me know as soon as possible.

Literature

I will follow my own lecture notes which were typed in TeX by Alex Nash in VT23. I may make some minor adjustments of the notes during the course. Chapter 7 about global Riemannian geometry is new and will updated during the course. Chapter 8 on differential forms and Stokes' theorem is not relevant for the exam, but kept for completeness and for the interested reader.

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	Reference in the Lecture notes
Jan 14	Topology, topological and smooth manifolds	Until Remark 1.2.4
Jan 17	Examples of smooth manifolds, smooth maps, tangent space	Example 1.2.5 to Definition 1.4.3
Jan 24	tangent map, immersions and submersions	Proposition 1.4.4 to Example 1.5.2
Jan 28	Submanifolds	Lemma 1.5.3 to Theorem 1.5.13
Feb 4	Vector fields and 1-forms	Definition 2.1.1 to Remark 2.2.9
Feb 11	Tensor fields	Example 2.2.10 to Notation 2.3.12
Feb 18	Scalar products and semi-Riemannian metrics	Definition 3.1.1 to Example 3.2.5
Feb 25	Riemannian & Lorentzian metrics, covariant derivative	Notation 3.3.1 to Definition 4.1.3
Mar 18	The covariant derivative of submanifolds	Remark 4.1.4 to Lemma 4.2.11
Mar 21	Geodesics	Definition 4.2.12 to Example 5.2.3 (ii)
Mar 28	The exponential map	Example 5.2.3 (iii) to Corollary 5.4.2
April 1	Riemannian and sectional curvature	Corollary 5.4.3 to Proposition 6.2.6
April 8	Ricci and scalar curvature	Corollary 6.2.7 to Remark 6.4.9
April 11	Metric spaces and length-minimizing geodesics	Lemma 7.1.1 to the proof of of Theorem 7.2.1

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 in my office at the following dates: May 19, 20, 26, 27, June 2, 3, 16, 17 with the following timeslots: 13:00, 13:40, 14:20, 15:00. 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)

Louis Yudowitz (teaching assisstant)