SF2722 VT25 Differential Geometry

Welcome to the course!

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 Download 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

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
April 29 Complete Riemannian manifolds, Hopf-Rinow Theorem Lemma 7.2.8 to Remark 7.3.4
May 6 Jacobi fields and conjugate points, some topology Lemma 7.4.1 to Example 7.5.7
May 13 Curvature and topology Lemma 7.6.1 to Remark 7.7.6, no proof of Theorem 7.7.4
May 16 The Bonnet-Myers theorem, topics in geometry Proof of Theorem 7.7.4, overview talk

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 Links to an external site. (teaching assisstant)