Differential Geometry I, WS 2020/21

News

Self-check exercise sheet: pdf. Help Blorx and enjoy the break!
The assignments for week 13 are online: pdf (All exercises are optional; submitted solutions count as bonus points!)
Please submit your solutions as pdf!
The lecture notes are updated (11.02.2021).
correction log.
If you are interested in working on a BSc thesis project with Prof. Ammann: details for a meeting on Feb 2 can be found on the GRIPS page of this course.
If you are interested in working on a project with me: check this page!
Registration in FlexNow/for the exams: Registration is now open; please read the details at the bottom of this page!
The evaluation results are available in GRIPS.
There are still free slots available in the seminar Lie groups, Lie algebras and their representations . Please contact Matthias Ludewig if you are interested in signing up.
In view of the COVID-19 pandemic, until further notice, this course will be taught remotely, based on
- guided self-study (via extensive lecture notes),
- interactive question sessions (via zoom and the GRIPS forum),
- remote exercise sessions (via zoom and the GRIPS forum).
More details: pdf.
Currently, all non-virtual teaching at UR is heavily restricted (until 04/2021): more information about dealing with 2019-nCoV at UR; in particular, I cannot offer any in-person office hours.
This course will be continued in SS 2021: Differential Geometry II by Prof. Dr. Bernd Ammann.
Interested in a project under my supervision? Here, you can find some information.

Differential Geometry I

Differential geometry is the study of geometric objects by analytic means. Geometric objects in this context usually are Riemannian manifolds, i.e., smooth manifolds with a Riemannian metric. This allows to define lengths, volumes, angles, ... in the language of (multilinear) analysis. A central concern of differential geometry is to define and compare (intrinsic) notions of curvature of spaces. A particularly fascinating aspect is that many local curvature constraints are reflected in the global shape.

Differential geometry has various applications in the formalisation of Physics, in medical imaging, and also in other fields of theoretical mathematics. For example, certain phenomena in group theory and topology can only be understood via the underlying geometry.

In this course, we will introduce basic notions of differential geometry. This includes, in particular, the Riemannian curvature tensor, sectional curvature, Ricci curvature, and scalar curvature. Moreover, we will study some first global obstructions to curvature constraints.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Time/Location

Wednesday, 8:30--10:00,
Thursday, 10:15--12:00.

In view of the COVID-19 pandemic, until further notice, this course will be taught remotely, based on:

guided self-study (via extensive lecture notes),
interactive question sessions (via zoom),
remote exercise sessions (via zoom).
an informal chat forum for discussing questions or sharing virtual meeting coordinates with fellow participants (the invite link is provided in GRIPS).

More details:

The first "lecture" will be on Wednesday, November 4, 08:30. In this "lecture", we will get acquainted with the video conferencing tool, we will discuss organisational matters, and I will give a brief overview of this course.
The last lecture is currently scheduled for Thursday, February 11.
Access information for the virtual meetings will be announced in GRIPS (and not publicly on the homepage).

Exercise class

There will be two groups:
Monday 8:00--10:00
Tuesday 8:00-10:00
We will organise the distribution via GRIPS (details).
The exercise classes start in the second week of the lecture period; in this first session, some basics will be recalled/discussed.
Details about the remote exercise sessions and the submission of solutions are explained in GRIPS.
Organisational matters

Read me! Lecture notes, schedules and assignments

The lecture notes will grow during the semester and will be continuously updated. The first upload is scheduled for November 4, after the first lecture. In subsequent weeks with remote teaching, I will always try to provide the material for the whole week n + 1 on the Thursday of week n.

Lecture notes: pdf
Correction log: here
More literature: pdf

Topics covered so far:

Guide to the literature
Introduction
- What is differential geometry?
- Why differential geometry?
- The Poincaré conjecture
- Overview of this course
Riemannian manifolds
- Smooth manifolds
  [Topological manifolds; Smooth manifolds; The category of smooth manifolds; Tangent spaces; Submanifolds]
- The tangent bundle
  [Smooth vector bundles; Constructing vector bundles; The tangent bundle; Tensor bundles]
- Riemannian manifolds
  [Riemannian metrics; Riemannian manifolds]
- Model spaces
  [Homogeneous spaces; Euclidean spaces; Spheres; Hyperbolic spaces; Group actions]
- Towards Riemannian geometry
  [Lengths of curves; The Riemannian distance function; Volume and orientation]
Curvature: Foundations
- The idea of curvature
- Connections
  [Connections; Local descriptions of connections; Covariant derivatives along curves; Geodesics; Parallel transport]
- The Levi-Civita connection
  [Connections on tensor bundles; The Levi-Civita connection]
- Curvature tensors
  [The Riemannian curvature tensor; Flat manifolds; Sectional curvature; Ricci curvature; Scalara curvature]
- Model spaces
  [Locally conformally flat manifolds; Symmetries and constant curvature; Sectional curvature of the model spaces]
Riemannian geodesics
- The exponential map
  [The exponential map; Normal coordinates]
- Riemannian geodesics
  [Variation of curves; Variation fields and the first variation formula; Minimising curves are geodesics; Geodesics are locally minimising; Riemannian isometries]
- Completeness
  [Two notions of completeness; The Hopf-Rinow theorem]
- Model spaces
  [Geodesics of the model spaces; Isometries of the model spaces]
- Jacobi fields
  [The second variation formula and the index form; Jacobi fields; Conjugate points]
Curvature: Local vs. global
- Local-global results
- Constant sectional curvature, locally
- Analytic and geometric comparison theorems
- Non-positive curvature
  [The Cartan-Hadamard theorem; Constant non-positive curvature]
- Positive curvature
  [The Bonnet-Myers theorem; Constant positive curvature]
- (The Svarc-Milnor lemma)
Appendix
- Categories and functors
- Partitions of unity
- Group actions

This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books. A list of suitable books can be found in the lecture notes.

Weekly assignments:

Week 1 (announced on November 4): lectures of November 4/5, exercise series 0 (no submission; will be discussed in the exercise class on November 9/10)
Week 2 (announced on November 5): lectures of November 11/12, exercise series 1 (submission before November 12, 10:00; will be discussed on November 16/17)
Week 3 (announced on November 12): lectures of November 18/19, exercise series 2 (submission before November 19, 10:00; will be discussed on November 23/24)
Week 4 (announced on November 19): lectures of November 25/26, exercise series 3 (submission before November 26, 10:00; will be discussed on November 30/December 1)
Week 5 (announced on November 26): lectures of December 2/3, exercise series 4 (submission before December 3, 10:00; will be discussed on December 7/8)
Week 6 (announced on December 3): lectures of December 9/10, exercise series 5 (submission before December 10, 10:00; will be discussed on December 14/15)
Week 7 (announced on December 10): lectures of December 16/17, exercise series 6 (submission before December 17, 10:00; will be discussed on December 21/22)
Week 8 (announced on December 17): lectures of December 23/January 7, exercise series 7 (submission before January 7, 10:00; will be discussed on January 11/12)
Week 8.5 (announced on December 18): 20 bonus points! (optional submission before January 7, 10:00)
Week 9 (announced on January 7): lectures of January 13/14, exercise series 8 (submission before January 14, 10:00; will be discussed on January 18/19)
Week 10 (announced on January 14): lectures of January 20/21, exercise series 9 (submission before January 21, 10:00; will be discussed on January 25/26)
Week 11 (announced on January 21): lectures of January 27/28, exercise series 10 (submission before January 28, 10:00; will be discussed on February 1/2)
Week 12 (announced on January 28): lectures of February 3/4, exercise series 11 (submission before February 4, 10:00; will be discussed on February 8/9)
Week 13 (announced on February 4): lectures of February 10/11, exercise series 12 (optional submission)
Week 14 (announced on February 11): exercise series 13 (no submission)

Try me! Quick checks and interactive tools

The quick checks in the lecture notes help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded. These quickchecks will have feedback implemented directly in the pdf.
This feature is based on PDF layers (not on JavaScript) and is supported by many PDF viewers, such as Acrobat Reader, Evince, Foxit Reader, Okular ("new" versions only), .... (Apple Preview seems to have difficulties with this, though.)
Moreover, there will be simple computational exercises on the assignment sheets (which also will not be submitted/graded). These will give the opportunity to practise basic computational techniques.

Ask me! Interactive sessions

There will be a forum and other communication tools linked on the GRIPS page of this course.

Notes of the current lecture: pdf
04.11.2020: notes
05.11.2020: notes
11.11.2020: notes
12.11.2020: notes
18.11.2020: notes
19.11.2020: notes
25.11.2020: notes
26.11.2020: notes
02.12.2020: notes
03.12.2020: notes
09.12.2020: notes
10.12.2020: notes
16.12.2020: notes
17.12.2020: notes (including what I recalled from the mysteriously lost page)
23.12.2020: notes
07.01.2021: notes
13.01.2021: notes
14.01.2021: notes
20.01.2021: notes
21.01.2021: notes
27.01.2021: notes
28.01.2021: notes
10.02.2021: notes
11.02.2021: notes

Is it for me? Prerequisites

All participants should have a firm background in Analysis I/II, in Linear Algebra I/II, in basic group theory (as covered in the lectures on Algebra), and in Analysis IV (e.g., as in Analysis auf Mannigfaltigkeiten; Prof. Ammann, SS 2020). In the first week, I will quickly recall basics on smooth manifolds.

Exams

Please read the information on organisation and formalities of this course.

Dates for the oral exams (25 minutes):

25.02.2021
24.03.2021
timeslots: 8:45--9:10, 9:20--9:55, 10:05--10:30, 10:40--11:05, 11:15--11:40, 11:50--12:15,
13:15--13:40, 13:50--14:15, 14:25--14:50, 15:00--15:25, 15:35--16:00

Registration details:

Please register in FlexNow for the oral exam. Registration deadline: Two weeks before the exam. (You can then un-register up to one week before the exam).
Moreover, for the oral exam, you also need to register with Ms. Bonn (rosina.bonn@ur.de) for a time slot! (Despite many efforts, I cannot take multiple oral exams at the same time ...)
Please send her an email with the desired date. If you have time restrictions on the desired date (e.g., because of other exams etc.), please indicate this in your email. You will then be assigned a timeslot.
Please only register in FlexNow for an exam date once you have a timeslot!
If applicable: For the Studienleistung (successful participation in the exercise classes), please register in FlexNow.
If none of the dates for the oral exams fits, please contact me directly by email.

. Concerning the oral exams:

As always: All of the material of this course is relevant for the exam (lectures, exercises).
We covered a lot of material. Of course, I don't expect you to remember all the technical arguments at a ridiculous level of detail; however, you will need to know the key ideas and recurring proof patterns.
It would be very helpful if you would have a means to screen-share handwritten notes during the exam (e.g., using a tablet, graphic tablet, document camera, mobile phone camera, ...). This is not strictly necessary, but it will simplify matters a lot.

Last change: February 11, 2021.