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Hyperbolic Geometry (Winter 2019-2020)


Course description

This is an introductory course on hyperbolic geometry. Both the lectures and exercise sessions are taught by myself in English.

Contents will include:

  1. Introduction to non-Euclidean geometry and curvature
  2. Minkowski geometry and the hyperboloid model
  3. Projective geometry and the Klein model
  4. Conformal geometry and the Poincaré models
  5. Hyperbolic trigonometry
  6. Möbius transformations, \(\mathbb{H}^2\) and \(\mathbb{H}^3\)
  7. Gromov hyperbolic spaces and classification of isometries
  8. Hyperbolic manifolds (time permits)


No prerequisites are required, but the more geometry you studied, the easier the course will be.


I will not base the lectures on one reference in particular. I suggest you do your own research to find references that you like. I can personally recommend the following:

  1. Ratcliffe's book Foundations of hyperbolic manifolds is a great reference for learning hyperbolic geometry for the first time and for future reference. It is very well written and self-contained.
  2. Thurston's book Three-dimensional geometry and topology is not specifically about hyperbolic geometry, but it is a great idea to read it regardless because it is a beautiful introduction to many geometric ideas. Chapter 2 provides an introduction to hyperbolic geometry.
  3. If you want to learn Riemannian geometry, which is not required for this course but nevertheless a good idea, I find that Lee's Riemannian manifolds is a very good textbook for first-time learners. A new edition came out in 2018.


Ask me in person.

Class time and location & Office hours

Lectures: Thursdays 15:20-17:00, room 301 or 409K in S2|15 (see schedule)
Exercise sessions: Fridays 15:20-17:00 (approximately every other week, see schedule), room 234 in S2|15
Office hours: By appointment

Note that the Lectures and Exercise sessions have been interchanged!

NB: Refer to the course schedule below for the precise dates.

Lecture notes and Exercise sheets (PDF)

Hyperbolic geometry is an introductory book to hyperbolic geometry based on a course taught at TU Darmstadt in the winter 2019-2020. Download (PDF).

Click on the exercise sheets below to view or download.

Course schedule

NB: The course schedule below is subject to updates.

Date Subject Location Remarks
Fri. 18.10.2019 Lecture #1 (Chap 1, Chap 2) S2|15 234
Thu. 24.10.2019 Lecture #2 (Chap 2) S2|15 409K
Thu. 31.10.2019 Lecture #3 (Chap 3) S2|15 409K
Fri. 01.11.2019 Exercise session #1 S2|15 234
Thu. 07.11.2019 Lecture #4 (Chap 4) S2|15 409K
Fri. 08.11.2019 Lecture #5 (Chap 4) S2|15 234 Exceptional date
Thu. 14.11.2019 Lecture #6 (Chap 5) S2|15 301
Fri. 15.11.2019 Exercise session #2 S2|15 234
Thu. 21.11.2019 - - Cancelled: moved to 08.11
Thu. 28.11.2019 Lecture #7 (Chap 5) S2|15 409K
Fri. 29.11.2019 Exercise session #3 S2|15 234
Thu. 05.12.2019 Lecture #8 (Chap. 6) S2|15 301
Thu. 12.12.2019 Lecture #9 (Chap. 6) S2|15 301
Fri. 13.12.2019 Exercise session #4 S2|15 234
Thu. 19.12.2019 Lecture #10 (Chap. 7) S2|15 301
Thu. 16.01.2020 Lecture #11 (Chap. 7) S2|15 301
Fri. 17.01.2020 Exercise session #5 S2|15 234
Thu. 23.01.2020 Lecture #12 (Chap. 8) S2|15 301
Thu. 30.01.2020 Lecture #13 (Chap. 9) S2|15 409K
Fri. 31.01.2020 Exercise session #6 S2|15 234
Thu. 06.02.2020 Lecture #14 (Chap. 10) S2|15 301
Thu. 13.02.2020 Lecture #15 (Chap. 11) S2|15 301
Fri. 14.02.2020 Exercise session #7 S2|15 234

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