Hyperbolic Geometry

Learning Outcomes

Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Calculate using the hyperbolic trigonometric laws
• Calculate hyperbolic length and distance in the upper half-plane and Poincaré disc models of the hyperbolic plane
• State, prove, and apply the Gauss-Bonnet theorem; present solutions to problems with a good standard of explanation

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Understand the action of the general Möbius group on the Riemann sphere and on the upper half-plane and Poincaré disc models of the hyperbolic plane
• Perform calculations with Möbius transformations and determine the multipliers and fixed point sets of Möbius transformations
• On successful completion of the module the students should be able to: understand the difference between hyperbolic and Euclidean geometry

Description of the upper half-plane H and Poincaré disc D models of the hyperbolic plane and the

connection between them.

• Description of the general Möbius group, including its subgroups preserving H and D.
• Derivation of the hyperbolic element of arc-length on H, and the use of hyperbolic length to define

hyperbolic distance.

• Characterization of the isometries of H.
• Discussion of the relationship between H and D, and a general method for constructing planar

models of the hyperbolic plane.

• Hyperbolic trigonometry.
• Hyperbolic area and the Gauss-Bonnet theorem, including a discussion of the non-existence of nonisometric

similarities of the hyperbolic plane.

In addition, a selection will be made from the following topics, and may form the basis of

students’ presentations and individual reports:

• Convexity and the characterization of convex sets in the hyperbolic plane
• Hyperbolic conic sections
• Further exploration of planar models of the hyperbolic plane arising from complex analytic methods
• Geometry of hyperbolic triangles
• The hyperbolic isoperimetric inequality
• Non-planar models of the hyperbolic plane

The instructor will give two lectures per week following a structured week-by week programme, based mainly on Anderson's book.

There will be two timetabled meeting per week, in which the lecturer will outline the topic for that week and be available to assist the students. The students are also expected to meet once a week by themselves, for a self-study tutorial session. They will each take responsibility for leading the session, based on guidance from the lecturer, supplemented as necessary. There will also be office-hour support from the lecturer. The assessment will be based on a portfolio of solved problems submitted periodically during the module. A high standard of explanation will be expected in the solutions, and this will form a part of the assessment criteria in addition to their mathematical correctness and the scope of the problems submitted.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.