The University of Southampton
Courses

# MATH6079 Hyperbolic Geometry

## Module Overview

This is a structured self-study module designed for MMath students in their fourth year. In this module, we take a guided tour of the hyperbolic plane, H, exploring its geometry and the group preserving its geometric structure. The hyperbolic plane arises in several contexts, most publicly in the etchings of MC Escher. We begin with a brief review of topology on Rn, to set the stage, before embarking on the actual tour of the hyperbolic plane. We will examine several realizations of H and explore the connections between them. We will classify the isometries of H, which are the maps from H to H preserving both angle and length, and close by considering area and trigonometry in H

### Aims and Objectives

#### Module Aims

In this module, we take a guided tour of the hyperbolic plane H, exploring its geometry and the group preserving its geometric structure.

#### Learning Outcomes

##### Knowledge and Understanding

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

• On successful completion of the module the students should be able to: understand the difference between hyperbolic and Euclidean geometry
• Perform calculations with Möbius transformations and determine the multipliers and fixed point sets of Möbius transformations
• 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
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

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

### Syllabus

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

### Learning and Teaching

#### Teaching and learning methods

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.

TypeHours
Teaching36
Independent Study114
Total study time150

KELLY, P and MATTHEWS, G. The Non-euclidean plane.

ANDERSON, J W. Hyperbolic Geometry.

GREENBERG, MJ. Euclidean and non-Euclidean geometries.

### Assessment

#### Summative

MethodPercentage contribution
Portfolio of Problems 100%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

#### Pre-requisites

To study this module, you will need to have studied the following module(s):

CodeModule
MATH2003Group Theory
MATH3079Metric Spaces and Topology

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.