2008/09 Undergraduate Module Catalogue

MATH3112 Differential Geometry 1

10 creditsClass Size: 100

Module manager: Professor R. Bielawski
Email: rb@maths.leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

Pre-requisites

MATH2051

Geometry of Curves and Surfaces

This module is mutually exclusive with

MATH5031M

Differential Geometry 2

This module is approved as an Elective

Module summary

This course follows on from MATH2051: Curves and Surfaces. It concentrates on (i) what properties of a surface (or curve) are intrinsic, i.e., can be measured by the inhabitants of the surface without going outside it; (ii) what properties are global, i.e. remain the same when the surface is deformed.

Objectives

On completion of this module, students should be able to:

(a) calculate envelopes and rotation indices;
(b) recognise when curves are congruent;
(c) recognize when a level set defines a surface and obtain its properties;
(d) understand and recognise properties of isometries, conformal mappings and map projections;
(e) appreciate the difference between intrinsic properties and extrinsic properties
(f) understand the abstract approach to manifolds and metrics;
(g) prove the major results of the module, where proofs have been given.

Syllabus

In the first case, we meet the Theorema Egregium of Gauss which says that the Gauss curvature of a surface is intrinsic, we contrast this with the mean curvature, which is zero for a soap film, but depends crucially on how that soap film lies in 3-space. More generally, we examine what properties are preserved by transformations, with applications to map projections of the surface of the earth. We finish with the celebrated Gauss-Bonnet theorem, which says that the total curvature of a surface is unchanged however much the surface is deformed, for example for any surface which "looks like" a sphere, it is 4?. A common theme is that of "curvature", this concept underpins much modern maths, for example, the curved universe of general relativity theory.
Topics include:
Plane curves: rotation index. Space curves: congruence. Submanifolds of Euclidean spaces as level sets. Gauss map,. Curves on a surface, geodesics. Transformations and map projections. Theorema Egregrium. Gauss-Bonnet theorem. Riemannian metrics on the plane. Introduction to abstract surfaces.

Teaching methods

Delivery type	Number	Length hours	Student hours
Example Class	6	1.00	6.00
Lecture	20	1.00	20.00
Private study hours			74.00
Total Contact hours			26.00
Total hours (100hr per 10 credits)			100.00

Opportunities for Formative Feedback

Regular problem sheets

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	2 hr	100.00
Total percentage (Assessment Exams)		100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Reading list

The reading list is available from the Library website

Last updated: 22/03/2010

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