2017/18 Undergraduate Module Catalogue

MATH3113 Differential Geometry

15 creditsClass Size: 40

Module manager: Dr Gerasim Kokarev
Email: G.Kokarev@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2017/18

Pre-requisites

MATH2051

Geometry of Curves and Surfaces

This module is mutually exclusive with

MATH5113M

Advanced Differential Geometry

This module is approved as a discovery module

Module summary

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

Objectives

On completion of this module, students should be able to:
(a) calculate rotation indices of plane curves
(b) recognise regularly homotopic curves
(c) recognise when curves are congruent
(d) recognize when a level set defines a surface and obtain its properties
(e) compute the Gauss curvature of hypersurfaces in n-space
(f) understand and recognise properties of isometries, conformal mappings and map projections
(g) appreciate the difference between intrinsic properties and extrinsic properties
(h) prove the major results of the module, where proofs have been given

Syllabus

We are interested in global properties of curves and surfaces and the relation between local quantities and global invariants.

In the first part of the course, we meet the Whitney-Graustein theorem, which says that two curves can be deformed into each other as soon as a single number (the rotation index) is the same for both curves. We give some applications of this result, before proceeding to study global properties of n-surfaces, such as isometries, shortest curves etc. 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, Whitney-Graustein theorem, turning tangent theorem
- 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

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	33	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

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 30 mins	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: 26/04/2017

Disclaimer

Browse Other Catalogues

• Undergraduate module catalogue
• Taught Postgraduate module catalogue
• Undergraduate programme catalogue
• Taught Postgraduate programme catalogue

Errors, omissions, failed links etc should be notified to the Catalogue Team.PROD