## 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 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

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

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