# 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

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

### Reading list

The reading list is available from the Library websiteLast updated: 22/03/2010

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- Undergraduate module catalogue
- Taught Postgraduate module catalogue
- Undergraduate programme catalogue
- Taught Postgraduate programme catalogue

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