MC442 Differential Geometry

Next: MC445 Ring Theory Up: Year 4 Previous: MC435 General Relativity

MC442 Differential Geometry

Credits: 20

Convenor: Dr. J. Hunton

Semester: 1

Prerequisites:	essential: MC240, MC241, MC224
Assessment:	Individual and group coursework: 15%	Three hour examination in January: 85%

Lectures:	32	Classes:	10
Tutorials:	none	Private Study:	104
Labs:	none	Seminars:	4
Project:	none	Other:	none
Total:	150

Explanation of Pre-requisites

This module will draw on some basic ideas from both algebra and analysis. Some facility with basic skills from linear algebra and both single and multivaraible calculus will be assumed. While this module is not a prerequisite for MC435 General Relativity, it may provide helpful background ideas.

Course Description

The Earth is round, but maps are drawn on flat pieces of paper. When the map is only a street plan of a city this doesn't matter much, but when the map is of a whole country there can be big discrepancies between the real geography and what the map appears to describe. In a similar manner, Einstein's theory of General Relativity tells us that the Universe is curved, but it's much easier to do the mathematics describing processes in physics as if, at least in local regions, the Universe was flat. The questions of how can we reconcile our ability to do mathematics (drawing curves, measuring distances, differentiating functions and so on) on flat surfaces and spaces with our need to deal with undulating land and curved space-time form the basis of Differential Geometry.

This is an introductory course and will concentrate on presenting some of the ideas of the subject through a study of low dimensional questions and simple geometric objects. It will begin with the study of the most simple geometric examples, namely curves in the plane, and will then proceed to elaborate the ideas needed for studying those to examine more complicated geometric entities. We will see that at each stage we can abstract and generalise the concepts we used before to the next level of complexity, thus the module ought to `reinforce' itself as we go along: the same ideas will keep cropping up again and again.

There will be a group project which will involve applying the concepts of the module to the mathematics of map-making (cartography).

Aims

The aim of this module is to introduce understanding of some of the methods, scope and results of Differential Geometry through a study of low dimensional questions and simple geometric objects. The module also aims to teach some of the subject's applications to cartography and know some of the geometric language that is utilised in other areas of science such as General Relativity. Additionally, the module demonstrates a process of mathematical abstraction and development.

Objectives

To know the definitions of and understand the key concepts introduced in this module.

To be able to understand, reproduce and apply the main results and proofs in this module.

To be able to solve routine problems in the subject matter covered.

To be aware of some of the applications of the subject.

To give the opportunity for participation in a group project and presentation.

Transferable Skills

The ability to present arguments and solutions in a coherent and logical form.

The ability to use the techniques taught within the module to solve problems.

Improved ability in group work and presentation skills.

Syllabus

After about three or four introductory lectures explaining the nature of the module and the subject matter, the module splits into four main sections.

§1 Smooth curves in ${\bf R}^2$ . Definition of smooth, regular curves in ${\bf R}^n$ with notions of speed and arc length parametrisation. For curves in ${\bf R}^2$ the necessary apparatus (signed curvature) to prove the Fundamental Theorem classifying such objects. Examples.

§2 Smooth curves in ${\bf R}^3$ . Frenet-Serret apparatus and equations and proof of the Fundamental Theorem for curves in ${\bf R}^3$ . Examples. Geometric interpretation of torsion.

§3 Surfaces in ${\bf R}^3$ . Definition of smooth surfaces, tangent planes and normals in ${\bf R}^3$ .Orientability. The first fundamental form and computations of arc length, angle and area on a surface. Curvature and Gauss's proof of Euler's theorem on principal curvatures. Computations and examples. Isometries.

§4 Surfaces not in ${\bf R}^3$ . Abstract manifolds, bundles, tangents. Partitions of unity and metrics.

Group Project. Elementary mathematical cartography.

Reading list

Background:

William Boothby, An introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press. Th. Bröcker and K. Jänich, Introduction to Differential Topology, Cambridge University Press. M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish.

Details of Assessment

There will be a series of pieces of work set during the semester which together will count for 10% of the final mark. There will be a short group project on aspects of the module run part way through the semester which will count for 5% of the final mark.

There will be six questions on the examination paper; any number of questions may be attempted but full marks may be gained from answers to four questions. All questions will carry equal weight. The examination will be worth 85% of the final mark.

Next: MC445 Ring Theory Up: Year 4 Previous: MC435 General Relativity

S. J. Ambler
11/20/1999