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MA4111 Differential Geometry

MA4111 Differential Geometry

Credits: 20 Convenor: Prof. J.R. Hunton Semester: 1 (weeks 1 to 12)
Prerequisites: essential: MA2002, MA2151 desirable: MA3151
Assessment: Coursework: 15% Examination: 85%
Lectures: 36 Problem Classes: 6
Tutorials: none Private Study: 108
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

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.

Learning Outcomes

To know the definitions of and understand the key concepts of low dimensional differnetial geometry as introduced in this module and to be able to understand, reproduce and apply the main results and proofs given. Students will also be able to solve routine problems in the subject matter covered and be aware of some of the applications of the subject.

Methods

Lectures, example classes, example sheets, group project.

Assessment

Marked problem sheets, written examination, group presentation.

Subject Skills

Aims

To see how one can extend classical `flat' mathematics to the study of the geometry of curved objects (curves, surfaces, and so on). To become more aware of the mathematical process of abstraction.

Learning Outcomes

Students will become familiar with the analysis of differentiable curves, surfaces and their generalisation, and be aware of their application to the mathematical theory of map making.

Methods

Lectures, examples classes, example sheets, group project.

Assessment

Marked problem sheets, written examination, group presentation.

Explanation of Pre-requisites

This module needs some knowledge of both analysis and the techniques of vector or multivariate calculus. The analysis needed requires the student to be comfortable with discussing reasonably straightforward topological spaces such as curves, spheres, tori, discs and so on. Conceptually, this is all more than covered in the Abstract Analysis course MA2151, but more familiarity with these conceps will be gained from the Topology module MA3151 which would be a good partner with this module.

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).

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

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

§3 Surfaces in $R^3$. Definition of smooth surfaces, tangent planes and normals in $R^3$. 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 $R^3$. Abstract manifolds, bundles, tangents. Partitions of unity and metrics.

Group Project. Elementary mathematical cartography.

Reading list

Recommended:

John McCleary, Geometry from a Differential Viewpoint, Cambridge University Press.

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.

Resources

Problem sheets, lecture rooms, lecturer, cartographical handouts.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA4121 Projective Curves Up: ModuleGuide03-04 Previous: MA4101 Algebraic Topology

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Last updated: 2004-02-21
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