The University of Southampton
Courses

# MATH6109 Differential Geometry and Lie Groups

## Module Overview

This is a compulsory lecture module for MMath students in their fourth year. The module will begin by looking at differential manifolds and the differential calculus of maps between manifolds. Manifolds are an abstraction of the idea of a smooth surface in Euclidean space. The module will then look at calculus on manifolds including the study of vector fields, tensor fields and the Lie derivative. The module will then go on to study Riemannian geometry in general by showing how the metric may be used to define geodesics and parallel transport, which in turn may be used to define the curvature of a Riemannian manifold. As another major application the module will investigate groups, such as the rotation group SO(3), which also have the structure of a manifold. Such objects are called Lie groups and play an important role in both theory and application of geometry. As an example of this we look at the symmetries of Riemannian manifolds. The isometries of a Riemannian metric form a group and the corresponding infinitesimal isometries form a Lie algebra. Another class of examples will be provided by matrix groups.

### Aims and Objectives

#### Learning Outcomes

##### Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• Understand the concept of a differentiable manifold
• Understand the basic results of calculus on manifolds
##### Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Be able to calculate the integral curve of a vector field and the Lie bracket of a pair of vector fields
• Calculate the connection and curvature of a metric on a manifold
##### Subject Specific Practical Skills

Having successfully completed this module you will be able to:

• Find infinitesimal isometries of simple metrics

### Syllabus

Syllabus: The derivative of functions from R^n to R^m The inverse function theorem and the implicit function theorem Tensors over a real vector space Abstract manifolds Vectors as derivative operators and the tangent space Covectors, tensors, tensor fields, and the commutator of vector fields The metric tensor Maps of manifolds, pushforward and pullback of tensor fields Integral curves of a vector and the Lie derivative Linear connections, torsion, curvature Riemannian geometry, geodesics Symmetry, Lie groups and Lie algebras Matrix Lie groups and their Lie algebras

### Learning and Teaching

#### Teaching and learning methods

Lectures, private study.

TypeHours
Lecture36
Tutorial12
Independent Study102
Total study time150

T. Aubin. A Course in Differential Geometry.

B Schutz. Geometrical Methods of Mathematical Physics.

M Lee. Manifords and Differential Geometry.

### Assessment

#### Summative

MethodPercentage contribution
Coursework 20%
Exam  (2 hours) 80%

#### Repeat Information

Repeat type: Internal & External

Pre-requisite: MATH2039

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.