| Unit name | Geometry of Manifolds |
|---|---|
| Unit code | MATHM0037 |
| Credit points | 20 |
| Level of study | M/7 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Viveka Erlandsson |
| Open unit status | Not open |
| Units you must take before you take this one (pre-requisite units) |
MATH20006 Metric Spaces and MATH20004 Introduction to Geometry |
| Units you must take alongside this one (co-requisite units) |
None |
| Units you may not take alongside this one |
N/A |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
Lecturers: Asma Hassannezhad and Viveka Erlandsson
Unit Aims
The aim of the unit is to study foundations of the geometry of manifolds including the concept of an abstract manifold, Riemannian metric and curvature, and to provide students with a firm grounding in the theory and techniques in this area and to offer students ample opportunity to build on their problem-solving ability.
Unit Description
The study of manifolds is fundamental in many important areas of modern mathematics. Manifolds provide a natural setting for research in various areas including geometry, analysis, partial differential equation and mathematical physics. It generalises the notion of curves and surfaces in R3 as well as many concepts from linear algebra. This unit builds a foundation of abstract differentiable and Riemannian manifolds. A differentiable manifold locally looks like the Euclidean space Rn and we can generalise the notion of the inner product in Rn to an inner product on the tangent space of the manifold. This gives rise to the definition of a Riemannian metric. A differentiable manifold equipped with a Riemannian metric is called a Riemannian manifold. In this unit, we study the various differentiable and geometric structures of a manifold including differential forms, Lie derivative, geodesics and curvatures.
Relation to Other unit
This unit can be considered as a continuation and as an advance version of the second-year unit MATH20004 Introduction to Geometry and to a lesser extent unit MATH21100 Linear Algebra 2. It is related to the 3rd-year unit MATH30018 Fields, Forms and Flows. It can be also complementary to the unit MATH30001/MATHM0008 Topics in Modern Geometry.
By the end of the unit, students should have developed an understanding of basic definitions and results regarding differentiable manifolds, like smooth maps and tangent spaces, Riemannian metrics and curvatures. They should be able to solve routine problems and to apply the techniques of the unit to unseen situations.
In addition, the unit is aimed to help students gain skills to articulate their mathematical ideas in the form of a presentation. This is an important skill for their future development.
Syllabus
Topics covered will include:
Lectures (3 hrs per week) and recommended problems.
The final mark is calculated as follows:
If this unit has a Resource List, you will normally find a link to it in the Blackboard area for the unit. Sometimes there will be a separate link for each weekly topic.
If you are unable to access a list through Blackboard, you can also find it via the Resource Lists homepage. Search for the list by the unit name or code (e.g. MATHM0037).
How much time the unit requires
Each credit equates to 10 hours of total student input. For example a 20 credit unit will take you 200 hours
of study to complete. Your total learning time is made up of contact time, directed learning tasks,
independent learning and assessment activity.
See the Faculty workload statement relating to this unit for more information.
Assessment
The Board of Examiners will consider all cases where students have failed or not completed the assessments required for credit.
The Board considers each student's outcomes across all the units which contribute to each year's programme of study. If you have self-certificated your absence from an
assessment, you will normally be required to complete it the next time it runs (this is usually in the next assessment period).
The Board of Examiners will take into account any extenuating circumstances and operates
within the Regulations and Code of Practice for Taught Programmes.
