Please note: Due to alternative arrangements for teaching and
assessment in place from 18 March 2020 to mitigate against the restrictions in
place due to COVID-19, information shown for 2019/20 may not always be accurate.
Please note: you are viewing unit and programme information
for a past academic year. Please see the current academic year for up to date information.
| 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. Hassannezhad |
| Open unit status |
Not open |
| Pre-requisites |
MATH20006 Metric Spaces and MATH20004 Introduction to Geometry
|
| Co-requisites |
None
|
| School/department |
School of Mathematics |
| Faculty |
Faculty of Science |
Description including Unit Aims
Unit Aims
The aim of the unit is to study foundations of geometry of manifolds including the concept of an abstract manifold, Riemannian metric and curvature.
Unit Description
- Differentiable Manifolds
- Tangent spaces; vector fields on manifolds
- Lie derivative and Frobenius integrability theorem
- Differential forms and Stoke’s Theorem
- Riemannian metric; covariant derivative
- Exponential map, geodesics and parallel transport
- Curvatures and spaces of constant curvature
Relation to Other Units
This unit replaces Fields, Forms and Flows 34 therefore students may not take this unit if they have already taken Fields, Forms and Flows 3. This can be considered as a continuation or as an advanced version of the second-year unit Introduction to Geometry. It is also complementary to the units Topics in Modern Geometry and Algebraic Geometry.
Intended Learning Outcomes
By the end of this course the students should be able to:
- understand the geometry of an abstract manifold including Stoke’s theorem
- understand the definition of metrics, geodesics and curvatures
- develop problem solving skills on an abstract manifold and apply the methods and definitions to more concrete examples
Teaching Information
3 hours of lecture per week, including a problem session every other week. The students will be given weekly or bi-weekly non-assessed homework assignments.
Assessment Information
The final mark is calculated as follows:
- 80% Exam
- 20% Assessed Coursework
Reading and References
Recommended
- Christian Bär, Elementary Differentiable Geometry, Cambridge University Press, 2010
- Manfredo Perdigão Do Carmo, Riemannian Geometry, Birkhäuser, 1992
- John M. Lee, Introduction to Smooth Manifolds, Springer, 2013
- John M. Lee, Riemannian Manifolds, Springer, 1997
