Last modified: 14 Nov 2025 14:16
One of the aims of the course is to understand the mathematical concept of curvature. We will do this by first studying the geometry of polygonal surfaces, and then by looking at smooth surfaces in Euclidean space.
Polygonal surfaces provide a set of very easy examples with which we can explore the new ideas and quantities. They also allow us to develop the intuition needed in the later part of the course.
| Study Type | Undergraduate | Level | 4 |
|---|---|---|---|
| Term | Second Term | Credit Points | 15 credits (7.5 ECTS credits) |
| Campus | Aberdeen | Sustained Study | No |
| Co-ordinators |
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This course is about geometry in the broadest sense. You will learn how familiar concepts from Euclidean geometry such as straight lines, angles and properties of triangles can be generalized to arbitrary metric spaces. The concepts of geodesics and of curvature will be explored through various examples, starting with polygonal complexes and moving through spherical and hyperbolic geometries. Examples and motivation from physics will be included wherever possible. The final part of the course serves as an introduction to differential and Riemannian geometry, focussing on surfaces in Euclidean 3-space. Come the end you will be able to appreciate Gauss’s Theorema Egregium (“Remarkable Theorem”), which states that the curvature of a surface at a point depends only on the metric near that point and is independent of how the surface is embedded in space.”
Information on contact teaching time is available from the course guide.
| Assessment Type | Summative | Weighting | 15 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback | ||||
| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Conceptual | Analyse | Reason with metric concepts such as (local) isometry, lengths of curves and induced and intrinsic metrics. |
| Conceptual | Understand | Understand the concept of curvature, and how it relates to topology via the Euler characteristic and the Gauss-Bonnet formula. |
| Procedural | Apply | Calculate lengths and angles in general metric spaces. |
| Procedural | Apply | Recognise, count and measure geodesics in general metric spaces. |
| Assessment Type | Summative | Weighting | 70 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Duration: 2 hours |
|||
| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Conceptual | Analyse | Reason with metric concepts such as (local) isometry, lengths of curves and induced and intrinsic metrics. |
| Conceptual | Understand | Understand the concept of curvature, and how it relates to topology via the Euler characteristic and the Gauss-Bonnet formula. |
| Conceptual | Understand | Understand the definition of a surface in 3-dimensional space. |
| Procedural | Apply | Compute the mean and Gauss curvatures of surfaces in 3-dimensional space. |
| Procedural | Apply | Calculate lengths and angles in general metric spaces. |
| Procedural | Apply | Recognise, count and measure geodesics in general metric spaces. |
| Assessment Type | Summative | Weighting | 15 | |
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback | ||||
| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Conceptual | Analyse | Reason with metric concepts such as (local) isometry, lengths of curves and induced and intrinsic metrics. |
| Procedural | Apply | Calculate lengths and angles in general metric spaces. |
| Procedural | Apply | Recognise, count and measure geodesics in general metric spaces. |
There are no assessments for this course.
| Assessment Type | Summative | Weighting | ||
|---|---|---|---|---|
| Assessment Weeks | Feedback Weeks | |||
| Feedback |
Best of (resit exam mark) or (resit exam mark with carried forward CA marks). Duration: 2 hours |
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| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
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|
||
| Knowledge Level | Thinking Skill | Outcome |
|---|---|---|
| Conceptual | Understand | Understand the concept of curvature, and how it relates to topology via the Euler characteristic and the Gauss-Bonnet formula. |
| Procedural | Apply | Recognise, count and measure geodesics in general metric spaces. |
| Procedural | Apply | Compute the mean and Gauss curvatures of surfaces in 3-dimensional space. |
| Conceptual | Analyse | Reason with metric concepts such as (local) isometry, lengths of curves and induced and intrinsic metrics. |
| Procedural | Apply | Calculate lengths and angles in general metric spaces. |
| Conceptual | Understand | Understand the definition of a surface in 3-dimensional space. |
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