Postgraduate Programme and Module Handbook 2020-2021 (archived)
Module MATH41320: Riemannian Geometry
Department: Mathematical Sciences
MATH41320: Riemannian Geometry
| Type | Tied | Level | 4 | Credits | 20 | Availability | Available in 2020/21 | Module Cap | None. |
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| Tied to | G1K509 |
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Prerequisites
- Differential Geometry.
Corequisites
- None
Excluded Combination of Modules
- Differential Geometry.
Aims
- Provide a knowledge of the intrinsic geometry of Riemannian manifolds. This is a significant generalisation of the metric geometry of surfaces in 3-space.
Content
- The metric geometry of Riemannian manifolds.
- Geodesics.
- Various notions of curvature, and their effect on the geometry of a Riemannian manifold.
- Second variation formula, global comparison theorems with applications.
Learning Outcomes
Subject-specific Knowledge:
- Have a knowledge and understanding of Riemannian geometry demonstrated through the following topic areas:
- Riemannian manifolds;
- geodesics;
- Levi-Civita connection;
- curvature;
- global comparison results.
Subject-specific Skills:
- Have developed advanced technical and scholastic skills in the area of the geometry of surfaces in 3-space.
Key Skills:
- Have developed an appreciation of high-level mathematical reasoning.
- Have developed the ability to present well-reasoned arguments and operate in complex and specialised contexts.
Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module
- Teaching is by lectures through which the main body of knowledge is made available.
- Students do regular formative work solving problems to gain insight into the details of the relevant theories and procedures.
- End of year examinations assess the learning.
Teaching Methods and Learning Hours
| Activity | Number | Frequency | Duration | Total/Hours | |
|---|---|---|---|---|---|
| Lectures | 42 | 2 per week for 20 weeks and 2 in term 3 | 1 Hour | 42 | |
| Problems Classes | 8 | Four in each of terms 1 and 2 | 1 Hour | 8 | |
| Preperation and Reading | 150 | ||||
| Total | 200 |
Summative Assessment
| Component: Examination | Component Weighting: 90% | ||
|---|---|---|---|
| Element | Length / duration | Element Weighting | Resit Opportunity |
| Written Examination | 3 hours | 100% | |
| Component: Continuous Assessment | Component Weighting: 10% | ||
| Element | Length / duration | Element Weighting | Resit Opportunity |
| Eight written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students. | 100% | ||
Formative Assessment:
■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University