Undergraduate Programme and Module Handbook 2024-2025
Module MATH4171: RIEMANNIAN GEOMETRY IV
Department: Mathematical Sciences
MATH4171: RIEMANNIAN GEOMETRY IV
| Type | Open | Level | 4 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
|---|
Prerequisites
- Mathematics modules to the value of 100 credits in Years 2 and 3, with at least 40 credits at Level 3 and including Differential Geometry III (MATH3021).
Corequisites
- None.
Excluded Combination of Modules
- None.
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 | |
| Preparation and Reading | 150 | ||||
| Total | 200 |
Summative Assessment
| Component: Examination | Component Weighting: 100% | ||
|---|---|---|---|
| Element | Length / duration | Element Weighting | Resit Opportunity |
| Written examination | 3 hours | 100% | |
Formative Assessment:
Eight assignments to be submitted.
■ 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