Undergraduate Programme and Module Handbook 2023-2024

# Module MATH3021: DIFFERENTIAL GEOMETRY III

## Department: Mathematical Sciences

### MATH3021: DIFFERENTIAL GEOMETRY III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2023/24 | Module Cap | Location | Durham |
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#### Prerequisites

- Analysis in Many Variables II (MATH2031).

#### Corequisites

- None.

#### Excluded Combination of Modules

- None.

#### Aims

- To provide a basic introduction to the theory of curves and surfaces, mostly in 3-dimensional Euclidean space.
- The essence of the module is the understanding of differential geometric ideas using a selection of carefully chosen interesting examples.

#### Content

- Curves.
- Surfaces in n-dimensional real space.
- First Fundamental Form.
- Mappings of surfaces.
- Geometry of the Gauss map.
- Intrinsic metric properties.
- Theorema Egregium.
- Geodesics.
- Minimal surfaces.
- Gauss-Bonnet Theorem.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve novel and/ or complex problems in Differential Geometry.
- have a systematic and coherent understanding of theoretical mathematics in the field of Differential Geometry.
- have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
- Curves and surfaces in Euclidean space.
- First and second fundamental form.
- Mappings of surfaces.
- Geometry of the gauss map, Gaussian and mean curvature.
- Intrinsic metric properties of surfaces: the Theorem Egregium.
- Curves of shortest length on a surface: geodesics.
- Gauss-Bonnet theorem.

Subject-specific Skills:

- In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Spatial Awareness.

Key Skills:

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
- Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### 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 written assignments to be assessed and returned. Other assignments are set for self-study and complete solutions are made available to students.

■ 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