# RIEMANNIAN GEOMETRY - 2025/6

Module code: MAT3044

## Module Overview

Riemannian geometry is the study of geometric properties of spaces, called manifolds, typically in higher dimensions, which are described smoothly by sets of well-defined co-ordinates. The emphasis is on how mathematically to understand the notion of distance via a metric on such spaces, and how this is used to investigate key associated geometric concepts.

Students are introduced to the key concepts of manifolds and metrics, with illustrative examples such as spheres, hyperbolic spaces and Lie groups. The module continues with geodesics, isometries, covariant derivatives, the Levi Civita connection, and concludes with curvature and its properties.

Modules in Curves and Surfaces and in Manifolds and Topology contain related material.

### Module provider

Mathematics & Physics

GUTOWSKI Jan (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 64

Lecture Hours: 33

Seminar Hours: 5

Guided Learning: 15

Captured Content: 33

Semester 2

None

## Module content

Indicative content includes:

• Manifolds and Vector Fields: Abstract manifolds, with examples such as spheres and Lie groups. Tangent vectors and vector fields, integral curves. The vector field commutator and its geometric interpretation. Jacobi identity. Lie derivatives.

• Metrics, geodesics and isometries: Metrics and pull-back Metrics, with examples. Isothermal co-ordinates. Derivation of geodesic equations. Examples of isometries and application to solving the geodesic equations.

• Covariant Derivative: Affine connections, torsion, Christoffel symbols, the Levi-Civita connection. Parallel transport with application to geodesics.

• Curvature:  Curvature tensor, algebraic and geometric properties of Riemann curvature. Low dimensional examples.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 minutes) 20
Examination End-of-Semester Examination (2 hours) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• The ability to understand and formulate statements about the geometry of various surfaces in higher dimensional space.

• Understanding and knowledge of important definitions, key theorems and propositions, and related mathematical concepts in Riemannian Geometry.

• The ability to apply subject knowledge to the analysis of unseen problems.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 and 4

• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1, 2, 3 and 4.

Formative assessment
There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

Feedback
Students will receive individual written feedback on both the formative unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from both courseworks and the in-semester test will assist students with preparation for the synoptic examination. Students also receive verbal and written feedback in tutorials and office hours, with tutorials taking place after each of the unassessed courseworks, and before and after the in-semester test.

## Module aims

• Introduce students to the geometry of manifolds of dimension greater than two, with the added feature of a metric on the manifold.
• Introduce students to the concepts of a covariant derivative and connection and their implications.
• Examine these concepts in the setting of examples such as spheres, hyperbolic manifolds, and some low-dimensional Lie groups.
• Help students to study geometric techniques such as stereographic projection and curvature of manifolds using appropriate Python packages.

## Learning outcomes

 Attributes Developed 001 Students will understand introductory key properties of Riemannian geometry such as the concept of a manifold and key properties of vector fields. KC 002 Students will understand more advanced concepts of Riemannian geometry, building on Learning Outcome 1, such as the use of a metric, the definition of a covariant derivative and its relation to a connection, isometries, geodesics, parallel transport and curvature. KC 003 Students will utilize key techniques, such as the Einstein summation convention, and both co-ordinate independent and component based calculations, to develop their understanding of the above theory. KCP 004 Students will have a clear idea of how to implement the module theory and techniques in appropriate examples. KCP 005 Students will develop digital capabilities through the utilization of Python code. KCPT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Be an introduction to the core ideas in Riemannian Geometry, concentrating in particular on metric, covariant derivative, and curvature, and how they can be used to investigate geometric properties associated with different types of manifolds. Students gain familiarity with the key mathematical techniques used to investigate these concepts through the examples presented in the lectures as well as the unassessed courseworks, which are also discussed in detail during seminars. This helps students to develop good mathematical practice.

The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures, together with supplementary Python code. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five one-hour tutorials per semester. These tutorials provide an opportunity for students to gain feedback and assistance with examples.

• Two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT3044

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourcefulness and Resilience and Sustainability. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Digital Capabilities: Students develop digital capabilities via examples of Python code. Some examples are illustrative, e.g. students will engage with Python coding to visualize the stereographic projection which is used in constructing co-ordinates on spheres. Other examples are computational, e.g. students will develop computational skills by using the Python package einsteinpy to calculate the components of the Riemann curvature in higher dimensional examples. The Python coding is described on the module virtual learning environment utilizing appropriate online resources.

Employability: Students enhance their employability by gaining additional experience utilizing Python code in the module to examine various geometric examples, as well as by developing abstract problem solving skills.

Global and Cultural Capabilities: Student enrolled in MAT3044 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: Students taking MAT3044 gain skills in problem solving and lateral thinking as they learn about the geometric properties of higher dimensional manifolds, assisted by the examples they investigate in the formative assessments. The module also helps to make connections and underpin key geometric aspects in the mathematics degree programmes, and assists in the scaffolding of these elements within the mathematics curriculum. In particular, it develops introductory concepts of differential geometry introduced in Curves and Surfaces by elucidating and generalizing the key mathematical structures. It also complements the more advanced structures associated with the topological aspects of differential forms and mathematical structures which appear in modules in Manifolds and Topology and in Lagrangian and Hamiltonian Dynamics.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MSc 2 Optional A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.