Module manager: Gerasim Kokarev
Email: G.Kokarev@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
MATH3130 Differential Geometry
| MATH3130 | Differential Geometry |
This module is not approved as an Elective
Riemannian geometry is the study of length, angle, volume, and curvature. It is a far-reaching generalisation of the theory of curves and surfaces. Famously, it formed the basis of Einstein's theory of general relativity, and its language and methods are ubiquitous in modern geometry and theoretical physics. In this module you will learn the basic concepts of Riemannian geometry, and study some beautiful theorems relating the geometry of a manifold to its topological properties.
We start with introducing the notion of manifold, and basics of calculus on manifolds. We proceed with the notion of Riemannian metric and the way it describes the geometry of an abstract manifold. Then we develop other analytical tools, by studying the Levi-Civita connection, the curvature tensor, and geodesics. Using calculus of variations and topological constructions, we study the relationships between topological and geometric properties of Riemannian manifolds.
Subject specific learning outcomes:
On successful completion of the module students will have
demonstrated the following learning outcomes relevant to the
subject:
1. Recall and explain main concepts of Riemannian
geometry;
2. Perform calculations involving vector fields and the Levi-Civita connection;
3. Define fundamental objects of the module, such as manifold, curvature tensor, geodesics, and recall their properties;
4. State theorems of Riemannian geometry and explain their proofs.
Skills learning outcomes:
On successful completion of the module students will be able to:
Solve related problems and give rigorous arguments; embark on further research in the related subjects.
Find and evaluate related digital resources to navigate and access information to enhance learning and problem solving.
Communicate with clarity and rigour.
Apply mathematical skills to model, generalise, formulate, and analyse problems.
Collect, process, and communicate related information in an ethical way.
1. Smooth manifolds
2. Vector fields and flows
3. Riemannian metrics and the Levi-Civita connection
4. Riemann curvature tensor and related quantities
5. Geodesics, the exponential map, and the Hopf-Rinow
Theorem
6. Variational theory of geodesics and its applications: the theorems of Cartan-Hadamard and Bonnet-Myers.
7. Riemannian coverings, closed geodesics and Synge’s theorem.
Additional topics that build on these may be covered as time
allows.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 33 | 1 | 33 |
| Private study hours | 117 | ||
| Total Contact hours | 33 | ||
| Total hours (100hr per 10 credits) | 150 | ||
117
Formative feedback will be provided on regular example sets or other similar learning activity.
Check the module area in Minerva for your reading list
Last updated: 12/05/2026
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