Mathematics: Specialised Course in Differential Geometry

This course provides an introduction to modern differential geometry, with a particular emphasis on the theory that grew out of Riemann’s groundbreaking 1854 lecture in Göttingen. Riemann’s ideas generalised the classical geometry of surfaces and laid the foundations for what we now call Riemannian geometry — the study of curved spaces using analytic and geometric tools. His perspective continues to influence both mathematics and physics, especially in the formulation of Einstein’s general relativity.

In this course, you will explore how geometric structures describe curvature, distance, and motion on manifolds, and how these concepts connect to physical phenomena such as gravity. You will also gain insight into the importance of differential geometry as a tool in other areas of mathematics and physics.

Topics include

• Differentiable manifolds, their tangent spaces and tangent bundles.
• Riemannian metrics and their unique Levi-Civita connection.
• Geodesics and the important Riemann curvature tensor and it influence on the local geometry.

The course is alternativecompulsory within the Master’s programme in Mathematics at Lund University and is also available as a standalone course.

Teaching consists of lectures and seminars. You will work in small groups on a compulsory assignment, which you will present orally. The course balances theoretical development with problemsolving and discussion, giving you multiple opportunities to present your reasoning and engage with peers — skills essential for both academic and professional environments.

The assessment consists of an oral presentation of group assignment during the course and an oral examination at the end of the course.

This course prepares you for further studies in mathematics and physics, especially in areas such as general relativity and geometric analysis. Differential geometry is foundational in theoretical physics, where it underpins models of spacetime and gravitational phenomena. It also appears in data science and machine learning through manifoldbased methods, in robotics and computer graphics for understanding shape and movement, and in engineering and space science where precise geometric modelling is essential.

By completing the course, you will acquire conceptual and technical tools that are valuable both for academic research and for applied fields requiring advanced mathematical modelling and reasoning.

Entry requirements

For admission to the course, English 6/B as well as at least 90 credits, with at least 60 credits in mathematics are required, including knowledge corresponding to the courses MATB22 Linear Algebra 2, 7.5 credits, MATB23 Analysis in Several Variables 2, 7.5 credits, and MATM33 Differential Geometry, 7.5 credits.

Selection criteria

Seats are allocated according to: ECTS (HPAV): 100 %.

Tuition fees for non-EU/EEA citizens

Citizens of countries outside:

• The European Union (EU)
• The European Economic Area (EEA) and
• Switzerland

are required to pay tuition fees. You pay an instalment of the tuition fee in advance of each
semester.

Tuition fees, payments and exemptions

Full programme/course tuition fee: SEK 23,125
First payment: SEK 23,125

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Note that you may also need to pay an application fee, or provide proof of exemption.

Application fee

No tuition fees for citizens of the EU, EEA and Switzerland

There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.