Mathematics: Differential Geometry

This course offers a solid introduction to classical differential geometry, a field essential for advanced studies in mathematics and theoretical physics. You will learn how geometry describes the shape and behaviour of curves and surfaces, and how these ideas connect to Einstein’s general theory of relativity through Riemannian geometry.

Topics covered include

• Geometry of curves in Euclidean space: curvature, torsion, and how they define the curve.
• Geometry of surfaces: first and second fundamental forms, Gauss map, principal curvatures, Gaussian and mean curvature.
• Theorema Egregium and in-depth analysis of geodesics — locally and globally.
• Gauss–Bonnet Theorem: two local versions and the famous global version.

The course is alternative-compulsory within the Master’s programme in Mathematics at Lund University. It can also be taken as an optional course within the Bachelor’s programme in Mathematics, or as a standalone course. 

Teaching combines lectures and seminars with a strong emphasis on problemsolving and mathematical communication. You will work actively with selected exercises and collaborate in small groups on a compulsory assignment, which is presented orally to the class.

Assessment is based on a written exam, complemented by an oral exam for those who pass the written part, and includes an oral group presentation during the course. Alternative examination formats may be arranged in consultation with Disability Support Services.

Differential geometry is a gateway to advanced mathematics and theoretical physics. After completing the course, you will be well-prepared for for advanced studies in mathematics and theoretical physics, particularly in geometry, topology, and mathematical physics. It develops analytical thinking, problemsolving skills, and the ability to communicate mathematics clearly, both orally and in writing. These abilities are essential for academic research and valuable in professional contexts where geometric modelling and advanced mathematical analysis play a central role.

Prerequisites

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

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.