Differential Geometry
MATM33 Differential Geometry, 7.5 credits, is an alternative-compulsory course at advanced level for a Master of Science degree in Mathematics. The course can be taken as an stand-alone course. The course is given at half-study pace during the first half of the autumn semester. The language of instruction is English.
Course Content
We show that a curve in R3 is, up to Euclidean motions, totally determined by its curvature and torsion. We study the second fundamental form of a surface, describing its shape in the ambient space R3. This leads to a fundamental object the curvature of the surface. Amongst many interesting results we prove the famous "Theorema Egregium" of Gauss which tells us that the curvature is an intrinsic object i.e. determined by the way we measure distances on the surface. Furthermore we prove the astonishing Gauss-Bonnet theorem. This implies that for a compact surface the curvature integrated over it is a topological invariant.
Teaching
The teaching consists of lectures and seminars. A compulsory assignment is included in the course. The assignment should be solved in smaller groups and the solutions should be presented orally to the entire student group.
Assessment
The examination consists of a written examination and an oral examination at the end of the course, as well as an oral presentation of group assignment during the course. The oral examination may only be taken by those students who pass the written examination.
Course Literature
- Sigmundur Gudmundsson, An Introduction to Gaussian Geometry, Lund University (2020) (main text).
- M. P. do Carmo, Differential geometry of curves and surfaces, Prentice Hall (1976) (recommended complementary reading).
- L.M. Woodward, J.Bolton, A First Course in Differential Geometry-Surfaces in Euclidean Space, Cambridge University Press (2019) (recommended complementary reading).
- Andrew Pressley, Elementary Differential Geometry (2010) (recommended complementary reading).
Official Course Description
Course Evaluation
Link to course evaluations on the department's website: