MATM43 Specialised Course in Differential Geometry, 7,5 credits, is an alternative-compulsory course for a Master of Science degree in mathematics. The course can be taken as an stand-alone course. The course is given at 25% study pace during each spring semester. The language of instruction is English.

Course Content
This course is an introduction to the beautiful theory of Riemannian Geometry, a subject with no lack of interesting examples. They are indeed the key to a good understanding of it and will therefore play a major role throughout the course. Of special interest are the classical Lie groups allowing concrete calculations of many of the abstract notions on the menu.

The course covers

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.

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 an oral examination at the end of the course, as well as an oral presentation of group assignment during the course.

Course Literature
No particular textbook will be used but the participants are recommended to have a look at some of the following:

M. P. do Carmo, Riemannian Geometry, Birkhäuser (1992)
D. Gromoll, W. Klingenberg, W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math. 55, Springer (1975)
S. Gudmundsson, An Introduction to Riemannian Geometry, Lund University (2021)
W. Klingenberg, Riemannian Geometry, de Gruyter (1995)
W. Kühnel, Differential Geometry: Curves - Surfaces - Manifolds, AMS (2006)
Serge Lang, Fundamentals of Differential Geometry, Springer (1999)
John M. Lee, Riemannian Manifolds, Springer (1997)
B. O'Neill, Semi-Riemannian Geometry, Academic Press (1983)
P. Petersen, Riemannian Geometry, Springer (2006)
T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs 149, AMS (1996).
M Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish (1979).

Official Course Description

Course Evaluation
Link to course evaluations on the department's website:

Course Evaluations, Mathematics at Faculty of Science