Lorentzian geometries are a special class of pseudo-Riemannian geometries which form the basis for general relativity. They are the natural analogue to describe curved space-times in the way that Riemannian geometry describes curved spaces. This course aims to be an introduction to the topic and it will be based on lecture notes by Christian Bär. There will also be an English version of these notes available before the commencement of the course. The two aims of the subject are (1) to treat the singularity theorems of Hawking and Penrose, and (2) understand the topological structure of globally hyperbolic Lorentzian manifolds through the classic theorem of Geroch's and its geometry through the recent theorem of Bernal and Sánchez.

The course will commence by recalling the basics concepts of differential geometry in the first lectures, before treating the important examples of Lorentzian geometry to help gain intuition for the general aspects of the subject. Then, we will treat causality, the soul of the subject, before looking at Cauchy hypersurfaces and global hyperbolicity. This is required to phrase and prove the singularity theorems of Hawking and Penrose, which are also of physical relevance. Then, we will consider the recent theorem due to Bernal and Sánchez which gives a product type decomposition for the metric of a globally hyperbolic manifold. The latter is a recent significant development in Lorentzian geometry which paves way for the use of mathematical analysis in studying physical problems.