This course should serve as an introduction to the field of Alexandrov geometry, the study of metric spaces with (a non-smooth analogue of) bounded sectional curvature. We discuss same basic metric geometry and introduce a variety of different formulations of these curvature bounds. We then go on to showcase several theorems, including: Hopf--Rinow--Cohn--Vossen, Majorisation, Gluing, and the Collision Theorem.
Very recently, the philosophy of Alexandrov geometry (and metric geometry in general) has found success also in Lorentzian signature. We present some basics from this theory as well and then plan to show the Toponogov Globalisation Theorem as well as the Bonnet--Myers Theorem.
There is some flexibility for the contents to be discussed. Only basic prerequisites on analysis and topology are necessary. Some knowledge of Riemannian (and Lorentzian) geometry increases the appreciation for motivation. Lecture notes will be provided on my homepage (https://sites.google.com/view/felixrott/) before the start of the lecture.
Concerning metric literature, the following sources are recommended:
M. R. Bridson, A. Haefliger. Metric spaces of non-positive curvature (1999).
D. Burago, Y. Burago, S. Ivanov. A course in metric geometry (2001).
S. Alexander, V. Kapovitch, A. Petrunin. An invitation to Alexandrov geometry: CAT(0) spaces (2019).
There is no established book yet on non-smooth Lorentzian geometry, however the following papers and surveys might be of interest:
M. Kunzinger, C. Sämann. Lorentzian length spaces (2018).
C. Sämann. A brief introduction to non-regular spacetime geometry (2024).
T. Beran, L. Napper, F. Rott. Alexandrov's Patchwork and the Bonnet--Myers theorem for Lorentzian length spaces (2025).
T. Beran, J. Harvey, L. Napper, F. Rott. A Toponogov globalisation result for Lorentzian length spaces (2025).
Concerning repetition of smooth knowledge, you may consult:
J. K. Beem, P. E. Ehrlich, K. L. Easley. Global Lorentzian geometry (1996).
J. M. Lee. Introduction to Riemannian manifolds (2018).