Appears in collection : Summer School 2021 - Curvature Constraints and Spaces of Metrics

The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds on the sectional curvature of a Riemannian manifold in term of properties of its distance function, and then to consider metric spaces with these properties. This approach turned out to be very fruitful and it found many applications, bringing geometric ideas to other settings.In this course we will introduce the metric spaces with a curvature upper-bound in the sense of Alexandrov, and derive some of their geometric properties. The subject is very vast and it is not possible to be exhaustive in the limited time of this course. We will concentrate on some properties, both local and global, emphasizing that these metric spaces share many properties with manifolds of bounded curvature.