An invitation to Alexandrov geometry: CAT(0) spaces

TL;DR

This work surveys Alexandrov geometry with a focus on CAT($0$) spaces, linking local curvature bounds to global structure. It develops core tools—Reshetnyak gluing, majorization, globalization, and two-convexity—and applies them to polyhedral spaces, subsets, and barycenters, while introducing the puff pastry construction for billiards. It connects metric geometry to geometric group theory through polyhedral constructions, flag and cubical complexes, and exotic aspherical manifolds, and provides practical criteria for CAT($0$) in complex assemblies. Overall, the text weaves together geometric constructions, comparison geometry, and topological consequences to illuminate how local curvature controls global topology and geometry.

Abstract

Our goal is to show the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. The topics include 1. Reshetnyak's gluing theorem, 2. Estimates on the number of collisions in billiards, 3. Reshetnyak's majorization theorem, 4. Hadamard--Cartan globalization theorem, 5. Polyhedral spaces, 6. Construction of exotic aspherical manifolds, 7. The geometry of two-convex sets in Euclidean space, 8. Barycenters and dimension theory.

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