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## Über dieses Buch

Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.

## Inhaltsverzeichnis

### Chapter 1. Preliminaries

Abstract
In this chapter we fix some conventions and recall the main definitions. The chapter may be used as a quick reference when reading the book. To learn background in metric geometry, the reader may consult the book of Dmitri Burago, Yuri Burago, and Sergei Ivanov [20].
Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

### Chapter 2. Gluing theorem and billiards

Abstract
In this chapter we define $$\mathrm{CAT}^{}(\upkappa )$$ spaces and give the first application, to billiards. Here “$$\text {CAT}$$” is an acronym for Cartan, Alexandrov, and Toponogov. It was coined by Mikhael Gromov in 1987. Originally, Alexandrov called these spaces “$$\mathfrak {R}_\upkappa$$ domain”; this term is still in use. Riemannian manifolds with nonpositive sectional curvature provide a motivating example. Specifically, a Riemannian manifold has nonpositive sectional curvature if and only if each point admits a $$\mathrm{CAT}^{}(0)$$ neighborhood.
Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

### Chapter 3. Globalization and asphericity

Abstract
In this chapter we introduce locally $$\mathrm{CAT}^{}(0)$$ spaces and prove the globalization theorem that provides a sufficient condition for locally $$\mathrm{CAT}^{}(0)$$ spaces to be globally $$\mathrm{CAT}^{}(0)$$. The theorem implies in particular that the universal metric cover of a proper length, locally $$\mathrm{CAT}^{}(0)$$ space is a proper length $$\mathrm{CAT}^{}(0)$$ space. It follows that any proper length, locally $$\mathrm{CAT}^{}(0)$$ space is aspherical; that is, its universal cover is contractible.
Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

### Chapter 4. Subsets

Abstract
In this chapter we give a partial answer to the question: Which subsets of Euclidean space, equipped with their induced length metrics, are $$\mathrm{CAT}^{}(0)$$?
Stephanie Alexander, Vitali Kapovitch, Anton Petrunin

### Backmatter

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