An Invitation to Alexandrov Geometry

CAT(0) Spaces

  • Stephanie Alexander
  • Vitali Kapovitch
  • Anton Petrunin

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
    Pages 1-16
  3. Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
    Pages 17-31
  4. Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
    Pages 33-48
  5. Stephanie Alexander, Vitali Kapovitch, Anton Petrunin
    Pages 49-65
  6. Back Matter
    Pages 67-88

About this book


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.


metric spaces Gromov–Hausdorff convergence Model angles and triangles. Space of directions and tangent space Geodesics Alexandrov geometry Gluing theorem and billiards Reshetnyak’s gluing theorem Reshetnyak’s puff pastry 4-point condition Polyhedral spaces Exotic aspherical manifolds ASPHERICITY Sets with smooth boundary Cubical complexes Two-convex hull Shefel’s theorem Riemannian geometry

Authors and affiliations

  • Stephanie Alexander
    • 1
  • Vitali Kapovitch
    • 2
  • Anton Petrunin
    • 3
  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Bibliographic information

  • DOI
  • Copyright Information The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-05311-6
  • Online ISBN 978-3-030-05312-3
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site
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