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9781470475369 Academic Inspection Copy

Alexandrov Geometry

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Alexandrov spaces are defined via axioms similar to those of the Euclid axioms but where certain equalities are replaced with inequalities. Depending on the signs of the inequalities, we obtain Alexandrov spaces with curvature bounded above (CBA) and curvature bounded below (CBB). Even though the definitions of the two classes of spaces are similar, their properties and known applications are quite different. The goal of this book is to give a comprehensive exposition of the structure theory of Alexandrov spaces with curvature bounded above and below. It includes all the basic material as well as selected topics inspired by considering Alexandrov spaces with CBA and with CBB simultaneously. The book also includes an extensive problem list with solutions indicated for every problem.
Stephanie Alexander, University of Illinois, Urbana-Champaign, IL. Vitali Kapovitch, University of Toronto, ON, Canada. Anton Petrunin, Pennsylvania State University, College State, PA.
Preliminaries Model plane Metric spaces Maps and functions Ultralimits Space of spaces The ghost of Euclid Dimension theory Fundamentals Fundamentals of curvature bounded below Fundamentals of curvature bounded above Kirszbraun revisited Warped products Polyhedral spaces Structure and tools First order differentiation Dimension of CAT spaces Dimension of CBB spaces Gradient flow Semisolutions Bibliography Index
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