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Full Description
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.
Contents
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
