This book serves as an introductory asset for learning metric geometry by delivering an in-depth examination of key constructions and providing an analysis of universal spaces, injective spaces, the Gromov-Hausdorff convergence, and ultralimits. This book illustrates basic examples of domestic affairs of metric spaces, this includes Alexandrov geometry, geometric group theory, metric-measure spaces and optimal transport.
Researchers in metric geometry will find this book appealing and helpful, in addition to graduate students in mathematics, and advanced undergraduate students in need of an introduction to metric geometry. Any previous knowledge of classical geometry, differential geometry, topology, and real analysis will be useful in understanding the presented topics.
Author(s): Anton Petrunin
Series: SpringerBriefs in Mathematics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 103
City: Cham
Tags: Metric Geometry, Universal Spaces, Injective Spaces, Gromov-Hausdorff Convergence, Metric Trees, Hausdorff Distance, Ultralimits, Geodesics, Gromov Selection Theorem, Universal Ambient Space
Preface
Contents
1 Definitions
A Metric spaces
B Topology
C Variations
D Maximal metric and gluing
E Completeness
F G-delta sets
G Compact spaces
H Proper spaces
I Geodesics
J Metric trees
K Length
L Length spaces
2 Universal Spaces
A Embedding in a normed space
B Extension property
C Universality
D Uniqueness and homogeneity
E Remarks
3 Injective Spaces
A Definition
B Admissible and extremal functions
C Equivalent conditions
D Space of extremal functions
E Injective envelope
F Remarks
4 Space of Subsets
A Hausdorff distance
B Hausdorff convergence
C An application
D Remarks
5 Space of Spaces
A Gromov–Hausdorff metric
B Approximations and almost isometries
C Optimal realization
D Convergence
E Uniformly totally bonded families
F Gromov selection theorem
G Universal ambient space
H Remarks
6 Ultralimits
A Faces of ultrafilters
B Ultralimits of points
C An illustration
D Ultralimits of spaces
E Ultrapower
F Tangent and asymptotic spaces
G Remarks
Semisolutions
Semisolutions
Bibliography
Index
