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2023 | Buch

Definitions Kapitel lesen Erstes Kapitel lesen

Pure Metric Geometry

verfasst von: Anton Petrunin

Verlag: Springer Nature Switzerland

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Definitions
Abstract
In this lecture, we remind several definitions related to metric spaces and fix some conventions.
Anton Petrunin
Chapter 2. Universal Spaces
Abstract
The Urysohn space is the main hero of this lecture. It shares some fundamental properties with classical spaces (spheres, Euclidean, and Lobachevsky spaces) but also has many counterintuitive properties.
Anton Petrunin
Chapter 3. Injective Spaces
Abstract
Injective hull is a useful construction that provides a canonical choice of a specially nice (injective) space that includes a given metric space. This construction is similar to the convex hull in Euclidean space. The following exercise gives a bridge from the latter to the former.
Anton Petrunin
Chapter 4. Space of Subsets
Abstract
In this lecture, we define and study Hausdorff metric on subsets of a given metric space.
Anton Petrunin
Chapter 5. Space of Spaces
Abstract
In this lecture, we define and study the so-called Gromov–Hausdorff metric on the isometry classes of compact metric spaces.
Anton Petrunin
Chapter 6. Ultralimits
Abstract
Ultralimits provide a very general way to pass to a limit. This procedure works for any sequence of metric spaces, its result reminds limit in the sense of Gromov–Hausdorff, but has some strange features; for example, the limit of a constant sequence of spaces \(\mathcal {X}_n=\mathcal {X}\) is not\(\mathcal {X}\) in general (see 6.13(b)).
Anton Petrunin
Backmatter
Metadaten
Titel
Pure Metric Geometry
verfasst von
Anton Petrunin
Copyright-Jahr
2023
Electronic ISBN
978-3-031-39162-0
Print ISBN
978-3-031-39161-3
DOI
https://doi.org/10.1007/978-3-031-39162-0

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