Topological Circle Planes And Topological Quadrangles
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Topological Circle Planes and Topological Quadrangles
This research note presents a complete treatment of the connection between topological circle planes and topological generalized quadrangles. The author uses this connection to provide a better understanding of the relationships between different types of circle planes and to solve a topological version of the problem of Apollonius. Topological Circle Planes and Topological Quadrangles begins with a foundation in classical circle planes and the real symmetric generalized quadrangle and the connection between them. This provides a solid base from which the author offers a more generalized exploration of the topological case. He also compares this treatment to the finite case. Subsequent chapters examine Laguerre, Möbius, and Minkowski planes and their respective relationships to antiregular quadrangles. The author addresses the Lie geometry of each and discuss the relationships of circle planes-the "sisters" of Möbius, Laguerre, and Minkowski planes - and concludes by solving a topological version of the problem of Apollonius in Laguerre, Möbius, and Minkowski planes. The treatment offered in this volume offers complete coverage of the topic. The first part of the text is accessible to anyone with a background in analytic geometry, while the second part requires basic knowledge in general and algebraic topology. Researchers interested in geometry-particularly in topological geometry-will find this volume intriguing and informative. Most of the results presented are new and can be applied to various problems in the field of topological circle planes. Features
Geometries on Surfaces
Author: Burkard Polster
language: en
Publisher: Cambridge University Press
Release Date: 2001-10-03
The projective, Möbius, Laguerre, and Minkowski planes over the real numbers are just a few examples of a host of fundamental classical topological geometries on surfaces. This book summarizes all known major results and open problems related to these classical point-line geometries and their close (nonclassical) relatives. Topics covered include: classical geometries; methods for constructing nonclassical geometries; classifications and characterizations of geometries. This work is related to many other fields including interpolation theory, convexity, the theory of pseudoline arrangements, topology, the theory of Lie groups, and many more. The authors detail these connections, some of which are well-known, but many much less so. Acting both as a reference for experts and as an accessible introduction for graduate students, this book will interest anyone wishing to know more about point-line geometries and the way they interact.
Elliptic Operators, Topology, and Asymptotic Methods
Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis. Retaining its concise presentation but offering streamlined analyses and expanded coverage of important exampl