Points and lines. Characterizing the classical geometries
type: Item
instance of: scholarly article
title: Points and lines. Characterizing the classical geometries
zbMATH Open document ID: 1213.51001
DOI: 10.1007/978-3-642-15627-4
author: E. E. Shult
published in: Universitext
publication date: 2010-10-19T00:00:00Z
review text: The book under review grew from the author's lecture notes to various courses over the years and his longstanding interest in and research into Lie incidence geometries. It is intended as a teaching book and thus is self-contained. Since the text contains complete proofs of many classical theorems scattered throughout the literature, it will be valuable to experts as a reference. Moreover, the later chapters, in particular, may be of interest due to new presentations of standard theories or some new results. The author's primary aim is to introduce the classical geometries of Lie type and then to characterize them by simple axioms on points and lines. The text is written very clearly and abounds with footnotes, historical and other remarks, explanations on where the author is going, forward references, all intended to give perspective to the reader. Each of the 18 chapters begins with an abstract. Almost every section has an introduction and most chapters end with a set of exercises. The book is divided into four parts, the first of which introduces the basics. The four chapters in this part deal with graphs, concepts of geometries (defined in terms of multipartite graphs as geometries over a typeset), point-line geometries, and Teirlinck's theory (of geometric hyperplanes and embeddings), respectively. The second part introduces the classical geometries: projective planes in chapter 5, and projective spaces (including proofs of the Veblen-Young theorem and the fundamental theorem of projective geometry), polar spaces (including the determination of non-degenerate polar spaces of polar rank at least 4) and near polygons (including Cameron's characterization of near polygons with strongly gated quads as classical dual polar spaces) in the following three chapters. Part III, called methodology, is where a unifying view of geometries of Lie type is presented and the tools for the characterizations in the final part are developed. Chapter 9 introduces chamber systems an…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51-01
Mathematics Subject Classification ID: 51-02
Mathematics Subject Classification ID: 51Axx
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 51N30
zbMATH DE Number: 5801912
zbMATH Keywords: point-line geometry
zbMATH Keywords: classical geometry
zbMATH Keywords: Lie incidence geometry
zbMATH Keywords: building
zbMATH Keywords: projective plane
zbMATH Keywords: projective space
zbMATH Keywords: Grassmann space: parapolar space
Wikidata QID: Q56020932
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-15627-4
OpenAlex ID: W2477359250
description: scientific article; zbMATH DE number 5801912