Points and Lines

Characterizing the Classical Geometries

  • Ernest Shult

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xxii
  2. Basics

    1. Front Matter
      Pages 1-1
    2. Ernest E. Shult
      Pages 3-41
    3. Ernest E. Shult
      Pages 43-58
    4. Ernest E. Shult
      Pages 59-77
  3. The Classical Geometries

    1. Front Matter
      Pages 103-103
    2. Ernest E. Shult
      Pages 105-133
    3. Ernest E. Shult
      Pages 135-165
    4. Ernest E. Shult
      Pages 167-249
    5. Ernest E. Shult
      Pages 251-287
  4. Methodology

    1. Front Matter
      Pages 289-289
    2. Ernest E. Shult
      Pages 291-397
    3. Ernest E. Shult
      Pages 399-413
    4. Ernest E. Shult
      Pages 415-440
    5. Ernest E. Shult
      Pages 441-453
  5. Applications to Other Lie Incidence Geometries

About this book

Introduction

The classical geometries of points and lines include not only the projective and polar spaces, but similar truncations of geometries naturally arising from the groups of Lie type. Virtually all of these geometries (or homomorphic images of them) are characterized in this book by simple local axioms on points and lines. Simple point-line characterizations of Lie incidence geometries allow one to recognize Lie incidence geometries and their automorphism groups. These tools could be useful in shortening the enormously lengthy classification of finite simple groups. Similarly, recognizing ruled manifolds by axioms on light trajectories offers a way for a physicist to recognize the action of a Lie group in a context where it is not clear what Hamiltonians or Casimir operators are involved. The presentation is self-contained in the sense that proofs proceed step-by-step from elementary first principals without further appeal to outside results. Several chapters have new heretofore unpublished research results. On the other hand, certain groups of chapters would make good graduate courses. All but one chapter provide exercises for either use in such a course, or to elicit new research directions.

Keywords

51A45, 51A50, 51A05 buildings parapolar space point-line geometry polar space

Authors and affiliations

  • Ernest Shult
    • 1
  1. 1.Kansas State UniversityManhattanUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-15627-4
  • Copyright Information Springer-Verlag Berlin Heidelberg 2011
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-642-15626-7
  • Online ISBN 978-3-642-15627-4
  • About this book