Translation Planes Foundations and Construction Principles
The book discusses various construction principles for translation planes and spreads from a general and unifying point of view and relates them to the theory of kinematic spaces. The book is intended for people working in the field of incidence geometry
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Norbert Knarr
Translation Planes Foundations and Construction Principles
Springer
Author Norbert Knarr Institut fur Analysis Technische Universitat Braunschweig Pockelsstr. 14 D-38 106 Braunschweig, Germany
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahrne Knarr, Norbert: Translation planes: foundations and construction principles / Norbert Knarr. - Berlin; Heidelberg; New York: Springer, 1995 (Lecture notes in mathernat ics ; 1611)
ISBN 3-540-60208-9 NE:GT
Mathematics Subject Classification (1991): 51A40, 5IR10, 51115
ISBN 3-540-60208-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10479552 46/3142-543210 - Printed on acid-free paper
Table of Contents
Introduction
1
1. Foundations
5
1. 2. 3. 4.
Translation Planes and Spreads Quasifields and Spread Sets Isomorphisms and Collineations of Translation Planes Some Topological Tools
5. Locally Compact Connected Translation Planes 6. The Transposition of Translation Planes
2. Spreads of 3-dimensional Projective Spaces 1. The Klein Quadric 2. Transversal Mappings 3. 4. 5. 6. 7.
Betten's Variant Indicator Sets Ll-indicator Sets The Kinematic Mapping Al-indicator Sets
3. Kinematic Spaces 1. 2. 3. 4. 5.
Stretch Translation Spaces Euclidean Motion Groups Minkowskian Motion Groups Elliptic Motion Groups Other Kinematic Spaces
5 " 10 15 20 21 23 25 25 26 " 29 31 33 36 37 40
41 42 43 45 46
VI
4. Examples and Supplements
48
1. Andre Planes
48
2. Remarks on the Collineation Group
49
3. Indicator Sets and Quasifields
51
4. Reguli and Flocks of Reguli
55
5. Locally Compact 4-dimensional Translation Planes
61
1. Topological Spreads of a 4-dimensional Real Vector Space
61
2. Flocks of Reguli in Topological Spreads
64
3. Locally Compact 4-dimensional Translation Planes with Large Collineation Group
65
6. Planes of Lenz Type V with Complex Kernel
78
1. Locally Compact Translation Planes with Complex Kernel
78
2. Locally Compact Planes of Lenz Type V with Complex Kernel
79
3. Proof of Theorem 6.6
84
7. Locally Compact Translation Planes of Higher Dimension .. 95 1. Contractions of Normed Algebras and Topological Spreads
95
2. Locally Compact Translation Planes with Large Collinea
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