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Chapter 2 Euclidean and Hyperbolic Geometry Classical Geometries in Modern Walter Benz Contexts Geometry of Real Inner Product Spaces Birkhäuser Verlag Basel • Boston • Berlin Authors: Walter Benz Fachbereich Mathematik Universität Hamburg Bundesstr. 55 20146 Hamburg Germany e-mail: [email protected] 2000 Mathematical Subject Classification 39B52, 51B10, 51B25, 51M05, 51M10, 83C20 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 3-7643-7371-7 Birkhäuser Verlag, Basel – Boston – Berlin 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, broadcast- ing, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2005 Birkhäuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover design: Micha Lotrovsky, CH-4106 Therwil, Switzerland Printed on acid-free paper produced from chlorine-free pulp. TCF °° Printed in Germany ISBN-10: 3-7643-7371-7 e-ISBN: 3-7643-7432-2 ISBN-13: 978-3-7643-7371-9 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Preface ix 1 Translation Groups 1 1.1Realinnerproductspaces....................... 1 1.2Examples................................ 2 1.3 Isomorphic, non-isomorphic spaces . 3 1.4InequalityofCauchy–Schwarz..................... 4 1.5 Orthogonal mappings . 5 1.6 A characterization of orthogonal mappings . 7 1.7Translationgroups,axis,kernel.................... 10 1.8Separabletranslationgroups...................... 14 1.9Geometryofagroupofpermutations................. 16 1.10Euclidean,hyperbolicgeometry.................... 20 1.11Acommoncharacterization...................... 21 1.12Otherdirections,acounterexample.................. 34 2 Euclidean and Hyperbolic Geometry 37 2.1Metricspaces.............................. 37 2.2ThelinesofL.M.Blumenthal..................... 38 2.3ThelinesofKarlMenger....................... 43 2.4Anotherdefinitionoflines....................... 45 2.5Balls,hyperplanes,subspaces..................... 46 2.6Aspecialquasi-hyperplane....................... 50 2.7 Orthogonality, equidistant surfaces . 51 2.8Aparametricrepresentation...................... 54 2.9 Ends, parallelity, measures of angles . 56 2.10Anglesofparallelism,horocycles................... 61 2.11Geometricalsubspaces......................... 63 2.12TheCayley–Kleinmodel........................ 66 2.13 Hyperplanes under translations . 70 2.14 Lines under translations . 72 2.15Hyperboliccoordinates......................... 74 vi Contents 2.16 All isometries of (X, eucl), (X, hyp)................. 75 2.17 Isometries preserving a direction . 77 2.18Acharacterizationoftranslations................... 78 2.19Differentrepresentationsofisometries................ 79 2.20Acharacterizationofisometries.................... 80 2.21Acounterexample........................... 85 2.22Anextensionproblem......................... 86 2.23Amappingwhichcannotbeextended................ 91 3 Sphere Geometries of M¨obius and Lie 93 3.1 M¨obiusballs,inversions........................ 93 3.2Anapplicationtointegralequations................. 96 3.3 A fundamental theorem . 98 3.4Involutions............................... 102 3.5 Orthogonality . 107 N 3.6 M¨obius circles, MN -andM -spheres................ 111 3.7Stereographicprojection........................ 120 3.8 Poincar´e’smodelofhyperbolicgeometry............... 123 3.9Spears,Laguerrecycles,contact.................... 133 3.10Separation,cyclographicprojection.................. 139 3.11 Pencils and bundles . 144 3.12 Lie cycles, groups Lie (X), Lag (X) ................. 150 3.13Liecyclecoordinates,Liequadric................... 154 3.14Lorentzboosts............................. 159 3.15 M (X)aspartofLie(X) ....................... 167 3.16 A characterization of Lag (X)..................... 170 3.17 Characterization of the Lorentz group . 172 3.18 Another fundamental theorem . 173 4 Lorentz Transformations 175 4.1Twocharacterizationtheorems.................... 175 4.2Causalautomorphisms......................... 177 4.3Relativisticaddition.......................... 181 4.4Lightlike,timelike,spacelikelines................... 184 4.5Lightcones,lightlikehyperplanes................... 186 4.6Characterizationofsomehyperplanes................ 191 4.7 L (Z) as subgroup of Lie (X) ..................... 193 4.8 A characterization of LM-distances . 194 4.9Einstein’scylindricalworld...................... 197 4.10Lines,null-lines,subspaces....................... 200 4.11 2-point invariants of C (Z),MC(Z) ................ 202 4.12DeSitter’sworld............................ 205 4.13 2-point invariants of Σ(Z),MΣ(Z) ................ 205 4.14 Elliptic and spherical distances . 208 Contents vii 4.15Points.................................. 210 4.16Isometries................................ 212 4.17 Distance functions of X0 ........................ 215 4.18 Subspaces, balls . 217 4.19Periodiclines.............................. 218 4.20Hyperbolicgeometryrevisited..................... 222 A Notations and symbols 231 B Bibliography 233 C Index 239 Preface The basic structure playing the key role in this book is a real inner product space (X, δ), i.e. a real vector space X together with a mapping δ : X × X → R,a so-called inner product, satisfying rules (i), (ii), (iii), (iv) of section 1 of chapter 1. In order to avoid uninteresting cases from the point of view of geometry, we will assume throughout the whole book that there exist elements a, b in X which are linearly independent. But, on the other hand, we do not ask for the exis- tence of a positive integer n such that every subset S of X containing exactly n elements is linearly dependent. In other words, we do not assume that X is a finite- dimensional vector space. So, when dealing in this book with different geometries like euclidean, hyperbolic, elliptic, spherical, Lorentz–Minkowskian geometry or M¨obius (Lie) sphere geometry over a real inner product space (X, δ), the reader might think of X = R2 or R3,ofX finite-dimensional, or of X infinite-dimensional. In fact, it plays no role, whatsoever, in our considerations whether the dimension of X is finite or infinite: the theory as presented does not depend on the dimension of X. In this sense we may say that our presentation in question is dimension-free. The prerequisites for a fruitful reading of this book are essentially based on the sophomore level, especially after mastering basic linear algebra and basic geometry of R2 and R3. Of course, hyperspheres are defined via the inner prod- uct δ. At the same time we also define hyperplanes by this product, namely by {x ∈ X | δ (a, x)=α}, or, as we prefer to write {x ∈ X | ax = α},with0= a ∈ X and α ∈ R. This is a quite natural and simple definition and familiar to everybody who learned geometry, say, of the plane or of R3. For us it means that we do not need to speak about the existence of a basis of X (see, however, section 2.6 where we describe an example of a quasi-hyperplane which is not a hyperplane) and, furthermore, that we do not need to speak about (affine) hyperplanes as images under translations of maximal subspaces = X of X (see R. Baer [1], p. 19), hence avoiding transfinite methods, which could be considered as somewhat strange in the context of geometries of Klein’s Erlangen programme. This programme was published in 1872 by Felix Klein (1849–1925) under the title Vergleichende Be- trachtungenuber ¨ neuere geometrische Forschungen, Programm zum Eintritt in die philosophische Facult¨at und den Senat der k. Friedrich-Alexander-Universit¨at zu Erlangen (Verlag von Andreas Deichert, Erlangen), and it gave rise to an ingenious x Preface and fundamental principle that allows distinguishing between different geometries (S, G) (see section 9 of chapter 1) on the basis of their groups G, their invariants and invariant notions (section 9). In connection with Klein’s Erlangen programme compare also Julian Lowell Coolidge, A History of Geometrical Methods, Claren- don Press, Oxford, 1940, and, for instance, W. Benz [3], p. 38 f. The papers [1] and [5] of E.M. Schr¨oder must be considered as pioneer work for a dimension-free presentation of geometry. In [1], for instance, E.M. Schr¨oder proved for arbitrary-dimensional X, dim X ≥ 2, that a mapping f : X → X satisfying f (0) = 0 and x1 − x2 = f (x1) − f (x2) for all x1,x2 ∈ X with x1 − x2 = 1 or 2 must be orthogonal. The methods of this result turned out to be important for certain other results of dimension-free geometry (see Theorem 4 of chapter 1 of the present book, see also W. Benz, H. Berens [1] or F. Rad´o, D. Andreescu, D. V´alcan [1]). The main result of chapter 1 is a common characterization of euclidean and hyperbolic geometry over (X, δ). With an implicit notion of a (separable) transla- tion group T of X with axis e ∈ X (see sections 7, 8 of chapter 1) the following theorem is proved (Theorem 7). Let d be a function, not identically zero, from X× X into the set R≥0 of all non-negative real numbers satisfying d (x, y)= d ϕ (x),ϕ(y) and, moreover, d (βe,0) = d (0,βe)=d (0,αe)+d (αe, βe) for all x, y ∈ X,allϕ ∈ T ∪ O (X)whereO (X) is the group of orthogonal bijections of X,andforallrealα, β with
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