GeometryandTopology Geometryprovidesawholerangeofviewsontheuniverse,servingastheinspiration,technical toolkitandultimategoalformanybranchesofmathematicsandphysics.Thisbookintroduces theideasofgeometry,andincludesageneroussupplyofsimpleexplanationsandexamples. Thetreatmentemphasisescoordinatesystemsandthecoordinatechangesthatgeneratesymme- tries.ThediscussionmovesfromEuclideantonon-Euclideangeometries,includingspherical andhyperbolicgeometry,andthenontoaffineandprojectivelineargeometries.Grouptheory isintroducedtotreatgeometricsymmetries,leadingtotheunificationofgeometryandgroup theoryintheErlangenprogram.Anintroductiontobasictopologyfollows,withtheMo¨bius strip, the Klein bottle and the surface with g handles exemplifying quotient topologies and thehomeomorphismproblem.Topologycombineswithgrouptheorytoyieldthegeometry oftransformationgroups,havingapplicationstorelativitytheoryandquantummechanics.A finalchapterfeatureshistoricaldiscussionsandindicationsforfurtherreading.Whilethebook requiresminimalprerequisites,itprovidesafirstglimpseofmanyresearchtopicsinmodern algebra,geometryandtheoreticalphysics. The book is based on many years’ teaching experience, and is thoroughly class tested. Therearecopiousillustrations,andeachchapterendswithawidesupplyofexercises.Further teachingmaterialisavailableforteachersviatheweb,includingassignableproblemsheets withsolutions. miles reidisaProfessorofMathematicsattheMathematicsInstitute,UniversityofWarwick bala´zs szendro´´iisaFacultyLecturerintheMathematicalInstitute,UniversityofOxford, andMartinPowellFellowinPureMathematicsatStPeter’sCollege,Oxford Geometry and Topology MilesReid MathematicsInstitute,UniversityofWarwick,CoventryCV47AL,UK Bala´zsSzendro´´i MathematicalInstitute,UniversityofOxford, 24–29StGiles,OxfordOX13LB,UK cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru,UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521848893 © Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 isbn-13 978-0-511-13733-4 eBook (NetLibrary) isbn-10 0-511-13733-8 eBook (NetLibrary) isbn-13 978-0-521-84889-3 hardback isbn-10 0-521-84889-x hardback isbn-13 978-0-521-61325-5 paperback isbn-10 0-521-61325-6 paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guaranteethatanycontentonsuchwebsitesis,orwillremain,accurateorappropriate. Contents Listoffigures pagex Preface xiii 1 Euclideangeometry 1 1.1 ThemetriconRn 1 1.2 LinesandcollinearityinRn 3 1.3 EuclideanspaceEn 4 1.4 Digression:shortestdistance 4 1.5 Angles 5 1.6 Motions 6 1.7 Motionsandcollinearity 7 1.8 Amotionisaffinelinearonlines 7 1.9 Motionsareaffinetransformations 8 1.10 Euclideanmotionsandorthogonaltransformations 9 1.11 Normalformofanorthogonalmatrix 10 1.11.1 The2×2rotationandreflectionmatrixes 10 1.11.2 Thegeneralcase 12 1.12 Euclideanframesandmotions 14 1.13 FramesandmotionsofE2 14 1.14 EverymotionofE2isatranslation,rotation,reflectionorglide 15 1.15 ClassificationofmotionsofE3 17 1.16 SampletheoremsofEuclideangeometry 19 1.16.1 Ponsasinorum 19 1.16.2 Theanglesumoftriangles 19 1.16.3 Parallellinesandsimilartriangles 20 1.16.4 Fourcentresofatriangle 21 1.16.5 TheFeuerbach9-pointcircle 23 Exercises 24 2 Composingmaps 26 2.1 Compositionisthebasicoperation 26 2.2 Compositionofaffinelinearmapsx(cid:2)→Ax+b 27 v vi CONTENTS 2.3 CompositionoftworeflectionsofE2 27 2.4 Compositionofmapsisassociative 28 2.5 Decomposingmotions 28 2.6 Reflectionsgenerateallmotions 29 2.7 AnalternativeproofofTheorem1.14 31 2.8 Previewoftransformationgroups 31 Exercises 32 3 Sphericalandhyperbolicnon-Euclideangeometry 34 3.1 Basicdefinitionsofsphericalgeometry 35 3.2 Sphericaltrianglesandtrig 37 3.3 Thesphericaltriangleinequality 38 3.4 Sphericalmotions 38 3.5 PropertiesofS2likeE2 39 3.6 PropertiesofS2unlikeE2 40 3.7 Previewofhyperbolicgeometry 41 3.8 Hyperbolicspace 42 3.9 Hyperbolicdistance 43 3.10 Hyperbolictrianglesandtrig 44 3.11 Hyperbolicmotions 46 3.12 IncidenceoftwolinesinH2 47 3.13 Thehyperbolicplaneisnon-Euclidean 49 3.14 Angulardefect 51 3.14.1 Thefirstproof 51 3.14.2 Anexplicitintegral 51 3.14.3 Proofbysubdivision 53 3.14.4 Analternativesketchproof 54 Exercises 56 4 Affinegeometry 62 4.1 Motivationforaffinespace 62 4.2 Basicpropertiesofaffinespace 63 4.3 Thegeometryofaffinelinearsubspaces 65 4.4 Dimensionofintersection 67 4.5 Affinetransformations 68 4.6 Affineframesandaffinetransformations 68 4.7 Thecentroid 69 Exercises 69 5 Projectivegeometry 72 5.1 Motivationforprojectivegeometry 72 5.1.1 Inhomogeneoustohomogeneous 72 5.1.2 Perspective 73 5.1.3 Asymptotes 73 5.1.4 Compactification 75 CONTENTS vii 5.2 Definitionofprojectivespace 75 5.3 Projectivelinearsubspaces 76 5.4 Dimensionofintersection 77 5.5 Projectivelineartransformationsandprojectiveframesofreference 77 5.6 ProjectivelinearmapsofP1andthecross-ratio 79 5.7 Perspectivities 81 5.8 AffinespaceAn asasubsetofprojectivespacePn 81 5.9 Desargues’theorem 82 5.10 Pappus’theorem 84 5.11 Principleofduality 85 5.12 Axiomaticprojectivegeometry 86 Exercises 88 6 Geometryandgrouptheory 92 6.1 Transformationsformagroup 93 6.2 Transformationgroups 94 6.3 Klein’sErlangenprogram 95 6.4 Conjugacyintransformationgroups 96 6.5 Applicationsofconjugacy 98 6.5.1 Normalforms 98 6.5.2 Findinggenerators 100 6.5.3 Thealgebraicstructureoftransformationgroups 101 6.6 Discretereflectiongroups 103 Exercises 104 7 Topology 107 7.1 Definitionofatopologicalspace 108 7.2 Motivationfrommetricspaces 108 7.3 Continuousmapsandhomeomorphisms 111 7.3.1 Definitionofacontinuousmap 111 7.3.2 Definitionofahomeomorphism 111 7.3.3 HomeomorphismsandtheErlangenprogram 112 7.3.4 Thehomeomorphismproblem 113 7.4 Topologicalproperties 113 7.4.1 Connectedspace 113 7.4.2 Compactspace 115 7.4.3 Continuousimageofacompactspaceiscompact 116 7.4.4 Anapplicationoftopologicalproperties 117 7.5 Subspaceandquotienttopology 117 7.6 Standardexamplesofglueing 118 7.7 TopologyofPn 121 R 7.8 Nonmetricquotienttopologies 122 7.9 Basisforatopology 124 viii CONTENTS 7.10 Producttopology 126 7.11 TheHausdorffproperty 127 7.12 Compactversusclosed 128 7.13 Closedmaps 129 7.14 Acriterionforhomeomorphism 130 7.15 Loopsandthewindingnumber 130 7.15.1 Paths,loopsandfamilies 131 7.15.2 Thewindingnumber 133 7.15.3 Windingnumberisconstantinafamily 135 7.15.4 Applicationsofthewindingnumber 136 Exercises 137 8 Quaternions,rotationsandthegeometryof transformationgroups 142 8.1 Topologyongroups 143 8.2 Dimensioncounting 144 8.3 Compactandnoncompactgroups 146 8.4 Components 148 8.5 Quaternions,rotationsandthegeometryofSO(n) 149 8.5.1 Quaternions 149 8.5.2 Quaternionsandrotations 151 8.5.3 Spheresandspecialorthogonalgroups 152 8.6 ThegroupSU(2) 153 8.7 Theelectronspininquantummechanics 154 8.7.1 Thestoryoftheelectronspin 154 8.7.2 Measuringspin:theStern–Gerlachdevice 155 8.7.3 Thespinoperator 156 8.7.4 Rotatethedevice 157 8.7.5 Thesolution 158 8.8 PreviewofLiegroups 159 Exercises 161 9 Concludingremarks 164 9.1 Onthehistoryofgeometry 165 9.1.1 Greekgeometryandrigour 165 9.1.2 Theparallelpostulate 165 9.1.3 Coordinatesversusaxioms 168 9.2 Grouptheory 169 9.2.1 Abstractgroupsversustransformationgroups 169 9.2.2 Homogeneousandprincipalhomogeneousspaces 169 9.2.3 TheErlangenprogramrevisited 170 9.2.4 Affinespaceasatorsor 171 CONTENTS ix 9.3 Geometryinphysics 172 9.3.1 TheGalileangroupandNewtoniandynamics 172 9.3.2 ThePoincare´ groupandspecialrelativity 173 9.3.3 Wigner’sclassification:elementaryparticles 175 9.3.4 TheStandardModelandbeyond 176 9.3.5 Otherconnections 176 9.4 Thefamoustrichotomy 177 9.4.1 Thecurvaturetrichotomyingeometry 177 9.4.2 Ontheshapeandfateoftheuniverse 178 9.4.3 Thesnackbarattheendoftheuniverse 179 AppendixA Metrics 180 Exercises 181 AppendixB Linearalgebra 183 B.1Bilinearformandquadraticform 183 B.2EuclidandLorentz 184 B.3Complementsandbases 185 B.4Symmetries 186 B.5OrthogonalandLorentzmatrixes 187 B.6Hermitianformsandunitarymatrixes 188 Exercises 189 References 190 Index 193