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Bibliographic Details
Main Authors:	Reid, Miles 1948- (Author), Szendrői, Balázs (Author)
Format:	Book
Language:	English
Published:	Cambridge [u.a.] Cambridge Univ. Press 2005
Edition:	1. publ.
Subjects:	Géométrie Topologie Geometry Topology Geometrie Lehrbuch
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014278832&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Physical Description:	XVIII, 196 S. graph. Darst.
ISBN:	9780521848893 9780521613255 0521613256 052184889x

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Record in the Search Index

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adam_text	GEOMETRY AND TOPOLOGY MILES REID MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK, COVENTRY CV4 7AL, UK BALAZS SZENDROI MATHEMATICAL INSTITUTE, UNIVERSITY OF OXFORD, 24-29 ST GILES, OXFORD 0X1 3LB, UK CAMBRIDGE UNIVERSITY PRESS CONTENTS LIST OF FIGURES PAGE X PREFACE XIII EUCLIDEAN GEOMETRY 1 1.1 THE METRIC ON W 1 1.2 LINES AND COLLINEARITY IN W 3 1.3 EUCLIDEAN SPACE E 4 1.4 DIGRESSION: SHORTEST DISTANCE 4 1.5 ANGLES Y 5 1.6 MOTIONS 6 1.7 MOTIONS AND COLLINEARITY 7 1.8 A MOTION IS AFFINE LINEAR ON LINES 7 1.9 MOTIONS ARE AFFINE TRANSFORMATIONS 8 1.10 EUCLIDEAN MOTIONS AND ORTHOGONAL TRANSFORMATIONS 9 1.11 NORMAL FORM OF AN ORTHOGONAL MATRIX 10 1.11.1 THE 2X2 ROTATION AND REFLECTION MATRIXES 10 1.11.2 THE GENERAL CASE 12 1.12 EUCLIDEAN FRAMES AND MOTIONS 14 1.13 FRAMES AND MOTIONS OF E 2 14 1.14 EVERY MOTION OF E 2 IS A TRANSLATION, ROTATION, REFLECTION OR GLIDE 15 1.15 CLASSIFICATION OF MOTIONS OF E 3 17 1.16 SAMPLE THEOREMS OF EUCLIDEAN GEOMETRY 19 1.16.1 PONS ASINORUM 19 1.16.2 THE ANGLE SUM OF TRIANGLES 19 1.16.3 PARALLEL LINES AND SIMILAR TRIANGLES 20 1.16.4 FOUR CENTRES OF A TRIANGLE 21 1.16.5 THE FEUERBACH 9-POINT CIRCLE 23 EXERCISES 24 COMPOSING MAPS 26 2.1 COMPOSITION IS THE BASIC OPERATION 26 2.2 COMPOSITION OF AFFINE LINEAR MAPS X I- AX + B 27 CONTENTS 2.3 COMPOSITION OF TWO REFLECTIONS OF E 2 27 2.4 COMPOSITION OF MAPS IS ASSOCIATIVE 28 2.5 DECOMPOSING MOTIONS 28 2.6 REFLECTIONS GENERATE ALL MOTIONS 29 2.7 AN ALTERNATIVE PROOF OF THEOREM 1.14 31 2.8 PREVIEW OF TRANSFORMATION GROUPS 31 EXERCISES 32 3 SPHERICAL AND HYPERBOLIC NON-EUCLIDEAN GEOMETRY 34 3.1 BASIC DEFINITIONS OF SPHERICAL GEOMETRY 35 3.2 SPHERICAL TRIANGLES AND TRIG 37 3.3 THE SPHERICAL TRIANGLE INEQUALITY 38 3.4 SPHERICAL MOTIONS 38 3.5 PROPERTIES OF S 2 LIKE E 2 39 3.6 PROPERTIES OF S 2 UNLIKE E 2 40 3.7 PREVIEW OF HYPERBOLIC GEOMETRY 41 3.8 HYPERBOLIC SPACE 42 3.9 HYPERBOLIC DISTANCE 43 3.10 HYPERBOLIC TRIANGLES AND TRIG 44 3.11 HYPERBOLIC MOTIONS 46 3.12 INCIDENCE OF TWO LINES IN 7I 2 47 3.13 THE HYPERBOLIC PLANE IS NON-EUCLIDEAN 49 3.14 ANGULAR DEFECT 51 3.14.1 THE FIRST PROOF 51 3.14.2 AN EXPLICIT INTEGRAL 51 3.14.3 PROOF BY SUBDIVISION 53 3.14.4 AN ALTERNATIVE SKETCH PROOF 54 EXERCISES 56 4 AFFINE GEOMETRY 62 4.1 MOTIVATION FOR AFFINE SPACE 62 4.2 BASIC PROPERTIES OF AFFINE SPACE 63 4.3 THE GEOMETRY OF AFFINE LINEAR SUBSPACES 65 4.4 DIMENSION OF INTERSECTION 67 4.5 AFFINE TRANSFORMATIONS 68 4.6 AFFINE FRAMES AND AFFINE TRANSFORMATIONS 68 4.7 THECENTROID 69 EXERCISES 69 5 PROJECTIVE GEOMETRY 72 5.1 MOTIVATION FOR PROJECTIVE GEOMETRY 72 5.1.1 INHOMOGENEOUS TO HOMOGENEOUS 72 5.1.2 PERSPECTIVE 73 5.1.3 ASYMPTOTES 73 5.1.4 COMPACTIFICATION 75 CONTENTS VII 5.2 DEFINITION OF PROJECTIVE SPACE 75 5.3 PROJECTIVE LINEAR SUBSPACES 76 5.4 DIMENSION OF INTERSECTION 77 5.5 PROJECTIVE LINEAR TRANSFORMATIONS AND PROJECTIVE FRAMES OF REFERENCE 77 5.6 PROJECTIVE LINEAR MAPS OF P 1 AND THE CROSS-RATIO 79 5.7 PERSPECTIVITIES 81 5.8 AFFINE SPACE A AS A SUBSET OF PROJECTIVE SPACE P 81 5.9 DESARGUES THEOREM 82 5.10 PAPPUS THEOREM 84 5.11 PRINCIPLE OF DUALITY 85 5.12 AXIOMATIC PROJECTIVE GEOMETRY 86 EXERCISES 88 6 GEOMETRY AND GROUP THEORY 92 6.1 TRANSFORMATIONS FORM A GROUP 93 6.2 TRANSFORMATION GROUPS 94 6.3 KLEIN S ERLANGEN PROGRAM 95 6.4 CONJUGACY IN TRANSFORMATION GROUPS 96 6.5 APPLICATIONS OF CONJUGACY 98 6.5.1 NORMAL FORMS 98 6.5.2 FINDING GENERATORS 100 6.5.3 THE ALGEBRAIC STRUCTURE OF TRANSFORMATION GROUPS 101 6.6 DISCRETE REFLECTION GROUPS 103 EXERCISES 104 7 TOPOLOGY 107 7.1 DEFINITION OF A TOPOLOGICAL SPACE 108 7.2 MOTIVATION FROM METRIC SPACES 108 7.3 CONTINUOUS MAPS AND HOMEOMORPHISMS 111 7.3.1 DEFINITION OF A CONTINUOUS MAP 111 7.3.2 DEFINITION OF A HOMEOMORPHISM 111 7.3.3 HOMEOMORPHISMS AND THE ERLANGEN PROGRAM 112 7.3.4 THE HOMEOMORPHISM PROBLEM 113 7.4 TOPOLOGICAL PROPERTIES 113 7.4.1 CONNECTED SPACE 113 7.4.2 COMPACT SPACE 115 7.4.3 CONTINUOUS IMAGE OF A COMPACT SPACE IS COMPACT 116 7.4.4 AN APPLICATION OF TOPOLOGICAL PROPERTIES 117 7.5 SUBSPACE AND QUOTIENT TOPOLOGY 117 7.6 STANDARD EXAMPLES OF GLUEING 118 7.7 TOPOLOGY OF PJ 121 7.8 NONMETRIC QUOTIENT TOPOLOGIES 122 7.9 BASIS FOR A TOPOLOGY 124 CONTENTS 7.10 PRODUCT TOPOLOGY 126 7.11 THE HAUSDORFF PROPERTY 127 7.12 COMPACT VERSUS CLOSED 128 7.13 CLOSED MAPS 129 7.14 A CRITERION FOR HOMEOMORPHISM 130 7.15 LOOPS AND THE WINDING NUMBER 130 7.15.1 PATHS, LOOPS AND FAMILIES 131 7.15.2 THE WINDING NUMBER 133 7.15.3 WINDING NUMBER IS CONSTANT IN A FAMILY 135 7.15.4 APPLICATIONS OF THE WINDING NUMBER 13 6 EXERCISES 137 QUATERNIONS, ROTATIONS AND THE GEOMETRY OF TRANSFORMATION GROUPS 142 8.1 TOPOLOGY ON GROUPS , 143 8.2 DIMENSION COUNTING 144 8.3 COMPACT AND NONCOMPACT GROUPS 146 8.4 COMPONENTS 148 8.5 QUATERNIONS, ROTATIONS AND THE GEOMETRY OF SO() 149 8.5.1 QUATERNIONS 149 8.5.2 QUATERNIONS AND ROTATIONS 151 8.5.3 SPHERES AND SPECIAL ORTHOGONAL GROUPS 152 8.6 THE GROUP SU(2) 153 8.7 THE ELECTRON SPIN IN QUANTUM MECHANICS 154 8.7.1 THE STORY OF THE ELECTRON SPIN 154 8.7.2 MEASURING SPIN: THE STERN-GERLACH DEVICE 155 8.7.3 THE SPIN OPERATOR 156 8.7.4 ROTATE THE DEVICE 157 8.7.5 THE SOLUTION 158 8.8 PREVIEW OF LIE GROUPS 159 EXERCISES 161 CONCLUDING REMARKS 164 9.1 ON THE HISTORY OF GEOMETRY 165 9.1.1 GREEK GEOMETRY AND RIGOUR 165 9.1.2 THE PARALLEL POSTULATE 165 9.1.3 COORDINATES VERSUS AXIOMS 168 9.2 GROUP THEORY 169 9.2.1 ABSTRACT GROUPS VERSUS TRANSFORMATION GROUPS 169 9.2.2 HOMOGENEOUS AND PRINCIPAL HOMOGENEOUS SPACES 169 9.2.3 THE ERLANGEN PROGRAM REVISITED 170 9.2.4 AFFINE SPACE AS A TORSOR 171 CONTENTS IX 9.3 GEOMETRY IN PHYSICS 172 9.3.1 THE GALILEAN GROUP AND NEWTONIAN DYNAMICS 172 9.3.2 THE POINCARE GROUP AND SPECIAL RELATIVITY 173 9.3.3 WIGNER S CLASSIFICATION: ELEMENTARY PARTICLES 175 9.3.4 THE STANDARD MODEL AND BEYOND 176 9.3.5 OTHER CONNECTIONS 176 9.4 THE FAMOUS TRICHOTOMY 177 9.4.1 THE CURVATURE TRICHOTOMY IN GEOMETRY 177 9.4.2 ON THE SHAPE AND FATE OF THE UNIVERSE 178 9.4.3 THE SNACK BAR AT THE END OF THE UNIVERSE 179 APPENDIX A METRICS 180 EXERCISES 181 APPENDIX B LINEAR ALGEBRA 183 B.I BILINEAR FORM AND QUADRATIC FORM 183 B. 2 EUCLID AND LORENTZ 184 B.3 COMPLEMENTS AND BASES 185 B.4 SYMMETRIES 186 B.5 ORTHOGONAL AND LORENTZ MATRIXES 187 B.6 HERMITIAN FORMS AND UNITARY MATRIXES 188 EXERCISES 189 REFERENCES 190 INDEX 193
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spellingShingle	Reid, Miles 1948- Szendrői, Balázs Geometry and topology Géométrie Topologie Geometry Topology Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd
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title	Geometry and topology
title_auth	Geometry and topology
title_exact_search	Geometry and topology
title_full	Geometry and topology Miles Reid ; Balázs Szendrői
title_fullStr	Geometry and topology Miles Reid ; Balázs Szendrői
title_full_unstemmed	Geometry and topology Miles Reid ; Balázs Szendrői
title_short	Geometry and topology
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topic	Géométrie Topologie Geometry Topology Geometrie (DE-588)4020236-7 gnd Topologie (DE-588)4060425-1 gnd
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