Geometries on surfaces:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2001
|
| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
84 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 490 S. graph. Darst. |
| ISBN: | 0521660580 |
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| 245 | 1 | 0 | |a Geometries on surfaces |c Burkard Polster and Günter Steinke |
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Record in the Search Index
| _version_ | 1804128812705775616 |
|---|---|
| adam_text | ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS GEOMETRIES ON SURFACES
BURKARD POLSTER MONASH UNIVERSITY AND GUNTER STEINKE UNIVERSITY OF
CANTERBURY CAMBRIDGE UNIVERSITY PRESS CONTENTS PREFACE PAGE XVII 1
GEOMETRIES FOR PEDESTRIANS 1 1.1 GEOMETRIES OF POINTS AND LINES 1 1.1.1
PROJECTIVE PLANES 2 1.1.2 AFFINE PLANES 5 1.1.3 BENZ PLANES*THE ORIGINAL
CIRCLE PLANES 6 1.1.4 ORTHOGONAL ARRAYS 8 1.2 GEOMETRIES ON SURFACES 9
1.2.1 (IDEAL) FLAT LINEAR SPACES 10 1.2.2 FLAT CIRCLE PLANES 13 1.2.3 A
NETWORK OF RELATIONSHIPS 15 1.2.4 INTERPOLATION AND THE AXIOM OF JOINING
16 1.2.5 CONVEXITY 17 1.2.6 TOPOLOGICAL GEOMETRIES ON SURFACES - . 18
1.2.7 CLASSIFICATION WITH RESPECT TO THE GROUP DIMENSION 19 1.3
DEFINITIONS OF FREQUENTLY USED TERMS 20 2 FLAT LINEAR SPACES 23 2.1
MODELS OF THE CLASSICAL FLAT PROJECTIVE PLANE 24 2.1.1 THE EUCLIDEAN
PLANE PLUS ITS LINE AT INFINITY 25 2.1.2 THE GEOMETRY OF GREAT CIRCLES
AND THE DISK MODEL 25 2.1.3 THE CLASSICAL POINT MOBIUS STRIP PLANE 27
2.2 CONVEXITY THEORY 28 2.2.1 CONVEX CURVES, ARCS, AND OVALS 31 2.2.2
DEPENDENCE OF THE AXIOMS OF JOINING AND INTERSECTION 34 2.3 CONTINUITY
OF GEOMETRIC OPERATIONS AND THE LINE SPACE 36 2.4 ISOMORPHISMS,
AUTOMORPHISM GROUPS, AND POLARITIES 44 2.5 TOPOLOGICAL PLANES AND FLAT
LINEAR SPACES 53 2.6 CLASSIFICATION WITH RESPECT TO THE GROUP DIMENSION
56 IX X CONTENTS 2.6.1 FLAT PROJECTIVE PLANES 57 2.6.2 R 2 -PLANES 59
2.6.3 MOBIUS STRIP PLANES 61 2.7 CONSTRUCTIONS 63 2.7.1 ORIGINAL MOULTON
PLANES 63 2.7.2 SEMI-CLASSICAL PLANES AND GENERALIZED MOULTON PLANES 65
2.7.3 RADIAL PLANES AND RADIAL MOULTON PLANES 66 2.7.4 SHIFT PLANES AND
PLANAR FUNCTIONS 67 2.7.5 ARC PLANES 70 2.7.6 SKEW HYPERBOLIC PLANES 76
2.7.7 CARTESIAN PLANES 77 2.7.8 STRAMBACH S SL 2 (R)-PLANE 78 2.7.9
INTEGRATED FOLIATIONS 79 2.7.10 DIFFERENT WAYS TO CUT AND PASTE 80
2.7.11 PASTED PLANES 88 2.7.12 SEMIOVAL PLANES 98 2.7.13 THE MODIFIED
REAL DUAL CYLINDER PLANE 103 2.8 PLANES WITH SPECIAL PROPERTIES 103
2.8.1 COMPACT GROUPS OF AUTOMORPHISMS 103 2.8.2 MORE RIGID PLANES 106
2.8.3 DIFFERENTIABLE PLANES 106 2.8.4 MAXIMAL FLAT STABLE PLANES AND THE
FIRST NONCLASSICAL FLAT LINEAR SPACE 107 2.9 OTHER INVARIANTS AND
CHARACTERIZATIONS 109 2.9.1 THE LENZ-BARLOTTI TYPES 110 2.9.2 GROUPS OF
PROJECTIVITIES *-. ILL 2.9.3 SEMIGROUPS OF CONTINUOUS LINEATIONS 115
2.10 RELATED GEOMETRIES 116 2.10.1 SHARPLY TRANSITIVE SETS 116 2.10.2
QUASI-SHARPLY-2-TRANSITIVE SETS AND ABSTRACT OVALS 120 2.10.3
SEMIBIPLANES 125 2.10.4 PSEUDOLINE ARRANGEMENTS, UNIVERSAL PLANES,
SPREADS 129 2.11 OPEN PROBLEMS 134 3 SPHERICAL CIRCLE PLANES 137 3.1
MODELS OF THE CLASSICAL FLAT MOBIUS PLANE 137 3.1.1 THE GEOMETRY OF
PLANE SECTIONS 137 3.1.2 THE GEOMETRY OF EUCLIDEAN LINES AND CIRCLES 138
3.1.3 PENTACYCLIC COORDINATES 140 3.1.4 THE GEOMETRY OF CHAINS 141 3.1.5
THE GEOMETRY OF THE GROUP OF FRACTIONAL LINEAR MAPS 143 CONTENTS XI 3.2
DERIVED PLANES AND TOPOLOGICAL PROPERTIES 144 3.2.1 DERIVED R 2 -PLANES
145 3.2.2 AFFINE PARTS 146 3.2.3 CONTINUITY OF GEOMETRIC OPERATIONS 146
3.2.4 TOPOLOGICAL MOBIUS PLANES 149 3.2.5 CIRCLE SPACE AND FLAG SPACE
151 3.3 CONSTRUCTIONS 154 3.3.1 OVOIDAL PLANES 154 3.3.2 EWALD S PLANES
157 3.3.3 SEMI-CLASSICAL FLAT MOBIUS PLANES 159 3.3.4 DIFFERENT WAYS TO
CUT AND PASTE 163 3.3.5 INTEGRALS OF R 2 -PLANES 168 3.4 GROUPS OF
AUTOMORPHISMS AND GROUPS OF PROJECTIVITIES 169 3.4.1 AUTOMORPHISMS AND
AUTOMORPHISM GROUPS 169 3.4.2 COMPACT GROUPS OF AUTOMORPHISMS 175 3.4.3
CLASSIFICATION WITH RESPECT TO GROUP DIMENSION 178 3.4.4 VON STAUDT S
POINT OF VIEW*GROUPS OF PROJECTIVITIES 183 3.5 THE HERING TYPES 185
3.5.1 Q- TRANSLATIONS 186 3.5.2 THE CLASSIFICATION 188 3.5.3 EXAMPLES
190 3.6 CHARACTERIZATIONS OF THE CLASSICAL PLANE 195 3.6.1 THE LOCALLY
CLASSICAL PLANE 196 3.6.2 THE MIQUELIAN PLANE 196 3.6.3 THE SYMMETRIC
PLANE 198 3.6.4 THE PLANE WITH TRANSITIVE GROUP 198 3.6.5 THE PLANE OF
HERING TYPE AT LEAST V 199 3.6.6 SUMMARY 200 3.7 PLANES WITH SPECIAL
PROPERTIES , 200 3.7.1 RIGID PLANES 201 3.7.2 DIFFERENTIABLE PLANES 201
3.8 SUBGEOMETRIES AND LIE GEOMETRIES 202 3.8.1 RECYCLED FLAT PROJECTIVE
PLANES 202 3.8.2 DOUBLE COVERS OF R 2 -PLANES AND FLAT PROJECTIVE PLANES
204 3.8.3 3-OVALS 206 3.8.4 FLOCKS AND RESOLUTIONS 206 3.9 OPEN PROBLEMS
208 4 TOROIDAL CIRCLE PLANES 212 4.1 MODELS OF THE CLASSICAL FLAT
MINKOWSKI PLANE 213 4.1.1 THE GEOMETRY OF PLANE SECTIONS 213 XII
CONTENTS 4.1.2 THE GEOMETRY OF EUCLIDEAN LINES AND HYPERBOLAS 214 4.1.3
THE PSEUDO-EUCLIDEAN GEOMETRY 217 4.1.4 PENTACYCLIC COORDINATES 219
4.1.5 THE GEOMETRY OF THE GROUP OF FRACTIONAL LINEAR MAPS 220 4.1.6 THE
GEOMETRY OF CHAINS 222 4.1.7 THE BECK MODEL 226 4.2 DERIVED PLANES AND
TOPOLOGICAL PROPERTIES 228 4.2.1 DERIVED R 2 -PLANES 229 4.2.2 AFFINE
PARTS 230 4.2.3 STANDARD REPRESENTATION AND SHARPLY 3-TRANSITIVE SETS
231 4.2.4 CONTINUITY OF THE GEOMETRIC OPERATIONS 232 4.2.5 TOPOLOGICAL
MINKOWSKI PLANES 234 4.2.6 CIRCLE SPACE AND FLAG SPACE 237 4.3
CONSTRUCTIONS 237 4.3.1 THE TWO HALVES OF A TOROIDAL CIRCLE PLANE 238
4.3.2 DIFFERENT WAYS TO CUT AND PASTE 241 4.3.3 INTEGRALS OF R 2 -PLANES
244 4.3.4 THE GENERALIZED HARTMANN PLANES 244 4.3.5 THE ARTZY-GROH
PLANES 247 4.3.6 MODIFIED CLASSICAL PLANES 251 4.3.7 PROPER TOROIDAL
CIRCLE PLANES 254 4.4 AUTOMORPHISM GROUPS AND GROUPS OF PROJECTIVITIES
255 4.4.1 AUTOMORPHISMS 256 4.4.2 GROUPS OF AUTOMORPHISMS 257 4.4.3 THE
KERNELS 258 4.4.4 PLANES ADMITTING 3-DIMENSIONAL KERNELS * 261 4.4.5 .
CLASSIFICATION WITH RESPECT TO THE GROUP DIMENSION 262 4.4.6 VON
STAUDT S POINT OF VIEW*GROUPS OF PROJECTIVITIES 266 4.5 THE KLEIN-KROLL
TYPES 268 4.5.1 G-TRANSLATIONS 269 4.5.2 ^-TRANSLATIONS 271 4.5.3 (P,
G)-HOMOTHETIES 272 4.5.4 SOME EXAMPLES 274 4.6 CHARACTERIZATIONS OF THE
CLASSICAL PLANE 275 4.6.1 THE LOCALLY CLASSICAL PLANE 275 4.6.2 THE
MIQUELIAN PLANE 275 4.6.3 THE PLANE WITH MANY DESARGUESIAN DERIVATIONS
276 4.6.4 THE PLANE IN WHICH THE RECTANGLE CONFIGURATION CLOSES 276
4.6.5 THE SYMMETRIC PLANE 277 4.6.6 THE PLANE WITH FLAG-TRANSITIVE GROUP
279 CONTENTS XIII 4.6.7 THE PLANE OF KLEIN-KROLL TYPE AT LEAST V, E, OR
21 279 4.6.8 SUMMARY 280 4.7 PLANES WITH SPECIAL PROPERTIES 281 4.7.1
RIGID PLANES 281 4.7.2 DIFFERENTIABLE PLANES 282 4.8 SUBGEOMETRIES AND
LIE GEOMETRIES 283 4.8.1 FLOCKS AND RESOLUTIONS 283 4.8.2 DOUBLE COVERS
OF DISK MOBIUS STRIP PLANES 286 4.9 OPEN PROBLEMS 287 5 CYLINDRICAL
CIRCLE PLANES 289 5.1 MODELS OF THE CLASSICAL FLAT LAGUERRE PLANE 290
5.1.1 THE GEOMETRY OF PLANE SECTIONS 290 5.1.2 THE GEOMETRY OF EUCLIDEAN
LINES AND PARABOLAS 290 5.1.3 THE GEOMETRY OF TRIGONOMETRIC POLYNOMIALS
292 5.1.4 THE GEOMETRY OF ORIENTED LINES AND CIRCLES 292 5.1.5
PENTACYCLIC COORDINATES 295 5.1.6 THE GEOMETRY OF CHAINS 296 5.2 DERIVED
PLANES AND TOPOLOGICAL PROPERTIES 299 5.2.1 DERIVED R 2 -PLANES 299
5.2.2 AFFINE PARTS 300 5.2.3 CONTINUITY OF THE GEOMETRIC OPERATIONS 301
5.2.4 TOPOLOGICAL LAGUERRE PLANES 302 5.2.5 CIRCLE SPACE AND FLAG SPACE
303 5.3 CONSTRUCTIONS 304 5.3.1 OVOIDAL PLANES 304 5.3.2 SEMI-CLASSICAL
FLAT LAGUERRE PLANES - 306 5.3.3 DIFFERENT WAYS TO CUT AND PASTE 312
5.3.4 INTEGRALS OF FLAT LINEAR SPACES 316 5.3.5 PLANES OF GENERALIZED
SHEAR TYPE 324 5.3.6 PLANES OF TRANSLATION TYPE 325 5.3.7 THE ARTZY-GROH
PLANES 326 5.3.8 PLANES OF SHIFT TYPE 328 5.4 AUTOMORPHISM GROUPS AND
GROUPS OF PROJECTIVITIES 329 5.4.1 AUTOMORPHISMS 329 5.4.2 THE KERNEL
332 5.4.3 PLANES ADMITTING AN AT LEAST 3-DIMENSIONAL KERNEL 333 5.4.4
CLASSIFICATION WITH RESPECT TO THE GROUP DIMENSION 336 5.4.5 VON
STAUDT S POINT OF VIEW*GROUPS OF PROJECTIVITIES * 341 5.5 THE
KLEINEWILLINGHOFER TYPES 342 5.5.1 C-HOMOLOGIES 343 XIV CONTENTS 5.5.2
LAGUERRE TRANSLATIONS 344 5.5.3 (P, G)-HOMOTHETIES 347 5.5.4 SOME
EXAMPLES 349 5.6 CHARACTERIZATIONS OF THE CLASSICAL PLANE 351 5.7 PLANES
WITH SPECIAL PROPERTIES 352 5.7.1 RIGID PLANES 352 5.7.2 DIFFERENTIABLE
PLANES 352 5.8 SUBGEOMETRIES AND LIE GEOMETRIES 353 5.8.1 RECYCLED FLAT
PROJECTIVE PLANES 353 5.8.2 DOUBLE COVERS OF R 2 -PLANES AND FLAT
PROJECTIVE PLANES 355 5.8.3 FLOCKS AND RESOLUTIONS 356 5.9 OPEN PROBLEMS
358 6 GENERALIZED QUADRANGLES 360 6.1 THE CLASSICAL ANTIREGULAR
3-DIMENSIONAL QUADRANGLE 361 6.2 BASIC PROPERTIES 364 6.3 FROM CIRCLE
PLANES TO GENERALIZED QUADRANGLES AND BACK 365 6.3.1 FLAT LAGUERRE
PLANES 6.3.2 FLAT MOBIUS PLANES 6.4 FLAT MINKOWSKI PLANES 6.5 SISTERS OF
CIRCLE PLANES 6.5.1 SISTERS OF FLAT LAGUERRE PLANES 6.5.2 SISTERS OF
FLAT MOBIUS PLANES 6.5.3 SISTERS OF (HALVES OF) FLAT MINKOWSKI PLANES
6.6 FLAT BIAFFINE PLANES AND FLAT HOMOLOGY SEMIBIPLANES 6.6.1 FLAT
BIAFFINE PLANES IN FLAT LAGUERRE PLANES 6.6.2 FLAT BIAFFINE PLANES IN
FLAT MOBIUS PLANES. 6.6.3 - FLAT HOMOLOGY SEMIBIPLANES IN FLAT LAGUERRE
PLANES 6.6.4 FLAT HOMOLOGY SEMIBIPLANES IN FLAT MOBIUS PLANES 6.6.5
SPLIT SEMIBIPLANES 6.7 DIFFERENT WAYS TO CUT AND PASTE 6.8 THE
APOLLONIUS PROBLEM 7 TUBULAR CIRCLE PLANES 7.1 UNISOLVENT SETS OF
FUNCTIONS 7.1.1 FIBRATED CIRCLE PLANES 7.1.2 MODELS OF THE CLASSICAL
TUBULAR CIRCLE PLANES 7.1.3 BASIC PROPERTIES 7.2 NESTED (PH)UNISOLVENT
SETS AND THEIR CIRCLE PLANES 7.2.1 UNRESTRICTED (PH)UNISOLVENT SETS
7.2.2 INTEGRATING UNISOLVENT SETS 7.2.3 NESTED TUBULAR CIRCLE PLANES
CONTENTS XV 7.2.4 AUTOMORPHISMS OF NESTED TUBULAR CIRCLE PLANES 416 7.3
CONVEXITY AND CUT-AND-PASTE CONSTRUCTIONS 420 7.3.1 CONVEXITY 421 7.3.2
DIFFERENT WAYS TO CUT AND PASTE 422 7.4 OPEN PROBLEMS 427 APPENDIX 1
TOOLS AND TECHNIQUES FROM TOPOLOGY AND ANALYSIS 429 APPENDIX 2 LIE
TRANSFORMATION GROUPS 444 BIBLIOGRAPHY 458 INDEX 483
|
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| id | DE-604.BV013963019 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:55:12Z |
| institution | BVB |
| isbn | 0521660580 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009556214 |
| oclc_num | 248415872 |
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| owner | DE-703 DE-355 DE-BY-UBR DE-20 DE-634 DE-11 |
| owner_facet | DE-703 DE-355 DE-BY-UBR DE-20 DE-634 DE-11 |
| physical | XXII, 490 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Polster, Burkard 1965- Verfasser (DE-588)113535384 aut Geometries on surfaces Burkard Polster and Günter Steinke 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2001 XXII, 490 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 84 Geometrie - Topologie - Fläche Surfaces Geometrie (DE-588)4020236-7 gnd rswk-swf Fläche (DE-588)4129864-0 gnd rswk-swf Fläche (DE-588)4129864-0 s Geometrie (DE-588)4020236-7 s DE-604 Steinke, Günter Verfasser aut Encyclopedia of mathematics and its applications 84 (DE-604)BV000903719 84 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009556214&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Polster, Burkard 1965- Steinke, Günter Geometries on surfaces Encyclopedia of mathematics and its applications Geometrie - Topologie - Fläche Surfaces Geometrie (DE-588)4020236-7 gnd Fläche (DE-588)4129864-0 gnd |
| subject_GND | (DE-588)4020236-7 (DE-588)4129864-0 |
| title | Geometries on surfaces |
| title_auth | Geometries on surfaces |
| title_exact_search | Geometries on surfaces |
| title_full | Geometries on surfaces Burkard Polster and Günter Steinke |
| title_fullStr | Geometries on surfaces Burkard Polster and Günter Steinke |
| title_full_unstemmed | Geometries on surfaces Burkard Polster and Günter Steinke |
| title_short | Geometries on surfaces |
| title_sort | geometries on surfaces |
| topic | Geometrie - Topologie - Fläche Surfaces Geometrie (DE-588)4020236-7 gnd Fläche (DE-588)4129864-0 gnd |
| topic_facet | Geometrie - Topologie - Fläche Surfaces Geometrie Fläche |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009556214&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000903719 |
| work_keys_str_mv | AT polsterburkard geometriesonsurfaces AT steinkegunter geometriesonsurfaces |