Hyperbolic geometry from a local viewpoint:
Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point...
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | London Mathamatical Society student texts
68 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.--From publisher description. |
| Beschreibung: | X, 271 S. graph. Darst. |
| ISBN: | 9780521863605 9780521682244 |
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| 520 | 3 | |a Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.--From publisher description. | |
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Datensatz im Suchindex
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| adam_text | HYPERBOLIC GEOMETRY FROM A LOCAL VIEWPOINT LINDA KEEN LEHMAN COLLEGE AND
GRADUATE CENTER CITY UNIVERSITY OF NEW YORK NIKOLA LAKIC LEHMAN COLLEGE
AND GRADUATE CENTER CITY UNIVERSITY OF NEW YORK CAMBRIDGE UNIVERSITY
PRESS CONTENTS INTRODUCTION PAGE 1 ELEMENTARY TRANSFORMATIONS OF THE
EUCLIDEAN PLANE AND THE-RIEMANN SPHERE 5 1.1 THE EUCLIDEAN METRIC 5 1.2
RIGID MOTIONS 6 1.2.1 SCALING MAPS 8 1.3 CONFORMAL MAPPINGS 9 1.4 THE
RIEMANN SPHERE 11 1.5 MOBIUS TRANSFORMATIONS AND THE CROSS RATIO 13
1.5.1 CLASSIFICATION OF MOBIUS TRANSFORMATIONS 18 1.6 MOBIUS GROUPS 22
1.7 DISCRETENESS OF MOBIUS GROUPS 24 1.8 THE EUCLIDEAN DENSITY 26 1.8.1
OTHER EUCLIDEAN TYPE DENSITIES 31 HYPERBOLIC METRIC IN THE UNIT DISK 32
2.1 DEFINITION OF THE HYPERBOLIC METRIC IN THE UNIT DISK 32 2.1.1
HYPERBOLIC GEODESIES 33 2.1.2 HYPERBOLIC TRIANGLES 39 2.2 PROPERTIES OF
THE HYPERBOLIC METRIC IN A 41 2.3 THE UPPER HALF PLANE MODEL 43 2.4 THE
GEOMETRY OF PSL{2, R) AND T 46 2.4.1 HYPERBOLIC TRANSFORMATIONS 46 2.4.2
PARABOLIC TRANSFORMATIONS 48 2.4.3 ELLIPTIC TRANSFORMATIONS 50 2.4.4
HYPERBOLIC REFLECTIONS 51 VN VIII CONTENTS 3 HOLOMORPHIC FUNCTIONS 53
3.1 BASIC THEOREMS 53 3.2 THE SCHWARZ LEMMA 55 3.3 NORMAL FAMILIES 58
3.4 THE RIEMANN MAPPING THEOREM 59 3.5 THE SCHWARZ REFLECTION PRINCIPLE
63 3.6 RATIONAL MAPS AND BLASCHKE PRODUCTS 64 3.7 DISTORTION THEOREMS 66
4 TOPOLOGY AND UNIFORMIZATION 68 4.1 SURFACES 68 4.2 THE FUNDAMENTAL
GROUP 70 4.3 COVERING SPACES 74 4.4 CONSTRUCTION OF THE UNIVERSAL
COVERING SPACE 78 4.5 THE UNIVERSAL COVERING GROUP 80 4.6 THE
UNIFORMIZATION THEOREM 81 5 DISCONTINUOUS GROUPS 83 5.1 DISCONTINUOUS
SUBGROUPS OF M 83 5.2 DISCONTINUOUS ELEMENTARY GROUPS 90 5.3
NON-ELEMENTARY GROUPS 94 6 FUCHSIAN GROUPS 96 6.1 AN HISTORICAL NOTE 96
6.2 FUNDAMENTAL DOMAINS 97 6.3 DIRICHLET DOMAINS AND FUNDAMENTAL
POLYGONS 101 6.4 VERTEX CYCLES OF FUNDAMENTAL POLYGONS 110 6.5
POINCAR6 S THEOREM 115 7 THE HYPERBOLIC METRIC FOR ARBITRARY DOMAINS 124
7.1 DEFINITION OF THE HYPERBOLIC METRIC 124 7.2 PROPERTIES OF THE
HYPERBOLIC METRIC FOR X 127 7.3 THE SCHWARZ-PICK LEMMA 130 7.4 EXAMPLES
133 7.5 CONFORMAL DENSITY AND CURVATURE 139 7.6 CONFORMAL INVARIANTS 141
7.6.1 TORUS INVARIANTS 141 7.6.2 EXTREMAL LENGTH 143 7.6.3 GENERAL
RIEMANN SURFACES 147 7.7 THE COLLAR LEMMA * 148 CONTENTS IX 8 THE
KOBAYASHI METRIC 153 8.1 THE CLASSICAL KOBAYASHI DENSITY 153 8.2 THE
KOBAYASHI DENSITY FOR ARBITRARY DOMAINS 154 8.2.1 GENERALIZED KOBAYASHI
DENSITY: BASIC PROPERTIES 155 8.2.2 EXAMPLES 161 9 THE CARATHEODORY
PSEUDO-METRIC 163 9.1 THE CLASSICAL CARATHEODORY DENSITY 163 9.2
GENERALIZED CARATHEODORY PSEUDO-METRIC 165 9.2.1 GENERALIZED
CARATHEODORY DENSITY: BASIC PROPERTIES 166 9.2.2 EXAMPLES 170 10
INCLUSION MAPPINGS AND CONTRACTION PROPERTIES 172 10.1 ESTIMATES OF
HYPERBOLIC DENSITIES 172 10.2 STRONG CONTRACTIONS 173 10.3 LIPSCHITZ
DOMAINS 175 10.4 GENERALIZED LIPSCHITZ AND BLOCH DOMAINS 180 10.4.1
KOBAYASHI LIPSCHITZ DOMAINS 180 10.4.2 KOBAYASHI BLOCH DOMAINS 182
10.4.3 CARATH6ODORY LIPSCHITZ DOMAINS 182 10.4.4 CARATHEODORY BLOCH
DOMAINS 184 10:5 EXAMPLES 184 11 APPLICATIONS I: FORWARD RANDOM
HOLOMORPHIC ITERATION 191 11.1 RANDOM HOLOMORPHIC ITERATION 191 11.2
FORWARD ITERATION 192 12 APPLICATIONS II: BACKWARD RANDOM ITERATION 19 5
12.1 COMPACT SUBDOMAINS 195 12.2 NON-COMPACT SUBDOMAINS: THE
CK-CONDITION 196 12.3 THE OVERALL PICTURE 198 13 APPLICATIONS III: LIMIT
FUNCTIONS * 201 13.1 UNIQUENESS OF LIMITS 201 13.1.1 THE KEY LEMMA 201
13.1.2 PROOF OF THEOREM 13.1.1 203 13.2 NON-BLOCH DOMAINS AND
NON-CONSTANT LIMITS 207 13.2.1 PREPARATORY LEMMAS 207 13.2.2 A NECESSARY
CONDITION FOR DEGENERACY 208 X CONTENTS 13.2.3 PROOF OF THEOREM 13.2.2
215 13.2.4 EQUIVALENCE OF CONDITIONS 217 14 ESTIMATING HYPERBOLIC
DENSITIES 219 14.1 THE SMALLEST HYPERBOLIC DENSITIES 219 14.2 A FORMULA
FOR P 01 220 14.3 A LOWER BOUND ON P 01 223 14.3.1 THE FIRST ESTIMATES
224 14.3.2 ESTIMATES OF P 0L NEAR THE PUNCTURES 229 14.3.3 THE
DERIVATIVES OF P 01 230 14.3.4 THE EXISTENCE OF A LOWER BOUND ON P 01
234 14.4 PROPERTIES OF THE SMALLEST HYPERBOLIC DENSITY 236 14.5
COMPARING POINCARE DENSITIES 240 15 UNIFORMLY PERFECT DOMAINS 245 15.1
SIMPLE EXAMPLES 246 15.2 UNIFORMLY PERFECT DOMAINS AND CRPSS RATIOS 247
15.3 UNIFORMLY PERFECT DOMAINS AND SEPARATING ANNULI 249 15.4 UNIFORMLY
THICK DOMAINS 253 16 APPENDIX: A BRIEF SURVEY OF ELLIPTIC FUNCTIONS 258
16.0.1 BASIC PROPERTIES OF ELLIPTIC FUNCTIONS 258 BIBLIOGRAPHY 264 INDEX
268
|
| adam_txt |
HYPERBOLIC GEOMETRY FROM A LOCAL VIEWPOINT LINDA KEEN LEHMAN COLLEGE AND
GRADUATE CENTER CITY UNIVERSITY OF NEW YORK NIKOLA LAKIC LEHMAN COLLEGE
AND GRADUATE CENTER CITY UNIVERSITY OF NEW YORK CAMBRIDGE UNIVERSITY
PRESS CONTENTS INTRODUCTION PAGE 1 ELEMENTARY TRANSFORMATIONS OF THE
EUCLIDEAN PLANE AND THE-RIEMANN SPHERE 5 1.1 THE EUCLIDEAN METRIC 5 1.2
RIGID MOTIONS 6 1.2.1 SCALING MAPS 8 1.3 CONFORMAL MAPPINGS 9 1.4 THE
RIEMANN SPHERE 11 1.5 MOBIUS TRANSFORMATIONS AND THE CROSS RATIO 13
1.5.1 CLASSIFICATION OF MOBIUS TRANSFORMATIONS 18 1.6 MOBIUS GROUPS 22
1.7 DISCRETENESS OF MOBIUS GROUPS 24 1.8 THE EUCLIDEAN DENSITY 26 1.8.1
OTHER EUCLIDEAN TYPE DENSITIES 31 HYPERBOLIC METRIC IN THE UNIT DISK 32
2.1 DEFINITION OF THE HYPERBOLIC METRIC IN THE UNIT DISK 32 2.1.1
HYPERBOLIC GEODESIES 33 2.1.2 HYPERBOLIC TRIANGLES 39 2.2 PROPERTIES OF
THE HYPERBOLIC METRIC IN A 41 2.3 THE UPPER HALF PLANE MODEL 43 2.4 THE
GEOMETRY OF PSL{2, R) AND T 46 2.4.1 HYPERBOLIC TRANSFORMATIONS 46 2.4.2
PARABOLIC TRANSFORMATIONS 48 2.4.3 ELLIPTIC TRANSFORMATIONS 50 2.4.4
HYPERBOLIC REFLECTIONS 51 VN VIII CONTENTS 3 HOLOMORPHIC FUNCTIONS 53
3.1 BASIC THEOREMS 53 3.2 THE SCHWARZ LEMMA 55 3.3 NORMAL FAMILIES 58
3.4 THE RIEMANN MAPPING THEOREM 59 3.5 THE SCHWARZ REFLECTION PRINCIPLE
63 3.6 RATIONAL MAPS AND BLASCHKE PRODUCTS 64 3.7 DISTORTION THEOREMS 66
4 TOPOLOGY AND UNIFORMIZATION 68 4.1 SURFACES 68 4.2 THE FUNDAMENTAL
GROUP 70 4.3 COVERING SPACES 74 4.4 CONSTRUCTION OF THE UNIVERSAL
COVERING SPACE 78 4.5 THE UNIVERSAL COVERING GROUP 80 4.6 THE
UNIFORMIZATION THEOREM 81 5 DISCONTINUOUS GROUPS 83 5.1 DISCONTINUOUS
SUBGROUPS OF M 83 5.2 DISCONTINUOUS ELEMENTARY GROUPS 90 5.3
NON-ELEMENTARY GROUPS 94 6 FUCHSIAN GROUPS 96 6.1 AN HISTORICAL NOTE 96
6.2 FUNDAMENTAL DOMAINS 97 6.3 DIRICHLET DOMAINS AND FUNDAMENTAL
POLYGONS 101 6.4 VERTEX CYCLES OF FUNDAMENTAL POLYGONS 110 6.5
POINCAR6'S THEOREM 115 7 THE HYPERBOLIC METRIC FOR ARBITRARY DOMAINS 124
7.1 DEFINITION OF THE HYPERBOLIC METRIC 124 7.2 PROPERTIES OF THE
HYPERBOLIC METRIC FOR X 127 7.3 THE SCHWARZ-PICK LEMMA 130 7.4 EXAMPLES
133 7.5 CONFORMAL DENSITY AND CURVATURE 139 7.6 CONFORMAL INVARIANTS 141
7.6.1 TORUS INVARIANTS 141 7.6.2 EXTREMAL LENGTH 143 7.6.3 GENERAL
RIEMANN SURFACES 147 7.7 THE COLLAR LEMMA * 148 CONTENTS IX 8 THE
KOBAYASHI METRIC 153 8.1 THE CLASSICAL KOBAYASHI DENSITY 153 8.2 THE
KOBAYASHI DENSITY FOR ARBITRARY DOMAINS 154 8.2.1 GENERALIZED KOBAYASHI
DENSITY: BASIC PROPERTIES 155 8.2.2 EXAMPLES 161 9 THE CARATHEODORY
PSEUDO-METRIC 163 9.1 THE CLASSICAL CARATHEODORY DENSITY 163 9.2
GENERALIZED CARATHEODORY PSEUDO-METRIC 165 9.2.1 GENERALIZED
CARATHEODORY DENSITY: BASIC PROPERTIES 166 9.2.2 EXAMPLES 170 10
INCLUSION MAPPINGS AND CONTRACTION PROPERTIES 172 10.1 ESTIMATES OF
HYPERBOLIC DENSITIES 172 10.2 STRONG CONTRACTIONS 173 10.3 LIPSCHITZ
DOMAINS 175 10.4 GENERALIZED LIPSCHITZ AND BLOCH DOMAINS 180 10.4.1
KOBAYASHI LIPSCHITZ DOMAINS 180 10.4.2 KOBAYASHI BLOCH DOMAINS 182
10.4.3 CARATH6ODORY LIPSCHITZ DOMAINS 182 10.4.4 CARATHEODORY BLOCH
DOMAINS 184 10:5 EXAMPLES 184 11 APPLICATIONS I: FORWARD RANDOM
HOLOMORPHIC ITERATION 191 11.1 RANDOM HOLOMORPHIC ITERATION 191 11.2
FORWARD ITERATION 192 12 APPLICATIONS II: BACKWARD RANDOM ITERATION 19 5
12.1 COMPACT SUBDOMAINS 195 12.2 NON-COMPACT SUBDOMAINS: THE
CK-CONDITION 196 12.3 THE OVERALL PICTURE 198 13 APPLICATIONS III: LIMIT
FUNCTIONS * ' 201 13.1 UNIQUENESS OF LIMITS 201 13.1.1 THE KEY LEMMA 201
13.1.2 PROOF OF THEOREM 13.1.1 203 13.2 NON-BLOCH DOMAINS AND
NON-CONSTANT LIMITS 207 13.2.1 PREPARATORY LEMMAS 207 13.2.2 A NECESSARY
CONDITION FOR DEGENERACY 208 X CONTENTS 13.2.3 PROOF OF THEOREM 13.2.2
215 13.2.4 EQUIVALENCE OF CONDITIONS 217 14 ESTIMATING HYPERBOLIC
DENSITIES 219 14.1 THE SMALLEST HYPERBOLIC DENSITIES 219 14.2 A FORMULA
FOR P 01 220 14.3 A LOWER BOUND ON P 01 223 14.3.1 THE FIRST ESTIMATES
224 14.3.2 ESTIMATES OF P 0L NEAR THE PUNCTURES 229 14.3.3 THE
DERIVATIVES OF P 01 230 14.3.4 THE EXISTENCE OF A LOWER BOUND ON P 01
234 14.4 PROPERTIES OF THE SMALLEST HYPERBOLIC DENSITY 236 14.5
COMPARING POINCARE DENSITIES 240 15 UNIFORMLY PERFECT DOMAINS 245 15.1
SIMPLE EXAMPLES 246 15.2 UNIFORMLY PERFECT DOMAINS AND CRPSS RATIOS 247
15.3 UNIFORMLY PERFECT DOMAINS AND SEPARATING ANNULI 249 15.4 UNIFORMLY
THICK DOMAINS 253 16 APPENDIX: A BRIEF SURVEY OF ELLIPTIC FUNCTIONS 258
16.0.1 BASIC PROPERTIES OF ELLIPTIC FUNCTIONS 258 BIBLIOGRAPHY 264 INDEX
268 |
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| id | DE-604.BV022218664 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:28:22Z |
| indexdate | 2024-07-09T20:52:38Z |
| institution | BVB |
| isbn | 9780521863605 9780521682244 |
| language | English |
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| series2 | London Mathamatical Society student texts |
| spelling | Keen, Linda Verfasser aut Hyperbolic geometry from a local viewpoint Linda Keen ; Nikola Lakic 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2007 X, 271 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier London Mathamatical Society student texts 68 Written for graduate students, this book presents topics in 2-dimensional hyperbolic geometry. The authors begin with rigid motions in the plane which are used as motivation for a full development of hyperbolic geometry in the unit disk. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. These ideas are developed in the context of what is used later. The authors then provide a detailed discussion of hyperbolic geometry for arbitrary plane domains. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems.--From publisher description. Géométrie hyperbolique Geometry, Hyperbolic Hyperbolische Geometrie (DE-588)4161041-6 gnd rswk-swf Hyperbolische Geometrie (DE-588)4161041-6 s DE-604 Lakic, Nikola Verfasser aut London Mathamatical Society student texts 68 (DE-604)BV000841726 68 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015429913&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Keen, Linda Lakic, Nikola Hyperbolic geometry from a local viewpoint London Mathamatical Society student texts Géométrie hyperbolique Geometry, Hyperbolic Hyperbolische Geometrie (DE-588)4161041-6 gnd |
| subject_GND | (DE-588)4161041-6 |
| title | Hyperbolic geometry from a local viewpoint |
| title_auth | Hyperbolic geometry from a local viewpoint |
| title_exact_search | Hyperbolic geometry from a local viewpoint |
| title_exact_search_txtP | Hyperbolic geometry from a local viewpoint |
| title_full | Hyperbolic geometry from a local viewpoint Linda Keen ; Nikola Lakic |
| title_fullStr | Hyperbolic geometry from a local viewpoint Linda Keen ; Nikola Lakic |
| title_full_unstemmed | Hyperbolic geometry from a local viewpoint Linda Keen ; Nikola Lakic |
| title_short | Hyperbolic geometry from a local viewpoint |
| title_sort | hyperbolic geometry from a local viewpoint |
| topic | Géométrie hyperbolique Geometry, Hyperbolic Hyperbolische Geometrie (DE-588)4161041-6 gnd |
| topic_facet | Géométrie hyperbolique Geometry, Hyperbolic Hyperbolische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015429913&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000841726 |
| work_keys_str_mv | AT keenlinda hyperbolicgeometryfromalocalviewpoint AT lakicnikola hyperbolicgeometryfromalocalviewpoint |