Metric spaces of non-positive curvature:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London
Springer
1999
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
319 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XXI, 643 Seiten Illustrationen |
| ISBN: | 3540643249 9783642083990 |
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| 100 | 1 | |a Bridson, Martin R. |d 1964- |e Verfasser |0 (DE-588)121157423 |4 aut | |
| 245 | 1 | 0 | |a Metric spaces of non-positive curvature |c Martin R. Bridson, André Haefliger |
| 264 | 1 | |a Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London |b Springer |c 1999 | |
| 300 | |a XXI, 643 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 319 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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| 650 | 7 | |a Espaces métriques |2 ram | |
| 650 | 4 | |a Géométrie différentielle | |
| 650 | 7 | |a Géométrie différentielle |2 ram | |
| 650 | 7 | |a Metrische ruimten |2 gtt | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Metric spaces | |
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Datensatz im Suchindex
| _version_ | 1804127288416010240 |
|---|---|
| adam_text | Table of Contents
Introduction VII
Part I. Geodesic Metric Spaces 1
1. Basic Concepts 2
Metric Spaces 2
Geodesies 4
Angles 8
The Length of a Curve 12
2. The Model Spaces MnK 15
Euclidean n Space E 15
The n Sphere § 16
Hyperbolic rc Space W 18
The Model Spaces M K 23
Alexandrov s Lemma 24
The Isometry Groups Isom(A/ ) 26
Approximate Midpoints 30
3. Length Spaces 32
Length Metrics 32
The Hopf Rinow Theorem 35
Riemannian Manifolds as Metric Spaces 39
Length Metrics on Covering Spaces 42
Manifolds of Constant Curvature 45
4. Normed Spaces 47
Hilbert Spaces 47
Isometries of Normed Spaces 51
V Spaces 53
5. Some Basic Constructions 56
Products 56
/c Cones 59
XVI Table of Contents
Spherical Joins 63
Quotient Metrics and Gluing 64
Limits of Metric Spaces 70
Ultralimits and Asymptotic Cones 77
6. More on the Geometry of M K 81
The Klein Model for W 81
The Mobius Group 84
The Poincare Ball Model for W 86
The Poincare Half Space Model for H 90
Isometries of M2 91
M as a Riemannian Manifold 92
7. MK Polyhedral Complexes 97
Metric Simplicial Complexes 97
Geometric Links and Cone Neighbourhoods 102
The Existence of Geodesies 105
The Main Argument 108
Cubical Complexes Ill
M^ Polyhedral Complexes 112
Barycentric Subdivision 115
More on the Geometry of Geodesies 118
Alternative Hypotheses 122
Appendix: Metrizing Abstract Simplicial Complexes 123
8. Group Actions and Quasi Isometries 131
Group Actions on Metric Spaces 131
Presenting Groups of Homeomorphisms 134
Quasi Isometries 138
Some Invariants of Quasi Isometry 142
The Ends of a Space 144
Growth and Rigidity 148
Quasi Isometries of the Model Spaces 150
Approximation by Metric Graphs 152
Appendix: Combinatorial 2 Complexes 153
Part II. CAT(k) Spaces 157
1. Definitions and Characterizations of CAT(k) Spaces 158
The CAT(k) Inequality 158
Characterizations of CAT(/c) Spaces 161
CATO) Implies CAT(*r ) if k k 165
Simple Examples of CAT(/f) Spaces 167
Table of Contents XVII
Historical Remarks 168
Appendix: The Curvature of Riemannian Manifolds 169
2. Convexity and Its Consequences 175
Convexity of the Metric 175
Convex Subspaces and Projection 176
The Centre of a Bounded Set 178
Flat Subspaces 180
3. Angles, Limits, Cones and Joins 184
Angles in CAT(k) Spaces 184
4 Point Limits of CAT(at) Spaces 186
Cones and Spherical Joins 188
The Space of Directions 190
4. The Cartan Hadamard Theorem 193
Local to Global 193
An Exponential Map 196
Alexandrov s Patchwork 199
Local Isometries and it Injectivity 200
Injectivity Radius and Systole 202
5. MK Polyhedral Complexes of Bounded Curvature 205
Characterizations of Curvature k 206
Extending Geodesies 207
Flag Complexes 210
Constructions with Cubical Complexes 212
Two Dimensional Complexes 215
Subcomplexes and Subgroups in Dimension 2 216
Knot and Link Groups 220
From Group Presentations to Negatively Curved 2 Complexes 224
6. Isometries of CAT(0) Spaces 228
Individual Isometries 228
On the General Structure of Groups of Isometries 233
Clifford Translations and the Euclidean de Rham Factor 235
The Group of Isometries of a Compact Metric Space
of Non Positive Curvature 237
A Splitting Theorem 239
7. The Flat Torus Theorem 244
The Flat Torus Theorem 244
Cocompact Actions and the Solvable Subgroup Theorem 247
Proper Actions That Are Not Cocompact 250
Actions That Are Not Proper 254
Some Applications to Topology 254
XVIII Table of Contents
8. The Boundary at Infinity of a CAT(O) Space 260
Asymptotic Rays and the Boundary dX 260
The Cone Topology on X = X U dX 263
Horofunctions and Busemann Functions 267
Characterizations of Horofunctions 271
Parabolic Isometries 274
9. The Tits Metric and Visibility Spaces 277
Angles in X 278
The Angular Metric 279
The Boundary (dX, Z) is a CAT(l) Space 285
The Tits Metric 289
How the Tits Metric Determines Splittings 291
Visibility Spaces 294
10. Symmetric Spaces 299
Real, Complex and Quaternionic Hyperbolic w Spaces 300
The Curvature of KH 304
The Curvature of Distinguished Subspaces of KH 306
The Group of Isometries of KH 307
The Boundary at Infinity and Horospheres in KH 309
Horocyclic Coordinates and Parabolic Subgroups for KH 311
The Symmetric Space P(n, R) 314
P(n, M) as a Riemannian Manifold 314
The Exponential Map exp: M(n, R) GL(n, R) 316
P(n, R) is a CAT(0) Space 318
Flats, Regular Geodesies and Weyl Chambers 320
The Iwasawa Decomposition of GL(n, R) 323
The Irreducible Symmetric space P(n, R) 324
Reductive Subgroups of GL(n, E) 327
Semi Simple Isometries 331
Parabolic Subgroups and Horospherical Decompositions of P(n, R) ... 332
The Tits Boundary of P(n, K), is a Spherical Building 337
dTP(n, R) in the Language of Flags and Frames 340
Appendix: Spherical and Euclidean Buildings 342
11. Gluing Constructions 347
Gluing CAT(k) Spaces Along Convex Subspaces 347
Gluing Using Local Isometries 350
Equivariant Gluing 355
Gluing Along Subspaces that are not Locally Convex 359
Truncated Hyperbolic Spaces 362
12. Simple Complexes of Groups 367
Stratified Spaces 368
Table of Contents XIX
Group Actions with a Strict Fundamental Domain 372
Simple Complexes of Groups: Definition and Examples 375
The Basic Construction 381
Local Development and Curvature 387
Constructions Using Coxeter Groups 391
Part III. Aspects of the Geometry of Group Actions 397
H. 5 Hyperbolic Spaces 398
1. Hyperbolic Metric Spaces 399
The Slim Triangles Condition 399
Quasi Geodesics in Hyperbolic Spaces 400
/t Local Geodesies 405
Reformulations of the Hyperbolicity Condition 407
2. Area and Isoperimetric Inequalities 414
A Coarse Notion of Area 414
The Linear Isoperimetric Inequality and Hyperbolicity 417
Sub Quadratic Implies Linear 422
More Refined Notions of Area 425
3. The Gromov Boundary of a ^ Hyperbolic Space 427
The Boundary dX as a Set of Rays 427
The Topology on X U dX 429
Metrizing dX 432
r. Non Positive Curvature and Group Theory 438
1. Isometries of CAT(O) Spaces 439
A Summary of What We Already Know 439
Decision Problems for Groups of Isometries 440
The Word Problem 442
The Conjugacy Problem 445
2. Hyperbolic Groups and Their Algorithmic Properties 448
Hyperbolic Groups 448
Dehn s Algorithm 449
The Conjugacy Problem 451
Cone Types and Growth 455
3. Further Properties of Hyperbolic Groups 459
Finite Subgroups 459
Quasiconvexity and Centralizers 460
Translation Lengths 464
Free Subgroups 467
The Rips Complex 468
XX Table of Contents
4. Semihyperbolic Groups 471
Definitions 471
Basic Properties of Semihyperbolic Groups 473
Subgroups of Semihyperbolic Groups 475
5. Subgroups of Cocompact Groups of Isometries 481
Finiteness Properties 481
The Word, Conjugacy and Membership Problems 487
Isomorphism Problems 491
Distinguishing Among Non Positively Curved
Manifolds 494
6. Amalgamating Groups of Isometries 496
Amalgamated Free Products and HNN Extensions 497
Amalgamating Along Abelian Subgroups 500
Amalgamating Along Free Subgroups 503
Subgroup Distortion and the Dehn Functions
of Doubles 506
7. Finite Sheeted Coverings and Residual Finiteness 511
Residual Finiteness 511
Groups Without Finite Quotients 514
C. Complexes of Groups 519
1. Small Categories Without Loops (Scwols) 520
Scwols and Their Geometric Realizations 521
The Fundamental Group and Coverings 526
Group Actions on Scwols 528
The Local Structure of Scwols 531
2. Complexes of Groups 534
Basic Definitions 535
Developability 538
The Basic Construction 542
3. The Fundamental Group of a Complex of Groups 546
The Universal Group FGQ?) 546
The Fundamental Group Ti (G(^), ct0) 548
A Presentation of n {Giy a0) 549
The Universal Covering of a Developable Complex of Groups 553
4. Local Developments of a Complex of Groups 555
The Local Structure of the Geometric Realization 555
The Geometric Realization of the Local Development 557
Local Development and Curvature 562
The Local Development as a Scwol 564
5. Coverings of Complexes of Groups 566
Definitions 566
Table of Contents XXI
The Fibres of a Covering 568
The Monodromy 572
A Appendix: Fundamental Groups and Coverings
of Small Categories 573
Basic Definitions 574
The Fundamental Group 576
Covering of a Category 579
The Relationship with Coverings of Complexes of Groups 583
Q. Groupoids of local Isometries 584
1. Orbifolds 585
Basic Definitions 585
Coverings of Orbifolds 589
Orbifolds with Geometric Structures 591
2. Etale Groupoids, Homomorphisms and Equivalences 594
Etale Groupoids 594
Equivalences and Developability 597
Groupoids of Local Isometries 601
Statement of the Main Theorem 603
3. The Fundamental Group and Coverings of Etale Groupoids .... 604
Equivalence and Homotopy of 5 Paths 604
The Fundamental Group n ({Q, X), jco) 607
Coverings 609
4. Proof of the Main Theorem 613
Outline of the Proof 613
? Geodesics 614
The Space X of 5 Geodesics Issuing from a Base Point 616
The Space X = X/Q 617
The Coverings : X ^ X 618
References 620
Index 637
|
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| author | Bridson, Martin R. 1964- Haefliger, André 1929-2023 |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:30:58Z |
| institution | BVB |
| isbn | 3540643249 9783642083990 |
| language | English |
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| spelling | Bridson, Martin R. 1964- Verfasser (DE-588)121157423 aut Metric spaces of non-positive curvature Martin R. Bridson, André Haefliger Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London Springer 1999 XXI, 643 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 319 Hier auch später erschienene, unveränderte Nachdrucke Espaces métriques Espaces métriques ram Géométrie différentielle Géométrie différentielle ram Metrische ruimten gtt Geometry, Differential Metric spaces Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Nichtpositive Krümmung (DE-588)4128763-0 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 s Nichtpositive Krümmung (DE-588)4128763-0 s DE-604 Haefliger, André 1929-2023 Verfasser (DE-588)121157377 aut Grundlehren der mathematischen Wissenschaften 319 (DE-604)BV000000395 319 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008581824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bridson, Martin R. 1964- Haefliger, André 1929-2023 Metric spaces of non-positive curvature Grundlehren der mathematischen Wissenschaften Espaces métriques Espaces métriques ram Géométrie différentielle Géométrie différentielle ram Metrische ruimten gtt Geometry, Differential Metric spaces Metrischer Raum (DE-588)4169745-5 gnd Nichtpositive Krümmung (DE-588)4128763-0 gnd |
| subject_GND | (DE-588)4169745-5 (DE-588)4128763-0 |
| title | Metric spaces of non-positive curvature |
| title_auth | Metric spaces of non-positive curvature |
| title_exact_search | Metric spaces of non-positive curvature |
| title_full | Metric spaces of non-positive curvature Martin R. Bridson, André Haefliger |
| title_fullStr | Metric spaces of non-positive curvature Martin R. Bridson, André Haefliger |
| title_full_unstemmed | Metric spaces of non-positive curvature Martin R. Bridson, André Haefliger |
| title_short | Metric spaces of non-positive curvature |
| title_sort | metric spaces of non positive curvature |
| topic | Espaces métriques Espaces métriques ram Géométrie différentielle Géométrie différentielle ram Metrische ruimten gtt Geometry, Differential Metric spaces Metrischer Raum (DE-588)4169745-5 gnd Nichtpositive Krümmung (DE-588)4128763-0 gnd |
| topic_facet | Espaces métriques Géométrie différentielle Metrische ruimten Geometry, Differential Metric spaces Metrischer Raum Nichtpositive Krümmung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008581824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT bridsonmartinr metricspacesofnonpositivecurvature AT haefligerandre metricspacesofnonpositivecurvature |