Metric structures for Riemannian and non-Riemannian spaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2007
|
| Edition: | Repr. of the 2001 ed. |
| Series: | Modern Birkhäuser classics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Orig. publ. as vol. 152 in the series Progress in mathematics |
| Physical Description: | XIX, 585 S. graph. Darst. |
| ISBN: | 9780817645823 0817645829 |
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MARC
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| 245 | 1 | 0 | |a Metric structures for Riemannian and non-Riemannian spaces |c Misha Gromov. With app. by M. Katz ... |
| 250 | |a Repr. of the 2001 ed. | ||
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XIX, 585 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1804136257166508032 |
|---|---|
| adam_text | Contents
Preface
to the French Edition
xi
Preface to the English Edition
xiii
Introduction: Metrics Everywhere
xv
Length Structures: Path Metric Spaces
1
A. Length structures
........................ 1
B. Path metric spaces
....................... 6
С
Examples of path metric spaces
................ 10
D. Arc-wise isometries
....................... 22
2
Degree and Dilatation
27
A. Topological review
....................... 27
B. Elementary properties of dilatations for spheres
....... 30
C. Homotopy counting Lipschitz maps
.............. 35
D. Dilatation of sphere-valued mappings
............. 41
E+ Degrees of short maps between compact and
noncompact manifolds
..................... 55
3
Metric Structures on Families of Metric Spaces
71
A. Lipschitz and Hausdorff distance
............... 71
B. The noncompact case
..................... 85
C. The Hausdorff-Lipschitz metric,
quasi-isometries, and word metrics
.............. 89
D+ First-order metric invariants
and ultralimits
......................... 94
E+ Convergence with control
................... 98
Sj^Convergence and Concentration of Metrics and Measures
ИЗ
A. A review of measures and mm spaces
............. 113
B.
Пд
-convergence
of mm spaces
................. 116
C. Geometry of measures in metric spaces
............ 124
ущ
Contents
D. Basic
geometry
of the space X
................ 129
E.
Concentration phenomenon
.................. 140
F. Geometric invariants of measures
related to concentration
.................... 181
G. Concentration, spectrum, and
the spectral diameter
...................... 190
H
Observable distance
Ηλ
on the space X and concentration
Xn
->
X
............................. 200
I. The Lipschitz order on X, pyramids, and asymptotic con¬
centration
............................ 212
J. Concentration versus dissipation
............... 221
4
Loewner Rediscovered
239
A. First, some history (in dimension
2) ............. 239
B. Next, some questions in dimensions
> 3........... 244
С
Norms on homology and Jacobi varieties
........... 245
D. An application of geometric integration theory
....... 261
E+ Unstable systolic inequalities and filling
........... 264
F+ Finer inequalities and systoles
of universal spaces
....................... 269
5
Manifolds with Bounded
Ricci
Curvature
273
A. Precompactness
......................... 273
B. Growth of fundamental groups
................ 279
C. The first
Betti
number
..................... 284
D. Small loops
........................... 288
E+ Applications of the packing inequalities
........... 294
F+ On the nilpotency of
πχ
.................... 295
G+ Simplicial volume and entropy
................. 302
H+ Generalized simplicial norms and
the metrization of homotopy theory
............. 307
1+
Ricci
curvature beyond coverings
............... 316
6
Isoperimetric Inequalities and Amenability
321
A. Quasiregular mappings
..................... 321
B. Isoperimetric dimension of a manifold
............ 322
C. Computations of isoperimetric dimension
.......... 327
D. Generalized quasiconformality
................. 336
E+ The Varopoulos isoperimetric inequality
........... 346
7
Morse Theory and Minimal Models
351
A. Application of Morse theory
to loop spaces
.......................... 351
B. Dilatation of mappings between
simply connected manifolds
..................357
Contents
ix
8+ Pinching and Collapse
365
A. Invariant classes of metrics
and the stability problem
................... 365
B. Sign and the meaning of curvature
.............. 369
C. Elementary geometry of collapse
............... 375
D. Convergence without collapse
................. 384
E. Basic features of collapse
.................... 390
A Quasiconvex Domains
in Rn
393
В
Metric Spaces and Mappings Seen
at Many Scales
401
I. Basic concepts and examples
................. 402
1.
Euclidean spaces, hyperbolic spaces, and ideas from analysis
402
2.
Quasimetrics, the doubling condition, and
examples of metric spaces
................... 404
3.
Doubling measures and regular metric spaces, deformations
of geometry, Riesz products and Riemann surfaces
..... 411
4.
Quasisymmetric mappings and deformations of geometry from
doubling measures
....................... 417
5.
Rest and recapitulation
.................... 422
II. Analysis on general spaces
................... 423
6.
Holder continuous functions on metric spaces
........ 423
7.
Metric spaces which are doubling
............... 430
8.
Spaces of homogeneous type
.................. 435
9.
Holder continuity and mean oscillation
............ 437
10.
Vanishing mean oscillation
................... 439
11.
Bounded mean oscillation
................... 443
III. Rigidity and structure
..................... 445
12.
Differentiability almost everywhere
.............. 445
13.
Pause for reflection
....................... 448
14.
Almost flat curves
....................... 448
15.
Mappings that almost preserve distances
........... 452
16.
Almost flat hypersurfaces
................... 455
17.
The Aao condition for doubling measures
........... 458
18.
Quasisymmetric mappings and doubling measures
..... 462
19.
Metric doubling measures
................... 464
20.
Bi-Lipschitz embeddings
.................... 468
21.
Αχ
weights
........................... 470
22.
Interlude: bi-Lipschitz mappings between Cantor sets
. . . 471
23.
Another moment of reflection
................. 471
24.
Rectifiability
.......................... 471
25.
Uniform rectifiability
...................... 475
x
Contents
26.
Stories from the past
...................... 477
27.
Regular mappings
....................... 479
28.
Big pieces of bi-Lipschitz mappings
.............. 480
29.
Quantitative smoothness for Lipschitz functions
....... 482
30.
Smoothness of uniformly rectifiable sets
........... 488
31.
Comments about geometric complexity
............ 490
IV. An introduction to real-variable methods
.......... 491
32.
The Maximal function
..................... 491
33.
Covering lemmas
........................ 493
34.
Lebesgue points
......................... 495
35.
Differentiability almost everywhere
.............. 497
36.
Finding Lipschitz pieces inside functions
........... 502
37.
Maximal functions and snapshots
............... 505
38.
Dyadic cubes
.......................... 505
39.
The
Calderón-Zygmund
approximation
............ 507
40.
The John-Nirenberg theorem
................. 508
41.
Reverse
Holder
inequalities
.................. 511
42.
Two useful lemmas
....................... 513
43.
Better methods for small oscillations
............. 515
44.
Real-variable methods and geometry
............. 517
С
Paul Levy s Isoperimetric Inequality
519
D Systolically
Free Manifolds
531
Bibliography
545
Glossary of Notation
575
Index
577
|
| adam_txt |
Contents
Preface
to the French Edition
xi
Preface to the English Edition
xiii
Introduction: Metrics Everywhere
xv
Length Structures: Path Metric Spaces
1
A. Length structures
. 1
B. Path metric spaces
. 6
С
Examples of path metric spaces
. 10
D. Arc-wise isometries
. 22
2
Degree and Dilatation
27
A. Topological review
. 27
B. Elementary properties of dilatations for spheres
. 30
C. Homotopy counting Lipschitz maps
. 35
D. Dilatation of sphere-valued mappings
. 41
E+ Degrees of short maps between compact and
noncompact manifolds
. 55
3
Metric Structures on Families of Metric Spaces
71
A. Lipschitz and Hausdorff distance
. 71
B. The noncompact case
. 85
C. The Hausdorff-Lipschitz metric,
quasi-isometries, and word metrics
. 89
D+ First-order metric invariants
and ultralimits
. 94
E+ Convergence with control
. 98
Sj^Convergence and Concentration of Metrics and Measures
ИЗ
A. A review of measures and mm spaces
. 113
B.
Пд
-convergence
of mm spaces
. 116
C. Geometry of measures in metric spaces
. 124
ущ
Contents
D. Basic
geometry
of the space X
. 129
E.
Concentration phenomenon
. 140
F. Geometric invariants of measures
related to concentration
. 181
G. Concentration, spectrum, and
the spectral diameter
. 190
H
Observable distance
Ηλ
on the space X and concentration
Xn
->
X
. 200
I. The Lipschitz order on X, pyramids, and asymptotic con¬
centration
. 212
J. Concentration versus dissipation
. 221
4
Loewner Rediscovered
239
A. First, some history (in dimension
2) . 239
B. Next, some questions in dimensions
> 3. 244
С
Norms on homology and Jacobi varieties
. 245
D. An application of geometric integration theory
. 261
E+ Unstable systolic inequalities and filling
. 264
F+ Finer inequalities and systoles
of universal spaces
. 269
5
Manifolds with Bounded
Ricci
Curvature
273
A. Precompactness
. 273
B. Growth of fundamental groups
. 279
C. The first
Betti
number
. 284
D. Small loops
. 288
E+ Applications of the packing inequalities
. 294
F+ On the nilpotency of
πχ
. 295
G+ Simplicial volume and entropy
. 302
H+ Generalized simplicial norms and
the metrization of homotopy theory
. 307
1+
Ricci
curvature beyond coverings
. 316
6
Isoperimetric Inequalities and Amenability
321
A. Quasiregular mappings
. 321
B. Isoperimetric dimension of a manifold
. 322
C. Computations of isoperimetric dimension
. 327
D. Generalized quasiconformality
. 336
E+ The Varopoulos isoperimetric inequality
. 346
7
Morse Theory and Minimal Models
351
A. Application of Morse theory
to loop spaces
. 351
B. Dilatation of mappings between
simply connected manifolds
.357
Contents
ix
8+ Pinching and Collapse
365
A. Invariant classes of metrics
and the stability problem
. 365
B. Sign and the meaning of curvature
. 369
C. Elementary geometry of collapse
. 375
D. Convergence without collapse
. 384
E. Basic features of collapse
. 390
A "Quasiconvex" Domains
in Rn
393
В
Metric Spaces and Mappings Seen
at Many Scales
401
I. Basic concepts and examples
. 402
1.
Euclidean spaces, hyperbolic spaces, and ideas from analysis
402
2.
Quasimetrics, the doubling condition, and
examples of metric spaces
. 404
3.
Doubling measures and regular metric spaces, deformations
of geometry, Riesz products and Riemann surfaces
. 411
4.
Quasisymmetric mappings and deformations of geometry from
doubling measures
. 417
5.
Rest and recapitulation
. 422
II. Analysis on general spaces
. 423
6.
Holder continuous functions on metric spaces
. 423
7.
Metric spaces which are doubling
. 430
8.
Spaces of homogeneous type
. 435
9.
Holder continuity and mean oscillation
. 437
10.
Vanishing mean oscillation
. 439
11.
Bounded mean oscillation
. 443
III. Rigidity and structure
. 445
12.
Differentiability almost everywhere
. 445
13.
Pause for reflection
. 448
14.
Almost flat curves
. 448
15.
Mappings that almost preserve distances
. 452
16.
Almost flat hypersurfaces
. 455
17.
The Aao condition for doubling measures
. 458
18.
Quasisymmetric mappings and doubling measures
. 462
19.
Metric doubling measures
. 464
20.
Bi-Lipschitz embeddings
. 468
21.
Αχ
weights
. 470
22.
Interlude: bi-Lipschitz mappings between Cantor sets
. . . 471
23.
Another moment of reflection
. 471
24.
Rectifiability
. 471
25.
Uniform rectifiability
. 475
x
Contents
26.
Stories from the past
. 477
27.
Regular mappings
. 479
28.
Big pieces of bi-Lipschitz mappings
. 480
29.
Quantitative smoothness for Lipschitz functions
. 482
30.
Smoothness of uniformly rectifiable sets
. 488
31.
Comments about geometric complexity
. 490
IV. An introduction to real-variable methods
. 491
32.
The Maximal function
. 491
33.
Covering lemmas
. 493
34.
Lebesgue points
. 495
35.
Differentiability almost everywhere
. 497
36.
Finding Lipschitz pieces inside functions
. 502
37.
Maximal functions and snapshots
. 505
38.
Dyadic cubes
. 505
39.
The
Calderón-Zygmund
approximation
. 507
40.
The John-Nirenberg theorem
. 508
41.
Reverse
Holder
inequalities
. 511
42.
Two useful lemmas
. 513
43.
Better methods for small oscillations
. 515
44.
Real-variable methods and geometry
. 517
С
Paul Levy's Isoperimetric Inequality
519
D Systolically
Free Manifolds
531
Bibliography
545
Glossary of Notation
575
Index
577 |
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| edition | Repr. of the 2001 ed. |
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| id | DE-604.BV022258176 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:41:51Z |
| indexdate | 2024-07-09T20:53:31Z |
| institution | BVB |
| isbn | 9780817645823 0817645829 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015468863 |
| oclc_num | 85804037 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 |
| owner_facet | DE-355 DE-BY-UBR DE-20 |
| physical | XIX, 585 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Modern Birkhäuser classics |
| spelling | Gromov, Mikhail 1943- Verfasser (DE-588)119289830 aut Structures métriques des variétés Riemanniennes Metric structures for Riemannian and non-Riemannian spaces Misha Gromov. With app. by M. Katz ... Repr. of the 2001 ed. Boston [u.a.] Birkhäuser 2007 XIX, 585 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Modern Birkhäuser classics Orig. publ. as vol. 152 in the series Progress in mathematics Metrische ruimten gtt Riemann-ruimten gtt Riemannian manifolds Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015468863&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gromov, Mikhail 1943- Metric structures for Riemannian and non-Riemannian spaces Metrische ruimten gtt Riemann-ruimten gtt Riemannian manifolds Metrischer Raum (DE-588)4169745-5 gnd |
| subject_GND | (DE-588)4169745-5 |
| title | Metric structures for Riemannian and non-Riemannian spaces |
| title_alt | Structures métriques des variétés Riemanniennes |
| title_auth | Metric structures for Riemannian and non-Riemannian spaces |
| title_exact_search | Metric structures for Riemannian and non-Riemannian spaces |
| title_exact_search_txtP | Metric structures for Riemannian and non-Riemannian spaces |
| title_full | Metric structures for Riemannian and non-Riemannian spaces Misha Gromov. With app. by M. Katz ... |
| title_fullStr | Metric structures for Riemannian and non-Riemannian spaces Misha Gromov. With app. by M. Katz ... |
| title_full_unstemmed | Metric structures for Riemannian and non-Riemannian spaces Misha Gromov. With app. by M. Katz ... |
| title_short | Metric structures for Riemannian and non-Riemannian spaces |
| title_sort | metric structures for riemannian and non riemannian spaces |
| topic | Metrische ruimten gtt Riemann-ruimten gtt Riemannian manifolds Metrischer Raum (DE-588)4169745-5 gnd |
| topic_facet | Metrische ruimten Riemann-ruimten Riemannian manifolds Metrischer Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015468863&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gromovmikhail structuresmetriquesdesvarietesriemanniennes AT gromovmikhail metricstructuresforriemannianandnonriemannianspaces |