Metric Structures for Riemannian and Non-Riemannian Spaces (eBook)

Contents 8
Preface to the French Edition 13
Preface to the English Edition 14
Introduction: Metrics Everywhere 16
1 Length Structures: Path Metric Spaces 21
A. Length structures 21
B. Path metric spaces 26
C Examples of path metric spaces 30
D. Arc- wise isometries 42
2 Degree and Dilatation 47
A. Topological review 47
B. Elementary properties of dilatations for spheres 50
C. Homotopy counting Lipschitz maps 55
D. Dilatation of sphere- valued mappings 61
E+ Degrees of short maps between compact and noncompact manifolds 75
3 Metric Structures on Families of Metric Spaces 91
A. Lipschitz and Hausdorff distance 91
B. The noncompact case 105
C. The Hausdorff- Lipschitz metric, quasi- isometries, and word metrics 109
D. First- order metric invariants quasi- isometries, and word metrics 114
E. Convergence with control 118
3 1/2 Convergence and Concentration of Metrics and Measures 133
A. A review of measures and mm spaces 133
B. D^^-convergence of mni- spaces 136
C. Geometry of measures in metric spaces 144
D. Basic geometry of the space 149
E. Concentration phenomenon 160
F. Geometric invariants of measures related to concentration 201
G. Concentration, spectrum, and the spectral diameter 210
H. Observable distance on the space and concentration 220
I. The Lipschitz order on A', pyramids, and asymptotic concentration 232
J. Concentration versus dissipation 241
4 Loewner Rediscovered 259
A. First, some history ( in dimension 2) 259
B. Next, some questions in dimensions > 3
C- Norms on homology and Jacobi varieties 265
D. An application of geometric integration theory 281
E+ Unstable systolic inequalities and filling 284
F+ Finer inequalities and systoles of universal spaces 289
5 Manifolds with Bounded Ricci Curvature 293
A. Precompactness 293
B. Growth of fundamental groups 299
C. The first Betti number 304
D. Small loops 308
E. Applications of the packing inequalities 314
G. Simplicial volume and entropy 322
H. Generalized simplicial norms and the metrization of homotopy theory 327
I. Ricci curvature beyond coverings 336
6 Isoperimetric Inequalities and Amenability 341
A. Quasiregular mappings 341
B. Isoperimetric dimension of a manifold 342
C. Computations of isoperimetric dimension 347
D. Generalized quasiconformality 356
E+ The Varopoulos isoperimetric inequality 366
7 Morse Theory and Minimal Models 371
A. Application of Morse theory to loop spaces 371
B. Dilatation of mappings between simply connected manifolds 377
8 Pinching and Collapse 385
A. Invariant classes of metrics and the stability problem 385
B. Sign and the meaning of curvature 389
C. Elementary geometry of collapse 395
D. Convergence without collapse 404
E. Basic features of collapse 410
Appendix A 413
Appendix B 421
I. Basic concepts and examples 422
II. Analysis on general spaces 443
III. Rigidity and structure 465
IV. An introduction to real-variable methods 511
Appendix C 539
Appendix D 551
Bibliography 565
Glossary of Notation 595
Index 597