000			03595cam a2200241 a 4500
001			vtls000001793
003			VRT
005			20250102224545.0
008			081110s1999 maua |b 001 0 eng
020			_a0817638989 (acidfree paper)
039		9	_a201402040100 _bVLOAD _c201007271208 _dmalmash _c200811101326 _dvenkatrajand _c200811100852 _dNoora _y200811100851 _zNoora
050	0	0	_aQA649 _b.G8313 1999
100	1		_aGromov, Mikhael, _d1943- _943675
240	1	0	_aEnglish
245	1	0	_aMetric Structures for Riemannian and Non-Riemannian Spaces / _cMikhail Gromov ; English translation by Sean Michael Bates ; with appendices by M. Katz, P. Pansu, and S. Semmes.
260			_aBoston : _bBirkhauser, _cc1999.
300			_axix, 585 p. : _bill. ; _c25 cm.
504			_aIncludes bibliographical references (p. [545]-574) and index.
505			_aPreface to the French Edition.- Preface to the English Edition.- Introduction: Metrics Everywhere.- Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.- Manifolds with Bounded Ricci Curvature.- Isoperimetric Inequalities and Amenability.- Morse Theory and Minimal Models.- Pinching and Collapse.- Appendix A: Quasiconvex Domains in Rn.- Appendix B: Metric Spaces and Mappings Seen at Many Scales.- Appendix C: Paul Levy's Isoperimetric Inequality.- Appendix D: Systolically Free Manifolds.- Bibliography.- Glossary of Notation.- Index.
520			_aMetric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov. The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov-Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy-Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity. The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous Green Book by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendicesoby Gromov on Levy's inequality, by Pansu on quasiconvex domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measuresoas well as an extensive bibliography and index round out this unique an
650		0	_aRiemannian manifolds. _943676
942			_2lcc _n0 _cBK
999			_c20031 _d20031