Metric structures for Riemannian and non-Riemannian spaces

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Metric theory has experienced a significant transformation in recent decades, shifting its focus from real analysis to Riemannian geometry, algebraic topology, infinite groups, and probability theory. This new direction began with influential papers by Svarc and Milnor on group growth and Mostow's proof of lattice rigidity. Gromov further advanced the field by developing the asymptotic metric theory of infinite groups. The structural metric approach to Riemannian geometry, rooted in Cheeger's thesis, centers on the Gromov–Hausdorff distance, which organizes Riemannian manifolds into a connected moduli space. This framework allows for dimension collapse and reveals complex geometries, as demonstrated by Cheeger, Fukaya, Gromov, and Perelman. Gromov also introduced metric structures in homotopy theory, leading to new invariants like the simplicial volume, which influences mapping degrees between manifolds. Concurrently, Banach spaces and probability theory underwent a geometric evolution, driven by the Levy–Milman concentration phenomenon and the law of large numbers for metric spaces. Initial developments were presented in Gromov's Paris course, which became the renowned "Green Book" by Lafontaine and Pansu (1979). The current English translation has been enhanced with new material and includes four appendices by Gromov, Pansu, Katz, and Semmes, along with an extensive bibliography and index, making it a unique and valuable res