Metric Measure Geometry [electronic resource] : Gromov’s Theory of Convergence and Concentration of Metrics and Measures / Takashi Shioya
Material type:
TextSeries: IRMA Lectures in Mathematics and Theoretical Physics (IRMA) ; 25Publisher: Zuerich, Switzerland : European Mathematical Society Publishing House, 2016Description: 1 online resource (194 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783037196588Subject(s): Differential & Riemannian geometry | Probability & statistics | Differential geometry | Measure and integration | Functions of a complex variable | Probability theory and stochastic processesOther classification: 53-xx | 28-xx | 30-xx | 60-xx Online resources: Click here to access online | cover image Summary: This book studies a new theory of metric geometry on metric measure spaces, originally developed by M. Gromov in his book “Metric Structures for Riemannian and Non-Riemannian Spaces” and based on the idea of the concentration of measure phenomenon due to Lévy and Milman. A central theme in this text is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov–Hausdorff topology and allows to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. This book makes advanced material accessible to researchers and graduate students interested in metric measure spaces.
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This book studies a new theory of metric geometry on metric measure spaces, originally developed by M. Gromov in his book “Metric Structures for Riemannian and Non-Riemannian Spaces” and based on the idea of the concentration of measure phenomenon due to Lévy and Milman. A central theme in this text is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov–Hausdorff topology and allows to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. This book makes advanced material accessible to researchers and graduate students interested in metric measure spaces.
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