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Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from Below.

By: Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2017Copyright date: ©2017Edition: 1st edDescription: 1 online resource (174 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470442668
Subject(s): Genre/Form: Additional physical formats: Print version:: Nonsmooth Differential Geometry-An Approach Tailored for Spaces with Ricci Curvature Bounded from BelowDDC classification:
  • 516.3/6
LOC classification:
  • QA641 .G545 2017
Online resources:
Contents:
Cover -- Title page -- Introduction -- Aim and key ideas -- Overview of the content -- Some open problems -- Chapter 1. The machinery of ^{ }(\mm)-normed modules -- 1.1. Assumptions and notation -- 1.2. Basic definitions and properties -- 1.3. Alteration of the integrability -- 1.4. Local dimension -- 1.5. Tensor and exterior products of Hilbert modules -- 1.6. Pullback -- Chapter 2. First order differential structure of general metric measure spaces -- 2.1. Preliminaries: Sobolev functions on metric measure spaces -- 2.2. Cotangent module -- 2.2.1. The construction -- 2.2.2. Differential of a Sobolev function -- 2.3. Tangent module -- 2.3.1. Tangent vector fields and derivations -- 2.3.2. On the duality between differentials and gradients -- 2.3.3. Divergence -- 2.3.4. Infinitesimally Hilbertian spaces -- 2.3.5. In which sense the norm on the tangent space induces the distance -- 2.4. Maps of bounded deformation -- 2.5. Some comments -- Chapter 3. Second order differential structureof \RCD( ,∞) spaces -- 3.1. Preliminaries: \RCD( ,∞) spaces -- 3.2. Test objects and some notation -- 3.3. Hessian -- 3.3.1. The Sobolev space ^{2,2}(\X) -- 3.3.2. Why there are many ^{2,2} functions -- 3.3.3. Calculus rules -- 3.3.3.1. Some auxiliary Sobolev spaces -- 3.3.3.2. Statement and proofs of calculus rules -- 3.4. Covariant derivative -- 3.4.1. The Sobolev space ^{1,2}_{ }( \X) -- 3.4.2. Calculus rules -- 3.4.3. Second order differentiation formula -- 3.4.4. Connection Laplacian and heat flow of vector fields -- 3.5. Exterior derivative -- 3.5.1. The Sobolev space ^{1,2}_{}(Λ^{ } *\X) -- 3.5.2. de Rham cohomology and Hodge theorem -- 3.6. Ricci curvature -- Bibliography -- Back Cover.
Summary: The author discusses in which sense general metric measure spaces possess a first order differential structure. Building on this, spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting the author to define Hessian, covariant/exterior derivatives and Ricci curvature.
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Cover -- Title page -- Introduction -- Aim and key ideas -- Overview of the content -- Some open problems -- Chapter 1. The machinery of ^{ }(\mm)-normed modules -- 1.1. Assumptions and notation -- 1.2. Basic definitions and properties -- 1.3. Alteration of the integrability -- 1.4. Local dimension -- 1.5. Tensor and exterior products of Hilbert modules -- 1.6. Pullback -- Chapter 2. First order differential structure of general metric measure spaces -- 2.1. Preliminaries: Sobolev functions on metric measure spaces -- 2.2. Cotangent module -- 2.2.1. The construction -- 2.2.2. Differential of a Sobolev function -- 2.3. Tangent module -- 2.3.1. Tangent vector fields and derivations -- 2.3.2. On the duality between differentials and gradients -- 2.3.3. Divergence -- 2.3.4. Infinitesimally Hilbertian spaces -- 2.3.5. In which sense the norm on the tangent space induces the distance -- 2.4. Maps of bounded deformation -- 2.5. Some comments -- Chapter 3. Second order differential structureof \RCD( ,∞) spaces -- 3.1. Preliminaries: \RCD( ,∞) spaces -- 3.2. Test objects and some notation -- 3.3. Hessian -- 3.3.1. The Sobolev space ^{2,2}(\X) -- 3.3.2. Why there are many ^{2,2} functions -- 3.3.3. Calculus rules -- 3.3.3.1. Some auxiliary Sobolev spaces -- 3.3.3.2. Statement and proofs of calculus rules -- 3.4. Covariant derivative -- 3.4.1. The Sobolev space ^{1,2}_{ }( \X) -- 3.4.2. Calculus rules -- 3.4.3. Second order differentiation formula -- 3.4.4. Connection Laplacian and heat flow of vector fields -- 3.5. Exterior derivative -- 3.5.1. The Sobolev space ^{1,2}_{}(Λ^{ } *\X) -- 3.5.2. de Rham cohomology and Hodge theorem -- 3.6. Ricci curvature -- Bibliography -- Back Cover.

The author discusses in which sense general metric measure spaces possess a first order differential structure. Building on this, spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting the author to define Hessian, covariant/exterior derivatives and Ricci curvature.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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