Lectures on nonsmooth differential geometry.(English)Zbl 1452.53002
SISSA Springer Series 2. Cham: Springer (ISBN 978-3-030-38612-2/hbk; 978-3-030-38613-9/ebook). xi, 204 p. (2020).
The aim of the book under review is to present some basic results in the field of nonsmooth geometric analysis. The content of the monograph is divided into 6 chapters. In Chapter 1, the authors review the basics of measure theory focussing on the space \(L^0(\mathfrak{m})\) of Borel functions considered up to \(\mathfrak{m}\)-almost everywhere equality. They discuss about continuous, absolutely continuous and geodesic curves on metric spaces, and collect the main results concerning Bochner integration.
Chapter 2 starts with the definition of the concept of Sobolev space \(W^{1,2}(X)\) and its basic properties. Next, two alternative definitions of Sobolev spaces are described: the original approach by relaxation of Cheeger and that of Shanmugalingam based on the notion of 2-modulus of a family of curves. Chapter 3 is devoted to exposing the theory of normed modulus over metric measure spaces. The main operations on normed modulus are discussed in detail. Chapter 4 develops a first-order differential structure on general metric measure spaces. Some key concepts like the tangent and cotangent modules, the differential of a Sobolev map and divergence operator are introduced. The last part of this chapter deals with transformations of metric measure spaces, called maps of bounded deformation. Such maps are naturally associated with a concept of differential. In Chapter 5, the authors prove the existence and uniqueness of the gradient flow associated to any convex and lower semicontinuous functional defined on a Hilbert space, and discuss its application to the study of the heat flow defined as the gradient flow in \(L^2(\mathfrak{m})\) of the Cheeger energy.
The last chapter deals with the class of those metric spaces that satisfy the Riemannian curvature-dimension condition, known as RCD spaces. Then, a second-order differential calculus over these structures is developed. Finally, the concepts of Hessian, covariant derivative and exterior derivative over any RCD\((K,\infty)\) space are defined and the main properties of Ricci curvature operator are established. The book ends with two appendices: The first contains some functional analytical tools, while the second reports the solutions of the problems scattered through the text.
The book is written in a clear and precise style. The notions are well motivated and many examples are given.In the reviewer’s opinion, this monograph will be of great interest to Ph.D. students and researchers working in the field of nonsmooth differential geometry.
Chapter 2 starts with the definition of the concept of Sobolev space \(W^{1,2}(X)\) and its basic properties. Next, two alternative definitions of Sobolev spaces are described: the original approach by relaxation of Cheeger and that of Shanmugalingam based on the notion of 2-modulus of a family of curves. Chapter 3 is devoted to exposing the theory of normed modulus over metric measure spaces. The main operations on normed modulus are discussed in detail. Chapter 4 develops a first-order differential structure on general metric measure spaces. Some key concepts like the tangent and cotangent modules, the differential of a Sobolev map and divergence operator are introduced. The last part of this chapter deals with transformations of metric measure spaces, called maps of bounded deformation. Such maps are naturally associated with a concept of differential. In Chapter 5, the authors prove the existence and uniqueness of the gradient flow associated to any convex and lower semicontinuous functional defined on a Hilbert space, and discuss its application to the study of the heat flow defined as the gradient flow in \(L^2(\mathfrak{m})\) of the Cheeger energy.
The last chapter deals with the class of those metric spaces that satisfy the Riemannian curvature-dimension condition, known as RCD spaces. Then, a second-order differential calculus over these structures is developed. Finally, the concepts of Hessian, covariant derivative and exterior derivative over any RCD\((K,\infty)\) space are defined and the main properties of Ricci curvature operator are established. The book ends with two appendices: The first contains some functional analytical tools, while the second reports the solutions of the problems scattered through the text.
The book is written in a clear and precise style. The notions are well motivated and many examples are given.In the reviewer’s opinion, this monograph will be of great interest to Ph.D. students and researchers working in the field of nonsmooth differential geometry.
Reviewer: Gabriel Eduard Vilcu (Ploieşti)
MSC:
| 53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |
| 58-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis |
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 58A99 | General theory of differentiable manifolds |
Keywords:
metric measure space;Bochner integration;Sobolev space;normed modulus;tangent module;cotangent module;Sobolev map;divergence operator;gradient flow;Hilbert space;heat flow;RCD spaceCite
Full Text:DOI
