Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- PART 1 SHORT COURSES
- 1 Introduction to optimal transport theory
- 2 Models and applications of optimal transport in economics, traffic, and urban planning
- 3 Logarithmic Sobolev inequality for diffusion semigroups
- 4 Lecture notes on variational models for incompressible Euler equations
- 5 Ricci flow: the foundations via optimal transportation
- 6 Lecture notes on gradient flows and optimal transport
- 7 Ricci curvature, entropy, and optimal transport
- PART 2 SURVEYS AND RESEARCH PAPERS
- References
7 - Ricci curvature, entropy, and optimal transport
from PART 1 - SHORT COURSES
Published online by Cambridge University Press: 05 August 2014
Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- PART 1 SHORT COURSES
- 1 Introduction to optimal transport theory
- 2 Models and applications of optimal transport in economics, traffic, and urban planning
- 3 Logarithmic Sobolev inequality for diffusion semigroups
- 4 Lecture notes on variational models for incompressible Euler equations
- 5 Ricci flow: the foundations via optimal transportation
- 6 Lecture notes on gradient flows and optimal transport
- 7 Ricci curvature, entropy, and optimal transport
- PART 2 SURVEYS AND RESEARCH PAPERS
- References
Summary
Abstract
This chapter comprises the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition.
Introduction
This chapter is extended notes based on the author's lecture series at the summer school at Université Joseph Fourier, Grenoble: “Optimal Transportation: Theory and Applications.” The aim of these five lectures (corresponding to Sections 7.3–7.7) was to review the recent impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey in the same spirit with a slightly different selection of topics.
Optimal transport theory is concerned with the behavior of transport between two probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically defined from the distance of the metric space. Optimal transport naturally inherits the geometric structure of the underlying space; in particular Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we obtain Jacobi fields as their variational vector fields. The behavior of these Jacobi fields is controlled by the Ricci curvature as is usual in comparison geometry. In this way, a lower Ricci curvature bound turns out to be equivalent to a certain convexity property of entropy in the space of probability measures.
- Type
- Chapter
- Information
- Optimal TransportTheory and Applications, pp. 145 - 200Publisher: Cambridge University PressPrint publication year: 2014
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