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Second order analysis on $(\mathscr P_{2}(M),W_{2})$

About this Title

Nicola Gigli, University of Bordeaux

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 216, Number 1018
ISBNs: 978-0-8218-5309-2 (print); 978-0-8218-8529-1 (online)
Published electronically: June 21, 2011
Keywords:Wesserstein distance, weak Riemannian structure
MSC: Primary 53C15, 49Q20

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Table of Contents


  • Introduction
  • Chapter 1. Preliminaries and notation
  • Chapter 2. Regular curves
  • Chapter 3. Absolutely continuous vector fields
  • Chapter 4. Parallel transport
  • Chapter 5. Covariant derivative
  • Chapter 6. Curvature
  • Chapter 7. Differentiability of the exponential map
  • Chapter 8. Jacobi fields
  • Appendix A. Density of regular curves
  • Appendix B. $C^1$ curves
  • Appendix C. On the definition of exponential map
  • Appendix D. A weak notion of absolute continuity of vector fields


We develop a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance $W_2$. Our discussion comprehends: definition of covariant derivative, discussion of the problem of existence of parallel transport, calculus of the Riemannian curvature tensor, differentiability of the exponential map and existence of Jacobi fields. This approach does not require any smoothness assumption on the measures considered.

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