#### How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

2. Complete and sign the license agreement.

3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046
Email: cust-serv@ams.org

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

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

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)
DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00619-2
Published electronically: June 21, 2011
Keywords:Wesserstein distance, weak Riemannian structure
MSC: Primary 53C15, 49Q20

View full volume PDF

View other years and numbers:

Chapters

• 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

### Abstract

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.