Memoirs of the American Mathematical Society 2012; 154 pp; softcover Volume: 216 ISBN10: 0821853090 ISBN13: 9780821853092 List Price: US$77 Individual Members: US$46.20 Institutional Members: US$61.60 Order Code: MEMO/216/1018
 The author develops a rigorous second order analysis on the space of probability measures on a Riemannian manifold endowed with the quadratic optimal transport distance \(W_2\). The discussion includes: 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. Table of Contents  Introduction
 Preliminaries and notation
 Regular curves
 Absolutely continuous vector fields
 Parallel transport
 Covariant derivative
 Curvature
 Differentiability of the exponential map
 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
 Bibliography
