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