Inserted: 16 mar 2009
Last Updated: 9 feb 2010
Journal: Memoirs of the American Mathematical Society
We develop a rigorous second order analysis on the space of probability measures on a Riemannian manifold $M$ 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.
Keywords: Wasserstein, Transport problem, Covariant derivative