Calculus of Variations and Geometric Measure Theory
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N. Gigli

Second order analysis on (P_2(M),W_2)

created by gigli on 16 Mar 2009
modified on 09 Feb 2010


Accepted Paper

Inserted: 16 mar 2009
Last Updated: 9 feb 2010

Journal: Memoirs of the American Mathematical Society
Year: 2009


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


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