My Question :

What is the definition of the energy of a curve in Riemannian manifold with density?

Thank you.

*If the energy of a curve c(t) in a classical Riemannian manifold is given by the integral of g(c'(t),c'(t)), then the energy with density f is given by the integral of f^2 g(c'(t),c'(t)) — fm*

the transportation metric then have any special structure ? ]]>

J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 169 (2009), 903–991.

*Thanks, updated.*

There is a generalization of the Ricci curvarure to Finsler manifolds by Shin-ichi Ohta.

*Thanks, one with Sturm on ArXiv. —FM*