  Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we
analyze the dynamical optimal mass transport problem where the cost is given by
the sum of the kinetic energy and the relative entropy with respect to a
reference volume measure $e^{-V}dx$. Under the only assumption that the
prescribed marginals lie in $L^1(M)$, and a lower bound on the Ricci curvature,
we characterize the minimal curves as unique weak solutions of the optimality
system coupling the continuity equation with a backward Hamilton-Jacobi
equation (with source given by $\log (m)$). We give evidence that the entropic
cost enhances diffusive effects in the evolution of the optimal densities,
proving $L^1\to L^\infty$ regularization in time for any initial-terminal data,
and smoothness of the solutions whenever the marginals are positive and smooth.
We use displacement convexity arguments (in the Eulerian approach) and gradient
bounds from quasilinear elliptic equations. We also prove the convergence of
optimal curves towards the classical Wasserstein geodesics, as the entropic
term is multiplied by a vanishing parameter, showing that this kind of
functionals can be used to build a smoothing approximation of the standard
optimal transport problem.
