The Holonomy of Optimal Mass Transport: The Smooth Case

We prove that, on a smooth $n$-dimensional Riemannian manifold without boundary, any vector field can be written as a linear combination of, at most, $\max \{6,6n-3\}$ depth one Lie brackets of pairs of gradient vector fields. Utilizing this along with the strong Trotter property, we show that, if the manifold is also compact and connected, the group generated by diffeomorphic optimal transport maps is dense in the identity component of the diffeomorphism group.