Isometric Representation of Lipschitz-Free Spaces over Connected Orientable Riemannian Manifolds

We show that the Lipschitz-Free Space over a connected orientable $n$-di\-men\-sio\-nal Riemannian manifold $M$ is isometrically isomorphic to a quotient of $L^1(M,TM)$, the integrable sections of the tangent bundle $TM$, if $M$ is either complete or lies isometrically inside a complete manifold $N$. Two functions are deemed equivalent in this quotient space if their difference has distributional divergence zero. This quotient is the pre-annihilator of the exact essentially bounded currents, and if $M$ is simply connected, one may replace ``exact'' with ``closed'' currents.