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On compact manifolds which are not simply connected, we prove the existenceof "fake" solutions to the optimal transportion problem. These maps preservevolume and arise as the exponential of a closed 1 form, hence appeargeometrically like optimal transport maps. The set of such solutions forms amanifold with dimension given by the first Betti number of the manifold. In theprocess, we prove a Hodge-Helmholtz decomposition for vector fields. The ideasare motivated by the analogies between special Lagrangian submanifolds andsolutions to optimal transport problems.

A McLean Theorem for the moduli space of Lie solutions to mass transport equations