The Wasserstein--Ebin Metric: A Geometric Lift of Unbalanced Optimal Transport to the space of Riemannian metrics

Abstract: We introduce dynamic and static formulations that formally extend unbalanced optimal transport from the space of positive densities to the space of Riemannian metrics. The first construction is based on a dynamic variational formulation in which the evolution of a Riemannian metric is driven by transport together with a source term. Choosing the $L^2$-metric to penalize the transport vector field and the Ebin metric to penalize the source component yields a new Riemannian metric on the manifold of Riemannian metrics, which we call the Wasserstein--Ebin metric. Our main result shows that the volume map defines a Riemannian submersion from the Wasserstein--Ebin metric to the Wasserstein--Fisher--Rao metric on the space of smooth densities. In addition, we construct a Riemannian submersion from the automorphism group of the tangent bundle onto the space of Riemannian metrics, providing a generalization of Otto's geometric description for the Wasserstein metric to the setting of the Wasserstein--Ebin metric. To propose a static formulation of unbalanced optimal Riemannian metric transport, we introduce two Kullback--Leibler-type divergences on the space of Riemannian metrics: one inspired by matrix information geometry, and another related, through the volume map, to the classical Kullback--Leibler divergence on densities. Establishing a link between the static and dynamic formulations remains an open direction for future work.