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.