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Using the dual formulation only, we show that regularity of unbalancedoptimal transport also called entropy-transport inherits from regularity ofstandard optimal transport. We then provide detailed examples of Riemannianmanifolds and costs for which unbalanced optimal transport is regular.Among allentropy-transport formulations, Wasserstein-Fisher-Rao metric, also calledHellinger-Kantorovich, stands out since it admits a dynamic formulation, whichextends the Benamou-Brenier formulation of optimal transport. Afterdemonstrating the equivalence between dynamic and static formulations on aclosed Riemannian manifold, we prove a polar factorization theorem, similar tothe one due to Brenier and Mc-Cann. As a byproduct, we formulate theMonge-Amp{\`e}re equation associated with Wasserstein-Fisher-Rao metric, whichalso holds for more general costs.

Regularity theory and geometry of unbalanced optimal transport