The last chapter is devoted to the metric diffusion along foliations. The Wasserstein distance, theory of foliations, and foliated heat diffusion lie beneath this notion. We propose a modification of the initial Riemannian metric using the notions of heat diffusion and Wasserstein distance as a metric defined as the Wasserstein distance of Dirac masses concentrated at given points x, y ∈ M diffused, by foliated diffusion of measures, at time t > 0. Further on, we study the topology of a manifold equipped with metric diffused along foliation. The main results of the paper, that is the sufficient condition for Wasserstein–Hausdorff convergence of the manifolds equipped with metric diffused along foliation of dimension one and the theorem about the convergence of the manifolds carrying a foliation with empty bad set close the paper.
KeywordsRiemannian Manifold Hausdorff Distance Compact Riemannian Manifold Dirac Mass Hausdorff Topology