Singular Riemannian foliations and $\mathcal{I}$-Poisson manifolds

Abstract: We recall the notion of a singular foliation (SF) on a manifold $M$, viewed as an appropriate submodule of $\mathfrak{X}(M)$, and adapt it to the presence of a Riemannian metric $g$, yielding a module version of a singular Riemannian foliation (SRF). Following Garmendia-Zambon on Hausdorff Morita equivalence of SFs, we define the Morita equivalence of SRFs (both in the module sense as well as in the more traditional geometric one of Molino) and show that the leaf spaces of Morita equivalent SRFs are isomrophic as pseudo-metric spaces. In a second part, we introduce the category of $\mathcal{I}$-Poisson manifolds. Its objects and morphisms generalize Poisson manifolds and morphisms in the presence of appropriate ideals $\mathcal{I}$ of the smooth functions on the manifold such that two conditions are satisfied: $(i)$ The category of Poisson manifolds becomes a full subcategory when choosing $\mathcal{I}=0$ and $(ii)$ there is a reduction functor from this new category to the category of Poisson algebras, which generalizes coistropic reduction to the singular setting. Every SF on $M$ gives rise to an $\mathcal{I}$-Poisson manifold on $T^*M$ and $g$ enhances this to an SRF if and only if the induced Hamiltonian lies in the normalizer of $\mathcal{I}$. This perspective provides, on the one hand, a simple proof of the fact that every module SRF is a geometric SRF and, on the other hand, a construction of an algebraic invariant of singular foliations: Hausdorff Morita equivalent SFs have isomorphic reduced Poisson algebras.