Concurring reduction schemes for Dirac structures

The notion of \emph{concurrence} was recently proposed as the natural compatibility relation between Dirac structures, generalizing the commutativity of two Poisson structures. We address the question of when a reduction scheme – that is, a way to induce a Dirac structure on a quotient of a submanifold – respects this relation. After characterizing the minimal scheme of \emph{Dirac reduction}, we prove that two concurring Dirac structures have concurring reductions whenever they share a common \emph{witness}, extending to Dirac geometry the reduction of the Marsden-Ra\cb{t}iu theorem. Two procedures for constructing such common witnesses are given, the second being the Dirac counterpart of Magri’s original recipe in bihamiltonian geometry. Examples drawn from Hamiltonian actions, Dirac-Nijenhuis manifolds, and complex Dirac structures conclude the paper and illustrate our methods.