Dirac products and concurring Dirac structures

We discuss in this note two dual canonical operations on Dirac structures L and R—the tangent product \(L \star R\) and the cotangent product \(L \circledast R\). Our first result gives an explicit description of the leaves of \(L \star R\) in terms of those of L and R, surprisingly ruling out the pathologies which plague general “induced Dirac structures.” In contrast to the tangent product, the more novel cotangent product \(L \circledast R\) need not be Dirac even if smooth. When it is, we say that L and R concur. Concurrence captures commuting Poison structures and refines the Dirac pairs of Dorfman and Kosmann–Schwarzbach, and it is our proposal as the natural notion of “compatibility” between Dirac structures. The rest of the paper is devoted to illustrating the usefulness of tangent- and cotangent products in general, and the notion of concurrence in particular. Dirac products clarify old constructions in Poisson geometry, characterize Dirac structures which can be pushed forward by a smooth map, and mandate a version of a local normal form. Magri and Morosi’s \(P\Omega \)-condition and Vaisman’s notion of two-forms complementary to a Poisson structures are found to be instances of concurrence, as is the setting for the Frobenius–Nirenberg theorem. We conclude the paper with an interpretation in the style of Magri and Morosi of generalized complex structures which concur with their conjugates.