Dirac products and concurring Dirac structures

TL;DR

The paper introduces tangent and cotangent Dirac products, $L \star R$ and $L \circledast R$, and defines concurrence as the property that $L \circledast R$ is a Dirac structure. It proves that when $L \star R$ is Dirac, the leaves of $L$ and $R$ meet cleanly and are described by their intersections, while $L \circledast R$ need not be Dirac, motivating concurrence as the natural compatibility notion. The framework unifies and clarifies many structures in Poisson geometry, including Libermann's theorem, Magri–Morosi $P\Omega$-structures, and various normal forms, and extends to complex and generalized complex structures, linking Dirac, PN, and $\Omega N$ structures through the concurrence relation. These results yield a cohesive language for studying compatibility between Dirac structures, with applications to push-forwards, normal forms, and the interplay between Poisson, foliations, and closed 2-forms. Overall, concurrence provides a canonical, geometry-driven notion of compatibility that subsumes and extends classical notions such as Dirac pairs and $P\Omega$-structures.

Abstract

We discuss in this note two dual canonical operations on Dirac structures $L$ and $R$ -- the \emph{tangent product} $L \star R$ and the \emph{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 contangent product $L \circledast R$ need not be Dirac even if smooth. When it is, we say that $L$ and $R$ \emph{concur}. Concurrence captures commuting Poison structures, refines the \emph{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 $PΩ$-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.

Theorems & Definitions (83)

• Lemma 1
• proof
• Remark 1
• Definition 1
• Lemma 2
• proof
• Lemma 3
• Example 1
• Lemma 4
• proof