Abstract. We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry [12] and Weinstein's splitting theorem [14]. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.

Abstract. In this note we discuss (weak) dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Diractheoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [19], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [13].

In this note, we examine the bundle picture of the pullback construction of Lie algebroids. The notion of submersions by Lie algebroids is introduced, which leads to a new proof of the local normal form for lie algebroid transversals of [7], and which we use to deduce that Lie algebroids transversals concentrate all local cohomology.The locally trivial version of submersions by Lie algebroids S is then discussed, and we show that this notion is equivalent to the existence of a complete Ehresmann connection for S, extending the main result in [11].Finally, we show that locally trivial version of submersions by Lie algebroids gives rise to a system of local coefficients, which is an integral part of a version of the homotopy invariance of de Rham cohomology in the context of Lie algebroids, and we apply such local systems to extend the localization theorem of [8].

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