A Γ-structure on a manifold is a maximal atlas whose changes of coordinates take values in a Lie pseudogroup Γ. Various geometric structures (e.g. symplectic, complex and contact structures) fit in this framework, but there is no general definition of almost Γ-structure (e.g. almost symplectic, almost complex and almost contact structures) in terms of Γ. In this talk we are going to fill this gap by introducing the general definition of an almost Γ-structure, and presenting a characterisation of its formal integrability. This will be obtained by introducing the concept of principal Pfaffian bundle. We will draw inspiration from the theory of PDEs, from Poisson geometry, as well as from similar results in the theory of G-structures, which we recover as particular cases. This is joint work with Marius Crainic.

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