Dorfman connections of Courant algebroids and the Atiyah class of Manin pairs
The aim of this work is to introduce the Atiyah class of Manin pairs. For this we construct an algebra of polydifferential operators on tensor products of a Courant algebroid $E$ with values in the endomorphism bundle of a smooth vector bundle $B$, predual of $E$, extending the construction and standard complex of the Courant-Dorfman algebra of $E$. It is shown that the Cartan calculus, curvatures of induced connections and basic differential geometric identities of a Dorfman connection of $E$ on $B$ make sense in this algebra. Starting from a flat Dorfman connection of a Dirac subbundle $L\subset E$ on $B$, we define compatible extensions to a Dorfman connection of $E$ on $B$ and construct an Atiyah class measuring the obstruction to compatibility. The Atiyah classes of foliations, Lie bialgebras and complex manifolds are shown to be contained in this construction.