Bimodule deformation of fibered manifolds and the HKR theorem

Abstract

We first want to consider the formal deformation of a fibered manifold $P \rightarrow M$ as a (bi-)module or subalgebra, where $M$ has a given differential star product. Consequently we want to find obstructions for the existence of a bimodule or subalgebra, which turns out to be the curvature of the fiber bundle. Since the order by order construction of this structures amounts to solving equations in the Hochschild cohomology of the smooth functions on $M$ with values in the differential operators on $P$, we proceed to computing this cohomology for the case of a smooth map $P \xrightarrow{p} M$ such that $p(P)$ is a closed submanifold of $M$.