Foliated cohomology and infinitesimal deformations of developable foliations

Let [Formula: see text] be a foliation on a connected manifold [Formula: see text]. Denote by [Formula: see text] the tangent bundle to [Formula: see text] and [Formula: see text] its normal bundle. We say that [Formula: see text] is developable if there exists a normal [Formula: see text]-covering [Formula: see text] such that the pull-back [Formula: see text] of [Formula: see text] by [Formula: see text] is a locally trivial fibration [Formula: see text] on which [Formula: see text] acts by automorphisms (and then on [Formula: see text]). The fiber of [Formula: see text] is denoted [Formula: see text]. Suppose [Formula: see text] compact. We prove: (i) if [Formula: see text], the space [Formula: see text] of infinitesimal deformations of [Formula: see text] is naturally identified to the first cohomology space [Formula: see text] of the discrete group [Formula: see text] with values in the [Formula: see text]-module [Formula: see text] of smooth vector fields on [Formula: see text]; (ii) if [Formula: see text] has trivial cohomology, the foliated cohomology [Formula: see text] of [Formula: see text] with values in [Formula: see text] is isomorphic to the cohomology [Formula: see text] of [Formula: see text] with values in [Formula: see text]. Some examples and explicit computations are given.