  We present a framework for the reduction of various geometric structures
extending the classical coisotropic Poisson reduction. For this we introduce
constraint manifolds and constraint vector bundles. A constraint Serre-Swan
theorem is proven, identifying constraint vector bundles with certain finitely
generated projective modules, and a Cartan calculus for constraint
differentiable forms and multivector fields is introduced. All of these
constructions will be shown to be compatible with reduction. Finally, we apply
this to obtain a reduction procedure for Lie (bi-)algebroids and Dirac
manifolds.
