Formal frames and deformations of affine connections

We will introduce formal frames of manifolds, which are a generalization of ordinary frames. Their fundamental properties are discussed. In particular, canonical forms are introduced, and torsions are defined in terms of them as a generalization of the structural equations. It will be shown that the vanishing of torsions are equivalent to the realizability of given formal frames as ordinary frames. We will also discuss deformations of linear connections on tangent bundles. An application to deformations of foliations are then given.