A generalized integrability problem for G-structures

Given an \tilde n-dimensional manifold \tilde M equipped with a \tilde G-structure \tilde π : \tilde P → \tilde M, there is a naturally induced G-structure π : P → M on any submanifold M ⊂ \tilde M that satisfies appropriate regularity conditions. We study generalized integrability problems for a given G-structure π : P → M, namely the questions of whether it is locally equivalent to induced G-structures on regular submanifolds of homogeneous \tilde G-structures \tilde π : \tilde P → \tilde H/ \tilde K. If \tilde π : \tilde P → \tilde H/\tilde K is flat k-reductive, we introduce a sequence of generalized curvatures taking values in appropriate cohomology groups and prove that the vanishing of these curvatures is a necessary and sufficient condition for the solution of the corresponding generalized integrability problem.