A generalized integrability problem for G-structures

Given an (n) over tilde -dimensional manifold (M) over tilde equipped with a (G) over tilde -structure (pi) over tilde : (P) over tilde -> (M) over tilde, there is a naturally induced G-structure pi : P -> M on any submanifold M subset of (M) over tilde that satisfies appropriate regularity conditions. We study generalized integrability problems for a given G-structure pi : P -> M, namely the questions of whether it is locally equivalent to induced G-structures on regular submanifolds of homogeneous (G) over tilde -structures (pi) over tilde : (P) over tilde -> (H) over tilde/(K) over tilde. If (pi) over tilde : (P) over tilde -> (H) over tilde/(K) over tilde 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.