A generalized integrability problem for G-structures

Given an n~ -dimensional manifold M~ equipped with a G~ -structure π~ : P~ → M~ , there is a naturally induced G-structure π: P→ M on any submanifold M⊂ 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 G~ -structures π~ : P~ → H~ / K~. If π~ : P~ → H~ / 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.