A generalized integrability problem for G-Structures

Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetilde\pi:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $\pi: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $\pi: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$. If $\widetilde\pi:\widetilde P\to \widetilde{H}/\widetilde{K}$ 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 are necessary and sufficient conditions for the solution of the corresponding generalized integrability problems.