We study stratified G-structures in ${ \cal N}=2$ compactifications of M-theory on eight-manifolds $M$ using the uplift to the auxiliary nine-manifold ${ \hat M}=M\times S^1$. We show that the cosmooth generalized distribution ${ \hat { \cal D}}$ on ${ \hat M}$ which arises in this formalism may have pointwise transverse or non-transverse intersection with the pull-back of the tangent bundle of $M$, a fact which is responsible for the subtle relation between the spinor stabilizers arising on $M$ and ${ \hat M}$ and for the complicated stratified G-structure on $M$ which we uncovered in previous work. We give a direct explanation of the latter in terms of the former and relate explicitly the defining forms of the $\mathrm{ SU}(2)$ structure which exists on the generic locus ${ \cal U}$ of $M$ to the defining forms of the $\mathrm{ SU}(3)$ structure which exists on an open subset ${ \hat { \cal U}}$ of ${ \hat M}$, thus providing a dictionary between the eight- and nine-dimensional formalisms.
Categories: hep-th math.DG