When collapsing Riemannian manifolds under suitable curvature conditions, two types of structure arise on the fibers: flat structures and nilpotent structures. This depends on the scale at which one is looking at the manifold, as explained for example in
Cheeger, Jeff. Structure theory and convergence in Riemannian geometry. Milan J. Math. 78 (2010), no. 1, 221–264.
Since these structures originate with the same (sequence of) manifolds, it is natural to ask if there is a relation between the (local) flat structure and nilpotent structure. Has this been made precise in the literature?
The relation at the level of a single fiber is clear; the question is whether there might be e.g., a global object incorporating both structures so that one could study the global relationship between them.