A fundamental result concerning collapsed manifolds withbounded sectional curvature is the existence of compatible local nilpotentsymmetry structures whose orbits capture all collapsed directions of thelocal geometry CFG. The underlying topological structure is called anN-structure of positive rank. We show that if a manifold M admits suchan N-structure N, then M admits a one-parameter family of metrics g_ewith curvature bounded in absolute value while injectivity radii and thediameters of N-orbits away from the singular set of N uniformly convergeto zero as e→ 0. Moreover, g_eis N-invariant away from the singular set.This result extends collapsing results in CG1, Fu3 and G.
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