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Stability of pure Nilpotent Structures on collapsed Manifolds

Collapsed manifolds with bounded sectional curvature (i.e., |sec|<1 and volume of every unit ball small) are characterized by Cheeger-Fukaya-Gromov's nilpotent structures. We focus on the stability problem on pure nilpotent structures.

We prove that if two metrics on a $n$-manifold of bounded sectional curvature are $L_0$-bi-Lipchitz equivalent and sufficient collapsed (depending on $L_0$ and $n$), then up to a diffeomorphism, the underlying nilpotent Killing structures coincide with each other or one is embedded into another as a subsheaf.

It improves Cheeger-Fukaya-Gromov's locally compatibility of pure nilpotent Killing structures for one collapsed metric of bounded sectional curvature to two Lipschitz equivalent metrics. As an application, we prove that those pure nilpotent Killing structures constructed by various smoothing method to a Lipschitz equivalent $\epsilon$-collapsed metric of bounded sectional curvature are uniquely determined by the original metric modulo a diffeomorphism.