Stability of pure Nilpotent Structures on collapsed Manifolds

The goal of this paper is to study the stability of pure nilpotent structures on a manifold associated to different collapsed metrics. 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 metric of bounded sectional curvature are uniquely determined by the original metric modulo a diffeomorphism.