Canonical nilpotent structure under bounded Ricci curvature and Reifenberg local covering geometry over regular limits

Abstract

It is known that a closed collapsed Riemannian [Formula: see text]-manifold [Formula: see text] of bounded Ricci curvature and Reifenberg local covering geometry admits a nilpotent structure in the sense of Cheeger–Fukaya–Gromov with respect to a smoothed metric [Formula: see text]. We study the nilpotent structures over a regular limit space with optimal regularities that describe the collapsing of original metric [Formula: see text], and prove that they are uniquely determined up to a conjugation by diffeomorphisms with bi-Lipschitz constant almost [Formula: see text], and are equivalent to nilpotent structures arising from other nearby metrics [Formula: see text] with respect to [Formula: see text]’s sectional curvature bound.