We study collapsed manifolds with Ricci bounded covering geometry i.e., Ricci curvature is bounded below and the Riemannian universal cover is non-collapsed or consists of uniform Reifenberg points. Via Ricci flows' techniques, we partially extend the nilpotent structural results of Cheeger-Fukaya-Gromov, on collapsed manifolds with (sectional curvature) local bounded covering geometry, to manifolds with (global) Ricci bounded covering geometry.

In the study of collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem ([Fu1], [CFG]), and the stability for isometric compact Lie group actions on manifolds ([Pa], [GK]). The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry. IntroductionsIn the study of collapsed manifolds with bounded sectional curvature (| sec | ≤ 1), the following two results provide basic tools: a (singular) fibration theorem ([Fu1], [CFG]), and the stability for isometric compact Lie group actions on manifoldsThe main purpose of this paper is to (partially) generalize the two results to manifolds with local bounded Ricci covering geometry (Definition 0.2). Our two generalized results have been used in a recent work by Rong ([Ro4]).A fibration theorem in [Fu1] ([CFG]) says:Theorem 0.1. Given constants, n, i 0 > 0, there exist constant δ(n, i 0 ) > 0 (depending on n and i 0 ) and C(n) > 0, such that if a compact n-manifold M and a compact k-manifold N , k ≤ n, satisfywhere inj N denotes the injectivity radius of N and d GH denotes the Gromov-Hausdorff distance, then there is a smooth fiber bundle map, f : M → N , such that (0.1.1) f is a Ψ(δ|n, i 0 )-GHA (Gromov-Hausdorff approximation), where the function Ψ(δ|n, i 0 ) → 0 as δ → 0 while n and i 0 are fixed.Note that (0.1.1) and (0.1.3) imply that the induced metric on an f -fiber is almost flat, and thus each f -fiber is diffeomorphic to an infra-nilmanifold ([Gr], [Ru]). If one replaces N by a compact length space X and δ = δ(X), then there is a singular fibration map, f : M → X ([Fu2], [CFG]). For a r-distance ball at x ∈ M , B r (x), let π : ( B r (x),x) → (B r (x), x), denote the (incomplete) Riemannian universal cover. Put vol(B r (x)) vol(B r (x)), called the local rewinding volume of B r (x) ([Ro2]). Then vol(B r (x)) ≥ vol(B r (x)) and "=" if and only if B r (x) is simply connected. Definition 0.2. An n-manifold M is said to satisfy a local (r, v)-bound Ricci covering geometry, if there are constants, r, v > 0, such that Ric M ≥ −(n − 1), vol(B r (x)) ≥ v, ∀ x ∈ M.

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