For ρ, v > 0, we say that an n-manifold M satisfies local (ρ, v)-bound Ricci covering geometry, if Ricci curvature RicM ⩾ − (n − 1), and for any x ∈ M, \(\text{vol}(B_{\rho}(\tilde{x})) \geqslant v>0\) , where \(\tilde{x}\) is an inverse image of x on the (local) Riemannian universal cover of the ρ-ball at x. In this paper, we present a new proof of a maximally collapsed manifold with Ricci bounded covering geometry, which implies Gromov’s almost flat manifolds theorem. Based on the techniques developed in the new proof, we extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed n-manifold M of bounded sectional curvature to M of local (ρ, v)-bound Ricci covering geometry, and M is close to a non-collapsed Riemannian manifold of lower dimension. We also give applications of the nilpotent fibration theorem.
