https://neckpinch.com/authors/xiaochun-rong/New arXiv papers by Xiaochun Rong, from Neckpinch Tue, 03 Nov 2026 08:41:12 GMT https://neckpinch.com/papers/2211.09998/arxiv:2211.09998Fri, 18 Nov 2022 00:00:00 GMTXiaochun Rong — For ρ, v>0, we say that an n-manifold M satisfies local (ρ,v)-bound Ricci covering geometry, if Ricci curvature Ric_M≥ −(n-1), and for all x∈ M, vol(B_ρ(tilde x))≥ 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 extend the nilpotent fiber bundle theorem of Cheeger-Fukaya-Gromov on a collapsed n-manifold M of bounded sectional curvature to M of a local (ρ,v)-bound Ricci covering geometry, and M is close to a non-collapsed Riemannian manifold of lower dimension. The nilpotent fiber bundle theorem significantly improves fiber bundle theorem in [Hu], and it strengthens a nilpotent fiber bundle seen from [NZ] and implies the torus bundle in [HW], which are obtained under additional local or global topological conditions, respectively. Our construction of a nilpotent fibration requires a new proof for a result in [HKRX]: if an n-manifold M with local (1,v)-bound Ricci covering geometry has diameter <ε(n,v), a constant depends on n and v, then M is diffeomorphic to an infra-nilmanifold. The proof in [HKRX] is to show that the Ricci flows produces an almost flat metric, thus the result follows from the Gromov's theorem on almost flat manifolds. The new proof is independent of the Gromov's theorem, thus has which as a corollary. If the first Betti number b₁(M)=n, then M satisfies a (1,v)-bound Ricci covering geometry, thus M is diffeomorphic to a standard torus ([Co2]).Ricci flow(Non)collapsingHomogeneous spaces