Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces

TL;DR

This paper introduces a new geometric structure over the limit space of collapsing Riemannian manifolds, enabling global analysis and extending key theorems like Gromov's Almost Flat Theorem and Ricci pinching results.

Contribution

It proposes a novel structure over the limit space that complements existing N-structures, facilitating analysis and generalizing important geometric theorems.

Findings

Generalization of Gromov's Almost Flat Theorem

New Ricci pinching theorems for collapsing manifolds

Topological implications of the new structure

Abstract

In this paper we study collapsing sequences M_{i}-> X of Riemannian manifolds with curvature bounded or bounded away from a controlled subset. We introduce a structure over X which in an appropriate sense is dual to the N-structure of Cheeger, Fukaya and Gromov. As opposed to the N-structure, which live over the M_{i} themselves, this structure lives over X and allows for a convenient notion of global convergence as well as the appropriate background structure for doing analysis on X. This structure is new even in the case of uniformly bounded curvature and as an application we give a generalization of Gromov's Almost Flat Theorem and prove new Ricci pinching theorems which extend those known in the noncollapsed setting. There are also interesting topological consequences to the structure.