Ricci Flow and Gromov Almost Flat Manifolds

TL;DR

This paper uses Ricci flow techniques to extend Gromov's almost flat manifold theorem, replacing a diameter and curvature bound with a weaker norm condition, thus broadening the class of manifolds covered.

Contribution

It introduces a generalized theorem for Gromov almost flat manifolds using Ricci flow, weakening the curvature conditions required.

Findings

The new theorem applies under weaker curvature conditions.

It broadens the class of manifolds satisfying the almost flat criteria.

The approach strengthens the connection between Ricci flow and geometric topology.

Abstract

We employ the Ricci flow to derive a new theorem about Gromov almost flat manifolds, which generalizes and strengthens the celebrated Gromov--Ruh Theorem. In our theorem, the condition $diam^2 |K| \leq \epsilon_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $\|Rm\|_{n/2}$ $ C_S^2 \leq \varepsilon_n$.