Ricci Flow and Gromov Almost Flat Manifolds

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 $diam² |K| \leq \epsilon_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $|Rm|_{n/2}$ $C_S² \leq \varepsilon_n$.