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 $dia{m}^2|K|\le {\epsilon }_n$ in the Gromov--Ruh Theorem is replaced by the substantially weaker condition $\Vert Rm{\Vert }_{n/2}$ $C_S^2\le {\epsilon }_n$.