Almost flat manifolds are the solutions of bounded size perturbations of the equation $\mathrm{Sec} = 0$ ($\mathrm{Sec}$ is the sectional curvature). In a celebrated theorem, Gromov proved that the presence of an almost flat metric implies a precise topological description of the underlying manifold.

During this talk we will explain how, under lower sectional curvature bounds, to impose an $L_1$-pinching condition on the curvature is surprisingly rigid, leading indeed to the same conclusion as in Gromov's theorem under more relaxed curvature conditions (in particular, so weak that we are not allowed to use Ricci flow in the proof). We will describe which alternative techniques lead us to a successful proof, ans this will be sketched in detail. This is a joint work with B. Wilking.