Ricci curvature and rigidity

One proves the following gap theorem, involving the volume and the Ricci curvature : For any integer $n \ge 3$ and $d > 0$, there exists $\epsilon(n, d) > 0 such that the following holds. Let $(X, g_0 )$ be a $n$-dimensional hyperbolic compact manifold with diameter $\le d$ and let $Y$ be a compact manifold which admits a continuous map $f : Y \rightarrow X$ of degree one. Then Y has a metric $g$ such that $Ric_g \geq -(n - 1)g$ and $vol_g (Y ) \leq (1 + \epsilon) vol_{g_0} (X )$ if and only if $f$ is homotopic to a diffeomorphism.