Differentiable rigidity under Ricci curvature lower bound

Abstract

In this article we prove a differentiable rigidity result. Let $(Y,g)$ and $(X,g_0)$ be two closed $n$-dimensional Riemannian manifolds ($n⩾3$), and let $f:Y\to X$ be a continuous map of degree $1$. We furthermore assume that the metric $g_0$ is real hyperbolic and denote by $d$ the diameter of $(X,g_0)$. We show that there exists a number $\epsilon :=\epsilon (n,d)>0$ such that if the Ricci curvature of the metric $g$ is bounded below by $-(n-1)g$ and its volume satisfies ${vol}_g\left(Y\right)⩽(1+\epsilon ){vol}_{g_0}\left(X\right)$, then the manifolds are diffeomorphic. The proof relies on Cheeger–Colding’s theory of limits of Riemannian manifolds under lower Ricci curvature bound.