TITLE: Nearly constant curvature implies "nearly isometric" to a space form?
QUESTION [7 upvotes]: It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally. 
My question is, is there a quantitative version of the above result? By this, I mean for example, given $\varepsilon>0$, there exists $\delta$ such that if $(M,g)$ satisfies 
 $\left|Rm\right|_g<\delta$ `, does there exist a local diffeomorphism $\phi$ to $\mathbb R^n$ such that $|\phi_*g-g_0|_{g_0}< \varepsilon$, where $g_0$ is the standard metric on $\mathbb R^n$?

REPLY [5 votes]: For the global question, and hyperbolic metrics, in dimension > 3 this is a result of Gromov, stated in his 1978 JDG paper, and in dimension 3 it is an unpublished result of Daryl Cooper, from the late nineties, and Gromov, independently, so while Tian might have a more general result, he is far from the first.