Gromov's Pinching Constant

Abstract

In early 80's M.Gromov showed that there exists a constant $\epsilon$ such that any compact Riemannian manifold $M^n$ with $|K|_{M^n} \cdot diam^2(M^n) \leq \epsilon$ can be finitely covered by a nilmanifold. The present paper illustrates by an explicit example that the pinching constant $\epsilon$ depends on the dimension $n$ of the manifold, in particular, it decreases with the dimension at least as $\frac{12}{n^2}.$