Gromov's Pinching Constant

TL;DR

This paper demonstrates that Gromov's pinching constant, which ensures a manifold is finitely covered by a nilmanifold under curvature and diameter constraints, depends on the dimension and decreases at least as fast as 12/n^2.

Contribution

It provides an explicit example showing the dimension dependence of Gromov's pinching constant, highlighting its decrease with increasing dimension.

Findings

Pinching constant depends on the dimension n.

The decrease of the constant is at least proportional to 1/n^2.

Explicit example illustrating the dimension dependence.

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 $\frac{12}{n^2}.$