Collapsing vs. positive pinching
In 1986 Klingenberg and Sakai stated the following conjecture: let \(M\) be a closed simply connected manifold of dimension \(m\) and \(0<\delta\leq 1\). Then there exists \(i_0=i_0(M,\delta)>0\) such that the injectivity radius \(i_g\) of any \(\delta\)-pinched metric \(g\) on \(M\), i.e., any Riemannian metric with sectional curvature \(\delta\leq K_g\leq 1\), is bounded from below by \(i_g\geq i_0\). This conjecture is known to be true in some special cases; on the other hand, for odd \(m>3\) and \(\delta>0\), it is completely open. In the reviewed paper the authors study this question by collapsing and Alexandrov space techniques. They prove the following bounded version of the Klingenberg-Sakai conjecture: let \(M\) be a closed (topological) manifold and \(d_0\) be a metric on \(M\) and \(0<\delta\leq 1\). Then there exists \(i_0=i_0(M,d_0,\delta)>0\) such that the injectivity radius \(i_g\) of any \(\delta\)-pinched \(d_0\)-bounded metric \(g\) on \(M\), i.e., any Riemannian manifold metric with sectional curvature \(\delta\leq K_g\leq 1\) and \(dist_g(x,y)\leq d_0(x,y)\), is bounded from below by \(i_g\geq i_0\). Moreover, the authors also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on \(M\) with positively uniformly pinched curvature cannot converge to a metric space of strictly lower dimension.
scientific article; zbMATH DE number 1383540