On manifold of positive Ricci curvature with large diameter (Q911067)

The author constructs a sequence of complete Riemannian metrics \(\{g_ i\}_{i\in {\mathbb{N}}}\) on \(S^{n-\ell}\times S^{\ell}\) for any integer \(n\geq 5\) and \(\ell =2,...,n-2\) such that the Ricci curvature \(Ric_{M_ i}\geq n-1,\) the volume \(vol(M_ i)\geq const>0\) and the diameter \(d(M_ i)\to \pi\) as \(i\to \infty\) for \(M_ i:=(S^{n-\ell}\times S^{\ell},g_ i).\) These examples show that the maximal diameter theorem for positive Ricci curvature [see \textit{S. Y. Cheng}, Math. Z. 143, 289-297 (1975; Zbl 0329.53035)] cannot be naturally extended to the diameter perturbed case without further restrictions. As an analogue of the above he constructs examples of a complete open Riemannian metric on \({\mathbb{R}}^{n-\ell}\times S^{\ell}\) with Ricci nonnegative, asymptotically nonnegative curvature and large volume growth.