Spaces on and beyond the boundary of existence
Given an integer \(n\geq 2\) and a real number \(k\leq K\), let \(M_k (n)\) (resp. \(M^K_k (n)\)) denote the class of complete, connected, pointed Riemannian \(n\)-manifolds with sectional curvature \(\geq k\) (resp. \(\geq k\) and \(\leq K\)). Then the description of the metric spaces in the Gromov-Hausdorff closure \(\overline {M_k (n)}\) and \(\overline {M^K_k (n)}\) may be a problem worthy to pursue. \textit{K. Grove} and \textit{P. Petersen} [J. Differ. Geom. 33, No. 2, 379-394 (1991; Zbl 0729.53045)]\ have shown that \(\overline {M_k (n)}\) is an Alexandrov space with curvature \(\geq k\) and dimension \(\leq n\), while \textit{K. Fukaya} [ibid. 28, No. 1, 1-21 (1988; Zbl 0652.53031)]\ proved that \(\overline {M^K_k (n)}\) looks like \(N/O(n)\) where \(N\) is a Riemannian manifold. After establishing the theorem stated below, the authors show in Theorems 2 and 3 that the converse of the above results is not true. The theorem mentioned above is as follows: Let \(M\) be a convex Riemannian manifold with bounded curvature on which a Lie group \(G\) acts as isometry in such a way that all orbits are closed. Then \(M/G\) with the natural quotient metric can be obtained as the limit of a sequence of Riemannian manifolds with a uniform lower sectional bound and fixed dimension in the pointed Gromov-Hausdorff topology.
scientific article; zbMATH DE number 828779