Collapsed manifolds with pinched positive sectional curvature.
Let \(M^{n}\) be a manifold of sectional curvature \(0<\delta \leq K_{M^{n}}\leq 1\), and let \(X\) be an Alexandrov space of curvature \(\geq -1\). If \(X\) has dimension \(<n\) and is not collapsed (i.e. the Hausdorff measure \(m_{H}(X)\) has a definite lower bound), \(S(X)\) is the set of singular point of \(X\) and the Gromov-Hausdorff distance \(d_{GH}(M^{n},X)\) of \(M^{n}\) and \(X\) is sufficiently small, depending only \(n\) and \(\delta \), then the problem that occurs is how the geometry and topology of \(M^{n}\) are reflected by \(\dim X\) and \(\text{codim} (S(X))\). Moreover, it is assumed that the Gromov-Hausdorff distance of \(M^{n}\) and \(X\) is less than \(\varepsilon \)(\(n,\delta \)) \(>0\). The author gives a characterization of a compact manifold \(M^{n}\) which admits a metric with \(0<\delta \leq K_{M^{n}}\leq 1\) such that the Gromov-Hausdorff distance \(d_{GH}(M^{n}\), \(X\)) is sufficiently small, depending on \(n\) and \(\delta \), where \(X\) is an Alexandrov space of the lowest dimension with \(n\) and \(\delta \) fixed. More exactly, the author proves the following result: if \(X\) has the lowest possible dimension \((n-1)/2\), then a covering space of \(M^{n}\) of order \(\leq (n+1)/2\) is diffeomorphic with a lens space \(S^{n}/Z_{q}\), such that \[ 0<c(n,\delta )[ \text{vol} (M^{n})] ^{-1}\leq q\leq \text{vol} (S_{\delta }^{n})[ \text{vol}(M^{n})] ^{-1}, \] where \(S_{\delta }^{n}\) is the sphere of constant curvature \(\delta \) and the constant \(c(n,\delta )\) bounds from below the volume of the pullback metric on \(S^{n}\). In fact, this theorem gives a characterization of a compact \(n\)-manifold which admits a \(\delta \)-pinched metric such that it is close to an Alexandrov space of the lowest possible dimension. Another main result of this paper is the following: Let \(M^{n}\) be a compact manifold of sectional curvature \(0<\delta \leq K_{M^{n}}\leq 1\), and let \(X\) be an Alexandrov space of curvature \(\geq -1\) and \(m_{H}(X)\geq m_{0}>0\). Suppose \(d_{GH}(M^{n},X)<\varepsilon (n,\delta ,m_{0})\). If \(X\) has non-empty boundary, then a covering space of \(M^{n}\) with order \(\leq (n+1)/2\) is diffeomorphic to a lens space \(S^{n}/Z_{q}\). Let \(M_{\delta }^{n}\) be the set of all compact manifolds with \(\delta \leq K\leq 1\) and \(M_{i}^{n}\) be a sequence in \(M_\delta^n\) such that \(M_{i}^{n}\rightarrow X\). Then \(X\) has empty boundary.
scientific article; zbMATH DE number 1465064