Geometric structures of collapsing Riemannian manifolds. II
This article is the sequel to [the authors, in: Surveys in geometric analysis and relativity. Dedicated to Richard Schoen in honor of his 60th birthday. Somerville, MA: International Press; Beijing: Higher Education Press. 439--466 (2011; Zbl 1261.53043)]. The authors consider the case where a sequence of \(n\)-dimensional Riemannian manifolds \(M_i\) converges in the Gromov-Hausdorff sense to a limit \(X\) so that the curvature of the \(M_i\) is bounded (perhaps only away from some controlled subset). Exploiting the convergence of the frame bundles over the \(M_i\), a structure is developed over the limit space \(X\) which allows for a notion of smooth convergence as well as for carrying out analysis on \(X\). This is the \(N^*\) bundle over \(X\), which is an \(\mathrm O(n)\)-vector bundle over an \(\mathrm O(n)\)-manifold \(Y\) such that \(X\) is homeomorphic to \(Y/\mathrm O(n)\). Among other applications, this structure is used to establish a generalization of Gromov's almost-flat theorem to the case of almost Ricci-flat manifolds (with sectional curvature bounds).
scientific article; zbMATH DE number 6971598
scientific article