Geometric structures of collapsing Riemannian manifolds. II. (English) Zbl 1403.53036
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).
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).
Reviewer: John Harvey (Cardiff)
MSC:
| 53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |
| 53C20 | Global Riemannian geometry, including pinching |
Keywords:
curvature bounds; Riemannian manifold; Riemannian orbifold; Gromov-Hausdorff limit; collapseCitations:
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