Stability of almost submetries

Xiaochun Rong, Shicheng Xu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


In this paper, we consider a triple of Gromov-Hausdorff convergence: and maps fi: Ai → Bi converge to a map f: A → B, where Ai are compact Alexandrov n-spaces and Bi are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψi: Ai → A, Φi: Bi → B such that f {ring operator} Ψi = Φi {ring operator} fi. When f is an ε-submetry with ε > 0, we obtain a sufficient condition for the stability in the case that Ai are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.

Original languageEnglish (US)
Pages (from-to)137-154
Number of pages18
JournalFrontiers of Mathematics in China
Issue number1
StatePublished - Jan 2011

All Science Journal Classification (ASJC) codes

  • Mathematics (miscellaneous)


  • (almost) submetry
  • Alexandrov space
  • Gromov-Hausdorff convergence
  • fiber bundle
  • stability


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