Stability of e ε-Lipschitz and co-Lipschitz maps in Gromov-Hausdorff topology

Xiaochun Rong, Shicheng Xu

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10 Scopus citations


A submetry is a metric analogue of a Riemannian submersion, and an e ε-Lipschitz and co-Lipschitz map is a metric analogue of an ε-Riemannian submersion. The stability of submetries from Alexandrov spaces to Riemannian manifolds in the Gromov-Hausdorff topology can be viewed as a parametrized version of Perelman's stability theorem in Alexandrov geometry. In this paper, we will study the stability of e ε-Lipschitz and co-Lipschitz maps. Our approach is based on controlled homotopy theory and semi-concave functions on Alexandrov spaces. As applications of our stability results, we generalize fiber bundle finiteness results on Riemannian submersions and partially generalize the stability of submetries.

Original languageEnglish (US)
Pages (from-to)774-797
Number of pages24
JournalAdvances in Mathematics
Issue number2
StatePublished - Oct 1 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


  • Alexandrov spaces
  • Controlled homotopy
  • Gromov-Hausdorff topology
  • Semi-concave functions
  • Stability of fiber bundles


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