Approximations of Lipschitz maps via Ehresmann fibrations and Reeb’s sphere theorem for Lipschitz functions

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Abstract

We show, as our main theorem, that if a Lipschitz map from a compact Riemannian manifold M to a connected compact Riemannian manifold N, where dim M ≥ dim N, has no singular points on M in the sense of Clarke, then the map admits a smooth approximation via Ehresmann fibrations. We also show the Reeb sphere theorem for Lipschitz functions, i.e., if a closed Riemannian manifold admits a Lipschitz function with exactly two singular points in the sense of Clarke, then the manifold is homeomorphic to the sphere.

Original languageEnglish
Pages (from-to)521-548
Number of pages28
JournalJournal of the Mathematical Society of Japan
Volume74
Issue number2
DOIs
Publication statusPublished - 2022

Keywords

  • convex analysis
  • Ehresmann fibration
  • Lipschitz map
  • nonsmooth analysis
  • Reeb’s sphere theorem
  • smooth approximation

ASJC Scopus subject areas

  • Mathematics(all)

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