Transnormal functions on a Riemannian manifold

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21 Citations (Scopus)

Abstract

We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge-Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function.

Original languageEnglish
Pages (from-to)130-139
Number of pages10
JournalDifferential Geometry and its Application
Volume31
Issue number1
DOIs
Publication statusPublished - 2013 Feb

Keywords

  • Isoparametric hypersurface
  • Singular foliation
  • Transnormal function
  • Transnormal system

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology
  • Computational Theory and Mathematics

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