Moment maps and isoparametric hypersurfaces of OT-FKM type: In Memory of Professor Zhengguo Bai (1916–2015)

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Abstract

Associated with a Clifford system on ℝ2l, a Spin(m + 1) action is induced on ℝ2l. An isoparametric hypersurface N in S2l−1 of OT-FKM (Ozeki, Takeuchi, Ferus, Karcher and Münzner) type is invariant under this action, and so is the Cartan-Münzner polynomial F(x). This action is extended to a Hamiltonian action on ℂ2l. We give a new description of F(x) by the moment map μ: ℂ2l→ k*, where k≅ o(m+ 1 ) is the Lie algebra of Spin(m + 1). It also induces a Hamiltonian action on ℂP2l−1. We consider the Gauss map G of N into the complex hyperquadric ℚ2l−2(ℂ) ⊂ ℂP2l−1, and show that G(N) lies in the zero level set of the moment map restricted to ℚ2l−2(ℂ).

Original languageEnglish
JournalScience China Mathematics
DOIs
Publication statusAccepted/In press - 2020

Keywords

  • 53C40
  • 53D20
  • Gauss map
  • isoparametric hypersurface
  • moment map
  • spin action

ASJC Scopus subject areas

  • Mathematics(all)

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