TY - JOUR

T1 - Moment maps and isoparametric hypersurfaces of OT-FKM type

T2 - In Memory of Professor Zhengguo Bai (1916–2015)

AU - Miyaoka, Reiko

PY - 2020

Y1 - 2020

N2 - 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).

AB - 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).

KW - 53C40

KW - 53D20

KW - Gauss map

KW - isoparametric hypersurface

KW - moment map

KW - spin action

UR - http://www.scopus.com/inward/record.url?scp=85091315178&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85091315178&partnerID=8YFLogxK

U2 - 10.1007/s11425-020-1746-2

DO - 10.1007/s11425-020-1746-2

M3 - Article

AN - SCOPUS:85091315178

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

ER -