TY - JOUR

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

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

AU - Miyaoka, Reiko

N1 - Funding Information:
This work was supported by Japan Society for the Promotion of Science (Grant No. 15H03616). The author appreciates the invitation to contribute this article to Science China Mathematics in Memory of Professor Zhengguo Bai.
Publisher Copyright:
© 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature.

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

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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 -