Moment maps and isoparametric hypersurfaces of OT-FKM type
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Special Issue on Differential Geometry
. ARTICLES .
Moment maps and isoparametric hypersurfaces of OT-FKM type In Memory of Professor Zhengguo Bai (1916–2015)
Reiko Miyaoka Institute of Liberal Arts and Sciences, Tohoku University, Sendai 9808576, Japan Email: [email protected] Received February 29, 2020; accepted July 17, 2020 Abstract
Associated with a Clifford system on R2l , a Spin(m + 1) action is induced on R2l . An isoparametric
hypersurface N in S 2l−1 of OT-FKM (Ozeki, Takeuchi, Ferus, Karcher and M¨ unzner) type is invariant under this action, and so is the Cartan-M¨ unzner 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 CP 2l−1 . We consider the Gauss map G of N into the complex hyperquadric Q2l−2 (C) ⊂ CP 2l−1 , and show that G(N ) lies in the zero level set of the moment map restricted to Q2l−2 (C). Keywords moment map, spin action, isoparametric hypersurface, Gauss map MSC(2010) 53C40, 53D20 Citation: Miyaoka R. Moment maps and isoparametric hypersurfaces of OT-FKM type. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1746-2
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Introduction
In [5], we give a moment map description of the Cartan-M¨ unzner polynomial F (x) for isoparametric hypersurfaces in S n+1 . For definition, see Section 4 of the present paper. Especially, for those of OT-FKM type associated with the representation of Clifford algebra, we use the moment map of the Spin(m + 1) action on T R2l ∼ = T ∗ R2l ∼ = C2l , 2l = n + 2. In this paper, we give a new description using the moment 2l ∗ map µ : C → k , k = o(m + 1) (see Theorem 5.2). Next, we projectify T ∗ R2l ∼ = C2l , and show that the Spin(m + 1) action is reduced to CP 2l−1 as a Hamiltonian action with the moment map µ : CP 2l−1 → k∗ given by (3.15). This Spin(m + 1) action on CP 2l−1 preserves the complex hyperquadric ∑ { } 2l 2 2l−1 Q2l−2 (C) = [z] ∈ CP z = 0 , j j=1
2l where Q2l−2 (C) is identified with the oriented two plane Grassmannian Gr+ 2 (R ), via [z] → x ∧ y, 2l 2l−2 2l−1 z = x + iy ∈ C \ {0}. For an isoparametric hypersurface N in S , we consider the Gauss map G : N → Q2l−2 (C) given by G(x) = x ∧ n where x ∈ N and n is the unit normal of N at x. Then we show that the Gauss image G(N ) is contained in the zero level set of its moment map ν : Q2l−2 (C) → k∗ , where ν = µ |Q2l−2 (C) (see Theorem 6.1). c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Preliminaries
The contents of the following two sections mainly follow from [6, Subsection 5.2]. When a Lie group G acts on a manifold M , and ζ is an element in the Lie algebra g of G, we have a vector field ζ˜ on M defined by d ˜ ζx = exp tζ · x, x ∈ M, (2.1) dt t=0
where exp : g → G denotes the exponential map and “·” denotes the action, which we omit later. A manifold M equipped with a closed no
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