Geometry of Lagrangian submanifolds in complex hyper ...ohnita/paper/2015OhnitaProcKNURIRCM.pdf ·...

Proceedings of The Nineteenth International Workshop on

Hermitian-Grassmannian Submanifolds and Its Applications 19(2015) 283-309

Geometry of Lagrangian submanifolds in complex hyper-quadrics and the Gauss images of isoparametric hypersurfaces

Yoshihiro OhnitaOsaka City University Advanced Mathematical Institute, & Department of Mathe-matics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585,JAPANe-mail: [email protected]

(2010 Mathematics Subject Classification : Primary: 53C40; Secondary: 53C42, 53D12.)

Abstract. In this article we discuss geometry of Lagrangian submanifolds in complex

hyperquadrics. The Gauss images, i.e. images of the Gauss map, of isoparametric hy-

persurfaces in the standard sphere provides a nice class of compact minimal Lagrangian

submanifolds embedded in the complex hyperquadrics Qn(C). It is an interesting problem

to investigate the relationship between properties of such Lagrangian submanifolds and

the structure of isoparametric hypersurfaces. In this article we also explain our recent

results in geometry of such Lagrangian submanifolds. This work is a part of author’s joint

works with H. Iriyeh, H. Ma and R. Miyaoka on the Hamiltonian non-displaceability and

the Lagrangian Floer theory of Gauss images of isoparametric hypersurfaces.

1 Introduction

It is a fundamental and interesting problem to study submanifold geometry inHermitian symmetric spaces. The complex hyperquadrics Qn(C) is canonically iso-metric to the real Grassmann manifolds of oriented 2-dimensional vector subspacesGr2(R), which are compact Hermitian symmetric spaces SO(n+2)/(SO(2)×SO(n))

Key words and phrases: isoparametric hypersurfaces, Lagrangian submanifolds, com-

plex hyperquadrics, Gauss images, Hamiltonian non-displaceability, Floer homology.

* This work is partly supported by JSPS Grant-in-Aid for Scientific Research (S)

No. 23224002, (C) No. 15K04851 and the JSPS Program for Advancing Strategic Inter-

national Networks to Accelerate the Circulation of Talented Researchers “Mathematical

Science of Symmetry, Topology and Moduli, Evolution of International Research Network

based on OCAMI”.

283

284 Y. Ohnita

of rank 2. Differential geometry of real hypersurfaces in complex hyperquadics wasinvestigated by J. Berndt and Y.-J. Suh ([3]).

In this article we discuss geometry of certain Lagrangian submanifolds in com-plex hyperquadrics. It is well-known that the Gauss map G of any oriented hy-persurface Nn immersed in the unit standard sphere Sn+1(1) ⊂ Rn+2 is always aLagrangian immersion of Nn into an n-dimensional complex hyperquadric Qn(C),and if Nn has constant principal curvatures of Sn+1(1), then the Gauss mapG : Nn → Qn(C) is a minimal immersion of Nn into Qn(C) in the sense of vanish-ing the mean curvature vector field. By the fundamental theory of E. Cartan andMunzner, any hypersurface of Sn+1(1) with constant principal curvatures in Sn+1(1)is known to be uniquely extended to a compact oriented hypersurface embedded inSn+1(1) with constant principal curvatures, which is a so-called isoparametric hy-persurface.

The Gauss images, i.e. images of the Gauss map Ln = G(Nn), of isoparamet-ric hypersurfaces Nn in Sn+1(1) provides a nice class of compact minimal (andthus monotone) Lagrangian submanifolds embedded in the complex hyperquadricsQn(C). The properties of so obtained Lagrangian submanifolds in complex hyper-quadrics have been investigated via the structure of isoparametric hypersurfaces([17], [18], [27], [30]).

In this article we also explain our recent results in geometry of such mono-tone Lagrangian submanifolds in complex hyperquadrics. This work is a part ofthe author’s joint works with H. Iriyeh, H. Ma and R. Miyaoka. on Hamiltoniannon-displaceability and Lagrangian Floer theory of Gauss images of isoparametrichypersurfaces ([16]).

This article is organized as follows: In Section 2, we begin with fundamentalstructures for differential geometry and topology of complex hyperquadrics, curva-ture, following the paper of Berndt and Y.-J. Suh, and we discuss an elementaryexample of Lagrangian submanifolds in complex hyperquadrics. In Section 3, weexplain the Gauss map construction of Lagrangian submanifolds in complex hyper-quadrics from oriented hypersurfaces of the standard unit sphere, and the funda-mental properties, construction and classification of isoparametric hypersurfaces inthe standard sphere. We mention elementary properties and known results on theGuass images of isoparametric hypersurfaces as Lagrangian submanifolds in complexhyperquadrics. In the final section we explain our recent results on the Hamiltoniannon-displace ability of the Gauss images of isoparametric hypersurfaces. Moreoverwe also discuss the extrinsic homology of the Guass images of isoparametric hyper-surfaces, which is inspired by the Hamiltonian non-displaceability problem.

Throughout this article all manifolds are smooth and connected.

2 Geometry and topology of complex hyperquadrics Qn(C)

The n-dimensional complex hyperquadric Qn(C) is by definition a complex al-gebraic hypersurface of the (n + 1)-dimensional complex projective space CPn+1

defined by the homogeneous quadratic equation (z0)2+(z1)

2+ · · ·+(zn+1)2 = 0. It

Geometry of Lagrangian Submanifolds in Qn(C) 285

is canonically isometric to the real Grassmann manifold Gr2(Rn+2) of all oriented2-dimensional vector subspaces of Rn+2 as

CPn+1 ⊃ Qn(C) ∋ [a+√−1b]←→ a ∧ b ∈ Gr2(Rn+2) ⊂

2∧Rn+2,

where {a,b} is an orthonormal basis of [W ] compatible with its orientation. It isnaturally isomorphic to a compact Hermitian symmetric space SO(n+2)/(SO(2)×SO(n)) of rank 2 if n ≥ 2. Note that Q1(C) = Gr2(R3) ∼= S2 if n = 1. We use thenatural identification

Qn(C) ∼= Gr2(Rn+2) ∼= SO(n+ 2)/(SO(2)× SO(n)).

For each point [W ] ∈ Gr2(Rn+2), the tangent vector space of Gr2(Rn+2) at [W ]is known to be expressed as

T[W ]Gr2(Rn+2) ∼= Hom(W,W⊥),

where Rn+2 = W ⊕W⊥ is an orthogonal direct sum decomposition of Rn+2.Let gstd denote the SO(n + 2)-invariant Einstein-Kahler metric on Qn(C) =

Gr2(Rn+2) induced from the standard Euclidean inner product of Rn+2, whoseEinstein constant is equal to n. Let gKC denote the SO(n+ 2)-invariant Einstein-

Kahler metric on Qn(C) = Gr2(Rn+2) induced from the Killing-Cartan form ofSO(n+ 2), whose Einstein constant is equal to 1/2.

The cohomology ring of Qn(C) is known as follows (see [32]):

Proposition 2.1. Let m ∈ N.

(1) The cohomology ring of Q2m(C) over Z is given as

H∗(Q2m(C);Z) ∼= Z[t, s]/(tm+1 − 2ts, s2),

where t ∈ H2 and s ∈ H2m.

(2) The cohomology ring of Q2m−1(C) over Z is given as

H∗(Q2m−1(C);Z) ∼= Z[t]/(t2m) ∼= H∗(CP 2m−1;Z),

where t ∈ H2.

It is known that Q2m−1(C) is a cohomological complex projective space and thehomology group of Qn(C) over Z has no torsion.

We shall use the description of differential geometry of complex hyperquadricsin Berndt and Suh ([3]) . Let J be the complex structure of Qn(C) induced from the

286 Y. Ohnita

standard complex structure of CPn+1 and gQn be the Kahler metric of Qn(C) in-duced from the Fubini-Study metric of CPn+1 with constant holomorphic sectionalcurvatures 4.

Let π : Cn+2 ⊃ S2n+3(1)→ CPn+1 be the Hopf fibration. For each [z] ∈ Qn(C),Then we have an Hermitian orthogonal direct sum decomposition

Cn+1 = [z]⊕ [z]⊕HQz and TzS

2n+3(1) = [z]⊕HQz

and thus (dπ)z : [z] ⊕ HQz → T[z]CPn+1, (dπ)z : HQ

z → T[z]Qn(C) and (dπ)z :

[z] → T⊥[z]Qn(C) are linear isometries over R. We see that HQ

z is invariant under

the standard conjugation ( ) of Cn+2 and the shape operator Az of Qn(C) in CPn+1

relative to the normal vector z is given by

Az(w) = w for each w ∈ HQz∼= T[z]Qn(C).

ThenA :=

∪[z]∈Qn(C)

{Aλz | λ ∈ C}

defines a rank two real vector subbundle of the tangent vector bundle End(TQn(C)),which is parallel relative to the Levi-Civita connection of Qn(C). Because Qn(C)has the parallel second fundamental form in CPn+1. We define the S1-subbundleof A as

R :=∪

[z]∈Qn(C)

{Aλz | λ ∈ S1 ⊂ C} ⊂ A ⊂ End(TQn(C)),

which is also parallel relative to the Levi-Civita connection of Qn(C). Each elementof R yields a real structure on a tangent vector space of Qn(C). A real structureA ∈ R induces an orthogonal direct sum decomposition

TpQn(C) = V (A)⊕ JV (A),

where V (A) = {v ∈ TpQn(C) | Av = v} and JV (A) = {v ∈ TpQn(C) | Av = −v}.V (A) is a real vector subspace of the complex vector space TpQn(C) relative to thereal structure A and particular it is a Lagrangian vector subspace of TpQn(C).

For each real structure A ∈ A, the Riemannian curvature tensor field of Qn(C)is expressed as follows ([3]):

RQn(X,Y )Z = g(Y, Z)X − g(X,Z)Y

+ g(JY, Z)JX − g(JX,Z)JY − 2g(JX, Y )JZ

+ g(AY,Z)AX − g(AX,Z)AY

+ g(JAY,Z)JAX − g(JAX,Z)JAY.

Note that its Einstein constant is equal to 2n+ 1. In particular, for each X,Y, Z ∈Vp(A) we have

RQn(X,Y )Z = 2(g(Y,Z)X − g(X,Z)Y ) ∈ Vp(A).


Thus Vp(A) is curvature-invariant, that is, a Lie triple system. Hence for any realstructure A ∈ R, Vp(A) generates an n-dimensional totally geodesic Lagrangiansphere Sn of Qn(C). They all are congruent each other in Qn(C) under the actionof SO(n + 2). This Lagrangian sphere will appear concretely in Example 1 ofSection 3 too. It can be characterized among Lagrangian submanifolds in complexhyperquadrics as follows:

Proposition 2.2. Assume that a Lagrangian submanifold Ln of Qn(C) whose tan-

gent space at each point of Ln is real with respect to a real structure of A. Then Ln

is minimal in Qn(C). Moreover, if n ≥ 3, then Ln is totally geodesic and congruent

to a Lagrangian sphere Sn of Qn(C).

In order to prove this result we use the following lemma:

Lemma 2.3 ([26], p.180, Lemma 5.2). Suppose that S is a curvature-invariant

submanifold immersed in a locally symmetric space M . Then the Riemannian cur-

vature tensor field RM of M and the second fundamental form B of S in M satisfy

the equation

B(X,RM (X,Y )Y ) +B(Y,RM (Y,X)X)

= RM (B(X,X), Y )Y +RM (B(Y, Y ), X)X

+RM (X,B(X,Y ))Y +RM (Y,B(Y,X))X

(2.1)

for each X,Y ∈ TpS,

Here a submanifold S ofM is said to be curvature-invariant if RM (u, v)w ∈ TxSfor each x ∈ S and each u, v, w ∈ TxS.

We shall prove Proposition 2.2 by using Lemma 2.2.

Proof. For each orthonormal X,Y ∈ TpLn,

RM (X,Y )Y = g(Y, Y )X − g(X,Y )Y

+ g(JY, Y )JX − g(JX, Y )JY − 2g(JX, Y )JY

+ g(AY, Y )AX − g(AX,Y )AY

+ g(JAY, Y )JAX − g(JAX, Y )JAY

= X + g(Y, Y )X − g(X,Y )Y + g(JY, Y )JX − g(JX, Y )JY

= 2X.

288 Y. Ohnita

The left-hand side of the equation (2.1) is

B(X,RM (X,Y )Y ) +B(Y,RM (Y,X)X) = 2 (B(X,X) +B(Y, Y )).

We compute the right-hand side of the equation (2.1).

RM (X,B(X,Y ))Y = 2 g(JB(X,Y ), X)JY.

and

RM (B(X,X), Y )Y = B(X,X)− 2 g(JB(X,X), Y )JY.

Hence

RM (X,B(X,Y ))Y +RM (Y,B(Y,X))X

+RM (B(X,X), Y )Y +RM (B(Y, Y ), X)X

= B(X,X)− 2 g(JB(X,X), Y )JY.+B(Y, Y )− 2 g(JB(Y, Y ), X)JX

+ 2 g(JB(X,Y ), X)JY + 2 g(JB(Y,X), Y )JX

= B(X,X) +B(Y, Y ).

Thus 2 (B(X,X) + B(Y, Y )) = B(X,X) + B(Y, Y ). Hence we obtain B(X,X) +

B(Y, Y ) = 0. Therefore S is minimal in M . If n ≥ 3, then for any orthonormal

X,Y, Z ∈ TpM , we have

B(X,X) +B(Y, Y ) = 0,

B(Y, Y ) +B(Z,Z) = 0,

−B(X,X)−B(Z,Z) = 0,

and thus 2B(Y, Y ) = 0, that is, B(Y, Y ) = 0. Hence S is totally geodesic in M .

Remark 2.4. In the case when n = 2, under the identification Q2(C) ∼= (S2, J)×

(S2,−J), a 2-dimensional Lagrangian submanifold in Q2(C) satisfying the assump-

tion of Proposition 2.2 is nothing but a 1-dimensional complex submanifold of

(S2, J)× (S2, J).


3 Gauss images of isoparametric hypersurfaces in the sphere

Let Nn be an oriented hypersurface immersed in the (n + 1)-dimensional unitstandard hypersphere Sn+1(1) ⊂ Rn+2. The Gauss map G : Nn → Qn(C) is definedby

G : Nn ∋ p 7−→ [x(p) +√−1n(p)] = x(p) ∧ n(p) ∈ Qn(C) = Gr2(C).

Notice that a normal deformation Nt of N is parallel in Sn+1(1) if and only if Nt

and N have the same Gauss map (cf. [18]).

Let (x, e1, · · · , en,n) be a local adapted orthonormal frame field on N , where xis the position vector of points on N , {e1, · · · , en} is an orthonormal frame tangentto N , and n is a unit normal vector field to N in Sn+1(1). The structure equationsare given by

(3.1)

dx = θiei,

dei = ωijej + hijθjn− δijθjx,

dn = −hijθjei.

Here h = hijθi ⊗ θj denotes the second fundamental form of Nn in Sn+1(1). Let{κ1, · · · , κn} denote the principal curvatures of of Nn in Sn+1(1). We may choose{ei} such that hij = κiδij . By (3.1), the differential of the Gauss map at p ∈ Nn

(3.2) (dG)p(ei(p)) ∈ T[W ]Qn(C) = Hom(W,W⊥) ([W ] = G(p) = x(p) ∧ n(p))

can be described as

[(dG)p(ei(p))](x(p),n(p)) = (e1(p), · · · , ei(p), · · · , en(p))

0 −h1i

......

0 −hi−1i

1 −hii

0 −hi+1i

......

0 −hni

= (e1(p), · · · , ei(p), · · · , en(p))

0 0...

...0 01 −κi

0 0...

...0 0

.

(3.3)

Then we know that

(1) G : Nn → Qn(C) is always a Lagrangian immersion.

290 Y. Ohnita

(2) The induced metric on Nn by the Gauss map G is

G∗gstd =n∑

i=1

(1 + κ2i ) θi ⊗ θi,

and thus {1√

1 + κ2i

ei | i = 1, · · · , n

}is an orthonormal frame on Nn relative to the metric G∗gstd.

(3) The mean curvature form αH of the Lagrangian immersion G : Nn → Qn(C)is expressed as follows (Palmer [30]):

αH = d

(Im

(log

n∏i=1

(1 +√−1κi)

))= −d

(n∑

i=1

ζi

).

where set κi = cot ζi (i = 1, · · · , n) with 0 < ζi < π.

Particularly, if Nn has constant principal curvatures, then the Gauss map G : Nn →Qn(C) is a minimal Lagrangian immersion.

Suppose that Nn is an oriented hypersurface of Sn+1(1) with constant principalcurvatures, which is a so-called isoparametric hypersurface of Sn+1(1). Let g denotethe number of distinct principal curvatures of Nn and (m1, · · · ,mg) denote theirmultiplicities. By the fundamental theory of isoparametric hypersurface in thestandard sphere, it is well-known that

Proposition 3.1 ([22], [23]). (1) mi = mi+2, where i ∈ Z (mod g).

(2) Nn can be uniquely extended to a compact oriented embedded hypersurface

Sn+1(1) ∩ F−1(s) (∃s ∈ (−1, 1)) in Sn+1(1) defined by the so-called Cartan-

Munzer polynomial F = F (x1, · · · , xn+2), which is a homogeneous real alge-

braic polynomial of degree g on Rn+2 satisfying the PDE∆F = c r2g−2,

∥dF∥2 = g2 r2g−2,

where r2 = ∥x∥2 = (x1)2 + · · ·+ (xn+2)

2 and c =g2(m2 −m1)

2.

(3) Let Nk (k = ±1) be the focal manifolds of Nn, Here let the ring of coefficients

R =

Z if N1 and N−1 are both orientable,

Z2 otherwise.


Let µ = m1 +m−1. Then

Hq(Nk, R) =

R for q ≡ 0 (mod µ), 0 < q < n

R for q ≡ m−k (mod µ), 0 < q < n

0 otherwise.

Furthermore,

Hq(N,R) =

R for q = 0 or n

Hq(N+)⊕Hq(N−) for 1 ≤ q ≤ n− 1.

(4) g must be 1, 2, 3, 4 or 6.

In general, using the Cartan-Munzer polynomial, we can express the Gauss mapfrom an isoparametric hypersurface Nn to a complex hyperquadric Qn(C) as

G : Nn = Sn+1(1) ∩ F−1(0) ∋ (x1, · · · , xn+2) 7−→[x1 +

√−1g

∂F

∂x1(x) : · · · : xn+2 +

√−1g

∂F

∂xn+2(x)

]∈ Qn(C) ⊂ CPn+1.

Example 1. The Cartan-Munzer polynomial of degree 1 on Rn+2 is given as

F (x1, · · · , xp+1, xp+2, · · · , xn+2) := xn+2.

Then Nn = Sn+1(1) ∩ F−1(0) is the great sphere Sn(1) in Sn+1(1). The Gauss

map from Nn to a complex hyperquadric Qn(C) can be expressed as

G : Nn = Sn+1(1)∩F−1(0) = Sn(1) ∋ (x1, · · · , xn+1, 0) 7−→

[x1 : · · · : xn+1 :√−1 ] ∈ Qn(C) ⊂ CPn+1.

This Gauss image is nothing but a totally geodesic Lagrangian sphere Sn appeared

in Section 2.

Example 2. Let p, q be positive integers with p + q = n. The Cartan-Munzer

polynomial of degree 2 on Rn+2 is given as

F (x1, · · · , xp+1, xp+2, · · · , xn+2)

:=− (x1)2 − · · · − (xp+1)

2 + (xp+2)2 + · · ·+ (xn+2)

2.

292 Y. Ohnita

Consider an real quadratic cone of Rn+2 defined by

−(x1)2 − · · · − (xp+1)

2 + (xp+2)2 + · · ·+ (xn+2)

2 = 0.

Then its section by Sn+1(1)

Nn = Sn+1(1) ∩ F−1(0)

is a smooth hypersurface embedded in Sn+1(1) isometric to the Riemannian product

Sp( 1√2) × Sq( 1√

2), which is the Clifford hypersurface. The Gauss map from Nn to

a complex hyperquadric Qn(C) can be expressed as

G : Nn = Sn+1(1) ∩ F−1(0) ∋ (x1, · · · , xp+1, xp+2, · · · , xn+2) 7−→

[√−1x1 : · · · :

√−1xp+1 : xp+2 : · · · : xn+2] ∈ Qn(C) ⊂ CPn+1.

For an isoparametric hypersurface Nn, we call the image of the Gauss mapG : Nn → Qn(C), the Gauss image of Nn. Notice that every isoperametric hyper-surface belonging to an isoparametric family in Sn+1(1) has the same Gauss image.The Gauss image of isoparametric hypersurface is known to have the followingproperties:

Proposition 3.2 ([17], [18], [19], [27]). Let Ln = G(Nn) be the Gauss image of an

isoparametric hypersurface Nn in Sn+1(1). Then

(1) Ln = G(Nn) is a compact minimal Lagrangian submanifold embedded in

Qn(C) and the Gauss map G yields the covering map

G : Nn → Ln = G(Nn) ∼= Nn/Zg

with the Deck transformation group Zg.

(2) Ln = G(Nn) is a compact monotone and cyclic Lagrangian submanifold em-

bedded in Qn(C) whose minimal Maslov number ΣL is given by

ΣL =2n

g=

m1 +m2, if g is even,

2m1, if g is odd.

Moreover, Ln = G(Nn) is orientable if and only if2n

gis even.


A submanifold N of a Riemannian manifold (M, g) is said to be homogeneous(more precisely, extrinsically homogeneous) if N is obtained as an orbit of a con-nected Lie subgroup of Aut(M, g) through a point of M .

We know the two methods to construct explicitly isoparametric hypersurfacesin the standard sphere as follows:

(a) Principal orbits of the isotropy representations of Riemannian symmetricspaces (U,K) of rank 2: They provide all homogeneous isoparametric hy-persurfaces in the standard sphere (Hsiang-Lawson, R. Takagi-T. Takahashi)

(b) Algebraic construction by representations of the Clifford algebras (Cliffordsystems) in the case when g = 4, which are so-called isoparametric hy-persurfaces of OT-FKM type: They provide countably infinite examples ofnon-homogeneous isoparametric hypersurfaces in the standard sphere (Ozeki-Takeuchi, Ferus-Karcher-Munzner)

At present the classification of isoparametric hypersurfaces in the standard sphereis completed except for only the remaining case when g = 4 and (m1,m2) = (7, 8).

(i) If g = 1, then Nn is an n-dimensional great sphere or small sphere Sn(r) ofSn+1(1).

(ii) If g = 2, then Nn is the Clifford hypersurface Sm1(r1)× Sm2(r2) of Sn+1(1)

with (r1)2 + (r2)

2 = 1.

(iii) If g = 3, then m1 = m2 = 1, 2, 4 or 8 and Nn is homogeneous (E.Cartan).

(iv) If g = 6, then m1 = m2 = 1 or 2 (Abresch [1]) and Nn is homogeneous(Dorfmeister-Nehr [10], R.Miyaoka [21]).

(v) If g = 4, then (m1,m2) = (2, 2), (4, 5) or m1 + m2 + 1 is a multiple of2ϕ(m1−1) ([1], [11], [31]), where ϕ(l) := ♯{s | 1 ≤ s ≤ l, s ≡ 0, 1, 2, 4 mod 8}.Moreover, if (m1,m2) = (7, 8), Nn is homogeneous or of OT-FKM type(Cecil-Chi-Jensen [6], Immervoll [14], Chi [7], [8]).

From Proposition 3.2 and the classification theorems of isoparametric hypersurfaceswith g = 4 (see also [31]), it follows that Ln = G(Nn) is non-orientable if and onlyif Nn is one of the following isoparametric hypersurfaces:

(1) g = 2 and Nn ∼= Sm1 × Sm2 is an odd-dimensional Clifford hypersurface ofSn+1(1).

(2) g = 4 and Nn is one of the following isoparametric hypersurfaces:

(a) a homogeneous isoparametric hypersurface N18 = U(5)SU(2)×SU(2)×T 2 with

(m1,m2) = (4, 5) which is associated to (U,K) = (SO(10), U(5)),

(b) an isoparametric hypersurface of OT-FKM type except for a homoge-

neous one N4l = SO(2)×SO(2l+1)Z2×SO(2l−1) with (m1,m2) = (1, 2l − 1) which is

associated to (U,K) = (SO(2l + 3), SO(2)× SO(2l + 1)),

(c) an isoparametric hypersurface with (m1,m2) = (7, 8).

294 Y. Ohnita

In this way the Gauss images of isoparametric hypersurfaces provide concreteand abundant examples of compact Lagrangian submanifolds in complex hyper-quadrics of several types. The purpose of the author’s joint work with Hui Ma wasto study geometry of Lagrangian submanifolds in complex hyperquadrics from thisviewpoint, and recently we have done

(1) Classification of all compact homogeneous Lagrangian submanifolds in Qn(C)([17]). In particular from this result we obtained that any compact homoge-neous minimal Lagrangian submanifolds in Qn(C) is obtained as the Gaussimage of a homogeneous isoparametric hypersurface in Sn+1(1).

(2) Determination of the Hamiltonian stability of Gauss images of all homoge-neous isoparametric hypersurfaces ([17], [19], [20]). As one of our results weobtained that the Gauss image Ln = G(Nn) of a homogeneous isoprametrichypersurface Nn corresponding to a rank 2 symmetric space (U,K) is Hamil-tonian stable if and only if the disparity between m1 and m2 is smaller than3 or (U,K) = (E6, U(1) · Spin(10)) ((m1,m2) = (6, 9)).

It is still an open problem to determine the Hamiltonian stability of Gaussimages of non-homogeneous isoparametric hypersurfaces of OT-FKM type.

4 Hamiltonian non-displaceability of Gauss images of isoparametric hy-

persurfaces

Recently in the joint work of H. Iriyeh, H.Ma, R.Miyaoka and the author ([16]),we studied the Hamiltonian non-displaceability of Gauss images of isoparametrichypersurfaces in the standard sphere. In this section we shall mention about resultsof this joint work.

Let (M,ω) be a symplectic manifold. A diffeomorphism ϕ of M is called aHamiltonian diffeomorphism of (M,ω) if there are time-dependent HamiltoniansHt (t ∈ [0, 1]) and an isotopy ϕt : M → M (t ∈ [0, 1]) of M with ϕ = ϕ1 satisfyingthe Hamiltonian equation

dϕt(x)

dt= (XHt)ϕt(x) and ϕ0(x) = x

for each x ∈ M . Here XHt denotes the Hamiltonian vector field corresponding tothe Hamiltonian Ht relative to ω. Let Haml(M,ω) be the set of all Hamiltoniandiffeomorphism of (M,ω). Then Haml(M,ω) is a subgroup of Symp0(M,ω), whichis the identity component of the symplectic diffeomorphism group of (M,ω). It isknown that Haml(M,ω) = Symp0(M,ω) if H1(M ;R) = {0}.

A Lagrangian submanifold L in (M,ω) is said to be Hamiltonian non-displaceable if

L ∩ ϕ(L) = ∅ for any ϕ ∈ Haml(M,ω).

Otherwise L is said to be Hamiltonian displaceable. It is a fundamental questionin symplectic topology to study whether a Lagrangian submanifold of a symplecticmanifold is Hamiltonian non-displaceable or not.


In our joint work (IMMO) we showed the following result:

Theorem 4.1 (Iriyeh-Ma-Miyaoka-O. [16]). Let Nn be a compact oriented isopara-

metric hypersurface of the standard sphere Sn+1(1) in Rn+2. Then its Gauss image

Ln = G(Nn) is Hamiltonian non-displaceable in (Qn(C), ωstd), except for the fol-

lowing few remaining cases:

g = 3, (m1,m2) = (1, 1), n = 3, N3 =SO(3)

Z2 + Z2,

g = 4, (m1,m2) = (1, k), n = 2k + 2, N2k+2 =SO(2)× SO(k + 2)

Z2 × SO(k)(k ≥ 1)

g = 6, (m1,m2) = (1, 1), n = 6, N6 =SO(4)

Z2 + Z2.

In particular, we obtained

Corollary 4.2 ([16]). The Gauss image Ln = G(Nn) of any non-homogeneous

isoparametric hypersurface Nn of Sn+1(1) is Hamiltonian non-displaceable in

(Qn(C), ωstd).

We study the Lagrangian Floer (co)homology of Gauss images of isoparametrichypersurfaces. Our main tool from symplectic geometry is to use the spectralsequence for Lagrangian Floer (co)homology due to Y.-G. Oh ([25]), Biran ([4]) andDamian ([9]). By definition the non-vanishing of their Floer homology or lifted Floerhomology implies the Hamiltonian non-displaceablility. Our method was inspiredby Iriyeh’s work on symplectic topology of Lagrangian submanifolds in complexprojective spaces CPn with parallel second fundamental form ([15]).

Let L be a closed monotone Lagrangian submanifold embedded in a closed sym-plectic manifold (M,ω) with minimal Maslov number ΣL ≥ 2. Let ϕ ∈ Haml(M,ω).Assume that L and ϕ(L) intersect transversely, denoted by the symbol L t ϕ(L).Let CF (L, ϕ) be the Z2-free module over the intersection points of L ∩ ϕ(L). Theboundary operator is defined by

∂J(p) :=∑

q∈L∩ϕ(L)

n(p, q) q ,

where n(p, q) denotes the number of isolated Floer trajectories (J-holomorphicstrips) connecting p and q in M . The homology HF (L) := H∗(CF (L, ϕ); ∂J ) of thecomplex (CF (L, ϕ), ∂J ) is called the Floer homology of L and it is independent ofa choice of generic J ∈ Jreg(M,ω) and ϕ ∈ Haml(M,ω) (Floer [12], Y.-G. Oh [24]).

Y.-G. Oh ([25]) showed that there exists a spectral sequence for the Floer co-homology of monotone Lagrangian submanifolds. Let f be a Morse function on L.

296 Y. Ohnita

Denote by Cf∗ the Morse chain complex of f . Then the Floer differential can be

decomposed as∂J = ∂0 + ∂1 + · · ·+ ∂ν ,

where ∂0 counts small trajectories in a Weinstein neighborhood of L in M and ∂0can be identified with the Morse differential of f , and ∂1+· · ·+∂ν is the contributionof big trajectories with the operators ∂ℓ : Cf

∗ → Cf∗+ℓΣL−1 (ℓ = 1, · · · , ν). . The

spectral sequence {Ep,qr , dr} satisfying the following properties (see Biran [4]):

1. Ep,q0 = Cf

p+q−pΣL⊗ ΛpΣL , d0 = [∂L

0 ]⊗ 1, where Λ := Z2[T, T−1] is called the

Novikov ring, deg(T ) = ΣL, Λi ⊂ Λ denotes the vector subspace consisting

of all elements of deg i.

2. Ep,q1 = Hp+q−pΣL

(L,Z2)⊗ ΛpΣL , d1 = [∂L1 ]⊗ T−ΣL , where

[∂L1 ] : Hp+q−pΣL

(L;Z2)→ Hp+q−1−(p−1)ΣL(L;Z2)

is induced by ∂L1 .

3. For any r ≥ 1, Ep,qr has the form Ep,q

r = V p,qr ⊗ ΛpΣL with

dr = δr ⊗ T−rΣL : Ep,qr → Ep−r,q+r−1

r ,

where each V p,qr is a vector space over Z2 and δr : V p,q

r → V p−r,q+r−1r is a

homomorphism defined for every p, q and satisfies δr ◦ δr = 0. Moreover,

V p,qr+1 =

Ker(δr : V p,qr → V p−r,q+r−1

r )

Im(δr : V p+r,q−r+1r → V p,q

r ).

In particular, V p,q0 = CL

p+q−pΣL, V p,q

1 = Hp+q−pΣL(L;Z2), δ1 = [∂L

1 ].

4. Ep,qr collapses to Ep,q

ν+1 = Ep,qν+2 = · · · = Ep,q

∞ at (ν+1)-step and for any p ∈ Zwe have ⊕

q∈ZEp,q

∞∼= HF (L; Λ),

where ν =

[dimL+ 1

ΣL

].

Back to the Gauss image of an isoparametric hypersurfaces, since ν =[dimL+ 1

ΣL

]=

[(n+ 1)g

2n

], we get

Corollary 4.3 ([16]). For each Gauss image Ln = G(Nn) ⊂ Qn(C) of an isopara-

metric hypersurface with g ≥ 3 and any p, q ∈ Z, we have

(0) Ep,q1 = Ep,q

∞ (ν = 0) if and only if g = 1 and n ≥ 2.

(1) Ep,q2 = Ep,q

∞ (ν = 1) if and only if (g, n) = (1, 1), g = 2 or (g,m1,m2) =

(3, 2, 2), (3, 4, 4), (3, 8, 8).


(2) Ep,q3 = Ep,q

∞ (ν = 2) if and only if (g,m1,m2) = (3, 1, 1) or g = 4.

(3) Ep,q4 = Ep,q

∞ (ν = 3) if and only if (g,m1,m2) = (6, 1, 1) or (6, 2, 2).

In the case when g = 3, that is, Nn is a Cartan hypersurface, we showed

Lemma 4.4 ([16]). The Gauss image Ln = G(Nn) of each isoparametric hypersur-

face with g = 3 is a Z2-homology sphere ( i.e. Hk(Ln;Z2) = 0 for each 0 < k < n )

satisfying H1(Ln;Z) ∼= Z3.

The Gauss images of Cartan hypersurfaces provide new examples of LagrangianZ2-homology spheres embedded in Hermitian symmetric spaces of compact type.

This result is crucial to the proof of Theorem 4.1 in the case of g = 3. In the casewhen g = 3 and m1 = m2 = 2, 4 or 8, it follows from Lemma 4.4 that ∂J = ∂0 inthe spectral sequence and thus HF (Ln) ∼= H∗(L

n;Z2), particularly the inequality

♯(L ∩ ϕ(L)) ≥ rankH∗(Ln;Z2) = 2

holds for every ϕ ∈ Haml(Qn(C), ωstd) with L t ϕ(L).In the case when g = 4 or 6, we used the cohomological information of Nn (see

Lemma 3.1 (3)) and the spectral sequence for the lifted Floer homology HF L(L)(Damian [9]) to our covering map G : L = N → L = G(N) in order to show thenon-vanishing of the lifted Floer homology which implies the Hamiltonian non-displaceability.

More generally the following is conjectured by Hajime Ono, Iriyeh, Ma, Miyaokaand the author (IMMOO):

Conjecture ([16]). Any compact minimal Lagrangian submanifold embedded in

an irreducible Hermitian symmetric space of compact type is Hamiltonian non-

displaceable.

Finally we shall discuss the extrinsic homology of the Guass images of isopara-metric hypersurfaces.

Suppose that M = Qn(C) is the n-dimensional complex hyperquadric. Let Nn

be a compact oriented isoparametric hypersurface embedded in the unit standardsphere Sn+1(1) with g distinct principal curvatures and multiplicities (m1,m2). LetG : Nn → Qn(C) be the Gauss map of Nn into Qn(C). The Gauss image L = G(Nn)is a compact minimal (and thus monotone) Lagrangian submanifold embedded inQn(C). Denote by ι : L = G(Nn)→M = Qn(C) the inclusion map.

From now we shall discuss the following question on the Gauss images of isopara-metric hypersurfaces:

Question. Determine whether the induced homology homomorphism over Z2

ι∗ : Hn(L;Z2) −→ Hn(M ;Z2)

vanishes or not.

298 Y. Ohnita

Here note that Hn(L;Z2) = Z2 [L], where [L] denotes the fundamental class ofHn(L;Z2), and we know that

Hn(Qn(C);Z2) =

{0} if n = 1,

Z2 ⊕ Z2 if n ≥ 2 is even,

{0} if n ≥ 3 is odd.

The following result in symplectic topology suggests us a relationship of theabove property with the Hamiltonian non-displaceability of monotone Lagrangiansubmanifolds.

Theorem 4.5 (Albers [2]). Let (M2n, ω) be a monotone closed symplectic manifold.

Let L be a monotone compact Lagrangian submanifold with minimal Maslov number

ΣL ≥ 3. If L is Hamiltonian displaceable, then the induced homology homomorphism

ιast : Hk(L;Z2) −→ Hk(M ;Z2) vanishes for degrees k > dimL + 1 − ΣL. In

particular, ι∗[L] = 0 in Hn(M ;Z2).

The Gauss map G : Nn → Ln = G(Nn) ⊂ Qn(C) induces a commutativediagram of homology groups as follows:

Hk(Nn,R) Hk(L

n,R) Hk(Qn(C),R)6 6 6

−→ −→

−→ −→Hk(Ln,Z) Hk(Qn(C),Z)

Hk(Ln,Z2) Hk(Qn(C),Z2)

Hk(Nn,Z)

? ? ?Hk(N

n,Z2) −→ −→

And the Gauss map G : Nn → Ln = G(Nn) ⊂ Qn(C) also induces a commutativediagram of cohomology groups as follows:


Hn(Nn,R) Hn(Ln,R) Hn(Qn(C),R)6 6 6

←− ←−

←− ←−Hn(Ln,Z) Hn(Qn(C),Z)

Hn(Ln,Z2) Hn(Qn(C),Z2)

Hn(Nn,Z)

? ? ?Hn(Nn,Z2) ←− ←−

Proposition 4.6. Assume that M = Qn(C) and Ln = G(Nn) is the Gauss image

of an isoparametric hypersurface Nn except for the case when g = 1 and n ≥ 2.

Then the induced Z-homology homomorphism

ι∗ : Hk(L;Z) −→ Hk(M ;Z)

and the induced Z2-homology homomorphism

ι∗ : Hk(L;Z2) −→ Hk(M ;Z2)

vanish for degrees dimL > k > dimL + 1 − ΣL. If g = 1 and n ≥ 2, then for

k = 0 > 1 − n = dimL + 1 − ΣL the corresponding homology homomorphisms do

not vanish.

Proof. Since ΣL ≥ 2, we have n = dimL > dimL+ 1−ΣL. Suppose that dimL+

1− ΣL < 0, i.e. (1− 2g )n+ 1 < 0. Hence g = 1 and −n+ 1 < 0, that is, n ≥ 2. If

g = 1 and n ≥ 2, then for k = 0 > dimL+ 1− ΣL = 1− n,

ι∗ : H0(L;Z) ∼= Z −→ H0(M ;Z) ∼= Z

and

ι∗ : H0(L;Z2) ∼= Z2 −→ H0(M ;Z2) ∼= Z2

both are isomorphisms, and thus they both do not vanish.

Recall that for each k < n,

Hk(Qn(C);Z) =

Z [ω]k/2 if k is even,

{0} if k is odd.

300 Y. Ohnita

Assume that g ≥ 2 or (g, n) = (1, 1), and thus dimL + 1 − ΣL ≥ 0. The induced

Z-homology homomorphism

ι∗ : Hk(L;Z) −→ Hk(M ;Z)

vanishes for each 0 < k < n. Indeed, if k is odd, then Hk(M ;Z) = {0} and thus

ι∗ : Hk(L;Z) −→ Hk(M ;Z) vanishes. Assume that k = 2l > 0 is even. Then for

any [c] ∈ Hk(L;Z),

⟨ ι∗[c], [ω]l ⟩ =∫c

∧lω = 0.

Because c is a smooth k-cycle in a Lagrangian submanifold L of Qn(C).

Therefore, except for the case when g = 1 and n ≥ 2, for each k with n > k >

dimL+ 1− ΣL ≥ 0 the induced Z-homology homomorphism

ι∗ : Hk(L;Z) −→ Hk(M ;Z)

vanishes and hence the induced Z2-homology homomorphism

ι∗ : Hk(L;Z2) −→ Hk(M ;Z2)

vanishes.

From now on we shall discuss the above diagram in the case when k = n.In the case of n = 1, since M = Q1(C) ∼= S2 and thus H1(Q1(C);Z) = {0},

ι∗ : H1(L;Z) −→ H1(Q1(C);Z) = {0}

vanishes and thus

ι∗ : H1(L;Z2) −→ H1(Q1(C);Z2) = {0}

vanishes.In the case when n ≥ 3 is odd,

ι∗ : Hn(L;Z) −→ Hn(Qn(C);Z) = {0}

vanishes and thus

ι∗ : Hn(L;Z2) −→ Hn((Qn(C);Z2) = {0}

vanishes. According to the classification of isoparametric hypersurfaces in spheres,all odd-dimensional isoparametric hypersurfaces Nn in Sn+1(1) are given as


(1) g = 1 and n is odd,

(2) g = 2 and only one of m1 and m2 is odd,

(3) g = 3, n = 3 and (m1,m2) = (1, 1).

If the Gauss image Ln = G(Nn) is non-orientable, that is, 2n/g = m1 +m2 isodd (g must be even and thus g = 2 or 4), then Hn(L;Z) = {0}. Hence

ι∗ : Hn(L;Z) = {0} −→ Hn(Qn(C);Z)

vanishes, but

ι∗ : Hn(L;Z2) −→ Hn(Qn(C);Z2) = {0}

does not necessarily vanish. We do not know when it vanish or not.

Example 3. Suppose that g = 2 and (m1,m2) = (1, n − 1). Then Ln = G(Nn)

is homotopic to zero in Qn(C) (cf. [17, p.775, Theorem 4.1(4)]). The inclusion

ι : Ln → Qn(C) induces the homomorphism ι∗ : Hn(Ln,Z2) → Hn(Qn(C),Z2).

When n is odd (k is even), we have Hn(Qn(C),Z2) = {0}. When n is even (k is

odd), we have ι∗(Hn(Ln,Z2)) = {0}. Therefore ι∗[L

n] = 0 in Hn(Qn(C),Z2).

Example 4. Suppose that g = 4 and (m1,m2) = (1, k). Set n = 2(k + 1). The

inclusion ι : Ln → Qn(C) induces the homology homomorphism ι∗ : Hn(Ln,Z2)→

Hn(Qn(C),Z2). In this case Ln = G(Nn) ∼= SO(2)×SO(k)Z2×Z4×SO(k) converges to RP k+1 (see

[17, p. 775, Theorem 4.1(5)]), more precisely the inclusion map ι : Ln → Qn(C) is

homotopic to a smooth map Ln → RP k+1 ⊂ Qn(C). By the homotopy invariance of

the homology homomorphism and n = 2(k+ 1) > k+ 1, we have ι∗(Hn(Ln,Z2)) =

{0}. Therefore ι∗[Ln] = 0 in Hn(Qn(C),Z2).

Assume that n is even and m1 + m2 is even. Let n = 2ℓ ≥ 2. We mayassume that the Gauss image Ln = G(Nn) is oriented compatible with the originalorientation of Nn, and thus Hn(L

n;Z) ∼= Z.We choose the standard (symplectic) Kahler form ω and an invariant 2ℓ-form Φ

of Qn(C) such that

[ω] ∈ H2(Q2ℓ(C);R) and [Φ] ∈ H2ℓ(Q2ℓ(C);R)

correspond to s ∈ H2(Q2ℓ(C);Z) and t ∈ H2ℓ(Q2ℓ(C);Z), respectively.Under the identification

T[z]Qn(C) ∼= T[W ]Gr2(Rn+2) ∼= Hom(W,W⊥) ∼= Hom(R2,Rn) ∼= M(2, n;R),

302 Y. Ohnita

choose a standard basis of the cotangent vector space of Qn(C) ∼= Gr2(Rn+2) at apoint

{ ε11, ε12, · · · , ε1n, ε21, ε22, · · · , ε2n}.

The action of

A(θ) :=

[cos θ − sin θsin θ cos θ

]∈ SO(2) (θ ∈ R)

on the cotangent vector space is defined by

A(θ) ε1i = cos θ ε1i + sin θ ε2i (i = 1, · · · , n).

Then we have

A(θ) (ε11 ∧ ε12 ∧ · · · ∧ ε1n)

= (A(θ) ε11) ∧ (A(θ) ε12) ∧ · · · ∧ (A(θ) ε1n)

= (cos θ ε11 + sin θ ε21) ∧ (cos θ ε12 + sin θ ε22) ∧ · · · ∧ (cos θ ε1n + sin θ ε2n).

Now define an SO(2)× SO(n)-invariant n-form Φ on T[z]Qn(C) ∼= M(2, n;R) by

Φ =

∫ 2π

0

A(θ) (ε11 ∧ ε12 ∧ · · · ∧ ε1n)dθ

=

n∑k=0

∫ 2π

0

cosn−k θ sink θ dθ

×∑

1≤i1<···<ik≤n

ε11 ∧ · · · ∧ ε2i1 ∧ · · · ∧ ε2i2 ∧ · · · ∧ ε2ik ∧ · · · ∧ ε1n

We can observe that Φ = 0 if n is odd.Suppose that n is even. Let n = 2m. Then Φ defines a non-zero SO(2m + 1)-

invariant 2m-form on Q2ℓ(C), denoted also by Φ.

Φ =2m∑k=0

∫ 2π

0

cos2m−k θ sink θ dθ

×∑

1≤i1<···<ik≤2m

ε11 ∧ · · · ∧ ε2i1 ∧ · · · ∧ ε2i2 ∧ · · · ∧ ε2ik ∧ · · · ∧ ε1n

=m∑

k=0

∫ 2π

0

cos2m−2k θ sin2k θ dθ

×∑

1≤i1<···<i2k≤2m

ε11 ∧ · · · ∧ ε2i1 ∧ · · · ∧ ε2i2 ∧ · · · ∧ ε2i2k ∧ · · · ∧ ε1n

Since Q2ℓ(C) is a symmetric space, Φ is parallel with respect to the Levi-Civitaconnection of Q2m(C). Notice that Φ defines a calibration on Q2m(C) and thecorresponding calibrated submanifolds are congruent to totally geodesic Lagrangianspheres in Q2m(C) in Example 1 ([13]).


Let {κi | i = 1, · · · , n} be the principal curvatures ofNn and {kα | α = 1, · · · , g}denote the distinct principal curvatures of Nn with k1 < · · · < kg. Let {ei | i =1, · · · , n} be an orthonormal frame on N2m with An(ei) = κi ei (i = 1, · · · , n). We

know that

{1√1+κ2

i

ei | i = 1, · · · , n}

is an orthonormal frame on Nn with respect

to the induced metric by the Gauss map G : N2m → Qn(C).Suppose that n is even. Let n = 2m. Then the pull-back form of Φ by the

Gauss map G : N2m → Qn(C) is a 2m-form on N2m described as follows:

Theorem 4.7.

(G∗Φ)

(1√

1 + κ21

e1, · · · ,1√

1 + κ22m

e2m

)

=

∫ π

−π

sinm1 θ sinm2

(θ +

π

g

)· · · sinmg

(θ +

(g − 1)π

g

)dθ .

Proof. For each α = 1, 2, · · · , g, we set

kα = cot ζα =cos ζαsin ζα

, 0 < ζα < π .

Here note that sin ζα > 0 and 0 < ζ1 < · · · < ζg < π. By the isoparametric property

of Nn ([22]) we know that ζα = ζ1 + (α− 1)π

g(α = 1, · · · , g). Since

cos θ + sin θ kα = cos θ + sin θcos ζαsin ζα

=cos θ sin ζα + sin θ cos ζα

sin ζα=

sin(θ + ζα)

sin ζα

and

1√1 + κ2

1

sin ζα=

1√1 +

cos2 ζα

sin2 ζα

1√sin2 ζα

=1√

sin2 ζα + cos2 ζα= 1,

we compute

Φ(e1, · · · , e2m)

=1√

(1 + κ21) · · · (1 + κ2

2m)

∫ 2π

0

(cos θ + sin θ κ1) · · · (cos θ + sin θ κ2m) dθ

=1√

(1 + k21)m1 · · · (1 + k2g)

mg

∫ 2π

0

(cos θ + sin θ k1)m1 · · · (cos θ + sin θ kg)

mg dθ

304 Y. Ohnita

=

∫ 2π

0

sin(θ + ζ1) sin(θ + ζ2) · · · sin(θ + ζ2m) dθ

=

∫ 2π

0

sinm1(θ) sinm2(θ +π

g) · · · sinmg (θ +

(g − 1)π

g) dθ

=

∫ π

−π


g) · · · sinmg (θ +

(g − 1)π

g) dθ .

If m1 and m2 both are even, then

(G∗Φ)(e1, · · · , e2m) > 0 .

Hence Ln = G(Nn) is not homologous to zero in Qn(C) over Z. However we do notknow when Ln = G(Nn) is homologous to zero in Qn(C) over Z2 or not.

If g = 2 and n = 2m = m1 +m2, then

(G∗Φ)(e1, · · · , e2m) =

∫ π

−π


2) dθ

=

∫ π

−π

sinm1(θ) cosm2(θ) dθ .

If g = 2, n = 2m and m1 is odd, then (G∗Φ)(e1, · · · , e2m) = 0. Hence Ln =G(Nn) is oriented and homologous to zero in Qn(C) over Z and thus Z2.

If g = 2, n = 2m and m1 is even, then

(G∗Φ)(e1, · · · , e2m) > 0.

Hence Ln = G(Nn) is not homologous to zero in Qn(C) over Z. . However we donot know when Ln = G(Nn) is homologous to zero in Qn(C) over Z2 or not.

If g = 3 and m1 = m2 = 2, 4 or 8, then m = 3, 6 or 12 and

(G∗Φ)(e1, · · · , e2m) =

∫ π

−π


3) sinm1(θ +

2π

3) dθ > 0.

Hence Ln = G(Nn) is not homologous to zero in Qn(C) over Z. However we do notknow when Ln = G(Nn) is homologous to zero in Qn(C) over Z2 or not.

If g = 3 and m1 = m2 = 1, then it becomes

(G∗Φ)(e1, · · · , e2m)

=

∫ π

−π

sin(θ) sin(θ +π

3) sin(θ +

2π

3) dθ

=1

22

(3

∫ π

−π

sin(θ) cos2(θ) dθ −∫ π

−π

sin3(θ) dθ

)= 0.


Hence Ln = G(Nn) is oriented and homologous to zero in Qn(C) over Z and thusZ2.

If g = 4, then m = 2(m1 +m2) and

(G∗Φ)(e1, · · · , e2m)

=

∫ π

−π


4) sinm1(θ +

2π

4) sinm2(θ +

3π

4) dθ

=1

2m1+m2

∫ π

−π

cosm1(2θ) sinm2(2θ) dθ .

If g = 4 and m1 or m2 is odd, then (G∗Φ)(e1, · · · , e2m) = 0. Hence Ln = G(Nn)is homologous to zero in Qn(C) over Z, and thus Z2 if m1 and m2 both are odd. . Ifg = 4 and m1,m2 both are even, then (G∗Φ)(e1, · · · , e2m) > 0. Hence Ln = G(Nn)is not homologous to zero in Qn(C) over Z. However we do not know whetherLn = G(Nn) is homologous to zero in Qn(C) over Z2 or not.

If g = 6 and m1 = m2 = 1 or 2, then m = 3 or 6 and

(G∗Φ)(e1, · · · , e2m)

=

∫ π

−π


6) sinm1(θ +

2π

6)

sinm2(θ +3π

6) sinm1(θ +

4π

6) sinm2(θ +

5π

6) dθ

=

∫ π

−π


6) sinm1(θ +

π

3)

sinm2(θ +π

2) sinm1(θ +

2π

3) sinm2(θ +

5π

6) dθ

=

∫ π

−π


6) sinm1(θ +

π

3)

cosm1(θ) cosm1(θ +π

6) cosm1(θ +

π

3) dθ .

=1

23m1

∫ π

−π

sinm1(2θ) sinm1(2θ +π

3) sinm1(2θ +

2π

3) dθ .

If g = 6 and m1 = m2 = 2, then (G∗Φ)(e1, · · · , e2m) > 0 and hence Ln = G(Nn)is not homologous to zero in Qn(C) over Z. However we do not know whetherLn = G(Nn) is homologous to zero in Qn(C) over Z2 or not.

If g = 6 and m1 = m2 = 1, then Ln = G(Nn) is oriented and

(G∗Φ)(e1, · · · , e2m) =1

23

∫ π

−π

sin(2θ) sin(2θ +π

3) sin(2θ +

2π

3) dθ

=1

23

∫ π

−π

sin(2θ) sin(2θ +π

3) sin(−2θ + π

3) dθ = 0.

Hence Ln = G(Nn) is homologous to zero in Qn(C) over Z and thus Z2.From those arguments we observe that

306 Y. Ohnita

Proposition 4.8. The induced Z2-homology homomorphism

ι∗ : Hn(Ln;Z2) −→ Hn(Qn(C);Z2)

vanishes if Ln = G(Nn) is the Gauss image of an isoparametric hypersurface Nn

in the following list:

g = 1, n is odd.

g = 2, n = m1 +m2, m1 or m2 is odd.

g = 3, (m1,m2) = (1, 1), n = 3, N3 =SO(3)

Z2 + Z2.

g = 4, (m1,m2) = (1, k), n = 2k + 2, N2k+2 =SO(2)× SO(k + 2)

Z2 × SO(k)(k ≥ 1).

g = 6, (m1,m2) = (1, 1), n = 6, N6 =SO(4)

Z2 + Z2.

In general we recall that

H2m(Q2m(C),Z) ∼= Z [S2m]⊕ Z [Qm(C)]

andH2m(Q2m(C),Z2) ∼= Z2 [S

2m]⊕ Z2 [Qm(C)]

(see [5]). Assume that m1 and m2 both are even. If we express the element 0 =ι∗[L

n] ∈ Hn(Qn(C),Z) as

ι∗[Ln] = p [S2m] + q [Qm(C)], p, q ∈ Z

in Hn(Qn(C),Z). As L2m and S2m are Lagrangian submanifolds in Qm(C), we have

0 = ⟨ι∗[L2m], [ω]m⟩= p ⟨[S2m], [ω]m⟩+ q ⟨[Qm(C)], [ω]m⟩= q ⟨[Qm(C)], [ω]m⟩

Since Qm(C) is a complex submanifold in Qm(C), we have q = 0 and thus ι∗[L2m] =

p [S2m]. If p is odd, then ι∗[L2m] = [S2m] = 0 in H2m(Q2m(C),Z2). If p is even,

then ι∗[L2m] = 0 in H2m(Q2m(C),Z2). Note that in H2m(Q2m(C),Z), since s2 ≡ 0,

we have [S2m] · [S2m] = 0 and thus ι∗[L2m] · [S2m] = 0.

Question. In the case when m1 and m2 both are even, we know that in

H2m(Q2m(C),Z), we have ι∗[L2m] = 0 and

ι∗[L2m] = p [S2m] for some nonzero p ∈ Z.

Then compute this integer p explicitly.


Acknowledgements. This article is a note based on my two talks at the 19th Inter-national Workshop on Hermitian-Grassmannian Submanifolds and Its Applications& the 10th RIRCM-OCAMI Joint Differential Geometry Workshop at NIMS in Dae-jeon on October 26–28 in 2015. The author would like to thank Professor YoungJin Suh (director of Research Institute of Real and Complex Manifolds, RIRCM)and Dr. Hyunjin Lee for their kind hospitality and and excellent organization. Atthis workshop OCAMI has contracted an agreement of the academic cooperationwith RIRCM on October 26 in 2015. He also would like to thank Professor MikiyaMasuda for valuable suggestion and useful advice on related algebraic topology andtransformation group theory.

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