GEOMETRY OF LAGRANGIAN SUBMANIFOLDSAND ISOPARAMETRIC HYPERSURFACESYOSHIHIRO OHNITA

Abstract. In this expository article we shall focus on a relationship between Lagrangiansubmanifold theory in symplectic geometry and isoparametric hypersurface theory in Riemannian geometry. First we explain some basic notions, invariants and results of Lagrangian submanifolds in symplectic manifolds and in Kähler manifolds. Secondly we
discuss theory of isoparametric hypersurfaces in spheres and some necessary results onthem. We give attention to Lagrangian submanifolds in complex hyperquadrics obtained
from isoparametric hypersurfaces in spheres. The “Gauss map”of an oriented hypersurfaces in a unit sphere provides Lagrangian submanifolds in complex hyperquadrics. The
Gauss images of isoparametric hypersurfaces in spheres are the main subject of the author’s recent joint work with Hui Ma (Tsingua University, Beijing). We obtain that theGauss image of a compact oriented isoparametric hypersurface with g distinct constantprincipal curvatures in a unit sphere S n+1 (1) is a compact minimal Lagrangian submanifold embedded in Qn (C) and a monotone, cyclic Lagrangian submanifold with minimal
Maslov index equal to 2n/g. Moreover we mention further related topics and problems.

Introduction
In this expository article we shall focus on a relationship between Lagrangian submanifold theory in symplectic geometry and isoparametric hypersurface theory in Riemannian
geometry.
In symplectic geometry in these twenty years there has been remarkable much progresssuch as Floer cohomology theory for intersections of Lagrangian submanifolds due to K.
Fukaya, Y.-G. Oh, H. Ohta and K. Ono, (FOOO [9], etc.). Inspired by such progress in
symplectic geometry, recently more several developments are made by diﬀerential geometers in the study of Lagrangian submanifolds in specific Kähler manifolds, such as complex
space forms, Hermitian symmetric spaces, generalized flag manifolds, toric manifolds andso on. Submanifold theory is an area of the longest history in Riemannian geometry, andit can provide various techniques and so many examples to such a study of Lagrangiansubmanifolds.
The isoparametric hypersurface is one of most fundamental and attractive subjects insubmanifold theory. An isoparametric hypersurface of a Riemannian manifold M is alevel hypersurface for a regular value of a smooth function f on M satisfying the partial
This article is based on my lectures at the 14th International Workshop on “Diﬀerential Geometry and Related
Fields”and the 3rd KNUGRG-OCAMI Diﬀerential Geometry Workshop at Kyungpook National University inNovember, 2010.
2010 Mathematics Subject Classification. Primary : 53C40; Secondary : 53D12, 53C42.
Key words and phrases. Lagrangian submanifold, isoparameric hypersurface, minimal Maslov number.
Partially supported by JSPS Grant-in-Aid for Scientific Research (A) No. 19204006 and and the Priority
Research of Osaka City University “Mathematics of knots and wide-angle evolutions to scientific objects ”.
1

diﬀerential equations:
(0.1)




 ∆¯ f = S ( f ), ∥∇¯ f ∥2 = T ( f ).

Such a function f is called an isoparametric function. It is well-known that an isoparametric hypersurface of a real space form M, i.e. a Riemannian manifold of constant sectional
curvatures, is nothing but a hypersurface with constant principal curvatures. Nowadays theisoparametric hypersurface theory is a well-developed area as an very interesting class ofnice smooth manifolds. The excellent survey articles on isoparametric hypersurfaces are
G. Thorbergsson [39], and T. E. Cecil [4].
In this article, first we recall some basic notions, invariants and results of Lagrangiansubmanifolds in general symplectic manifolds and Lagrangian subamifolds in Kähler manifolds. Secondly we review theory of isoparametric hypersurfaces in spheres and we prepare some fundamental results which are applied to our research.
Now we should notice the construction of Lagrangian submanifolds related to isoparametric hypersurfaces. Especially we shall give attention to Lagrangian submanifolds incomplex hyperquadrics obtained from isoparametric hypersurfaces in spheres.
The n-dimensional complex hyperquadrics Qn (C) is a compact algebraic hypersurface
of a complex projective space CPn+1 defined by the homogeneous quadratic equation(z0 )2 + (z1 )2 + · · · + (zn+1 )2 = 0. It can be identified with the real Grassmann manifold
˜ 2 (Rn+2 ) of oriented 2-dimensional vector subspaces of Euclidean vector space Rn+2Gr˜ 2 (Rn+2 ) in the natural way. It has the homogeneous space expression Qn (C)  Gr
S O(n+2)/(S O(2)×S O(n)) and it is a compact rank 2 Hermitian symmetric space equippedwith the standard Einstein-Kähler manifold of positive Einstein constant. Q1 (C)  S 2 andQ2 (C)  S 2 × S 2 and Qn (C) is irreducible if n ≥ 3. Note that the complex hyperquadrics
Qn (C) is considered as a compactification of the tangent vector bundle T S n (1) of the unit
standard sphere S n (1).
The “Gauss map”G of oriented hypersurfaces in spheres provides Lagrangian submanifolds in complex hyperquadrics (see Subsection 4.2). The Gauss images of isoparametric
hypersurfaces in spheres are the main subject of my recent joint work with Hui Ma (Tsingua University, Beijing). We obtain that the Gauss image G(N n ) of a compact oriented
isoparametric hypersurface N n with g distinct constant principal curvatures in a unit sphereS n+1 (1) is a compact minimal Lagrangian submanifold embedded in Qn (C) covered by N nwith Deck transformation group Zg (see Theorem 4.1). Moreover, G(N n ) is a compact
monotone, cyclic Lagrangian submanifold with minimal Maslov index equal to a natural
number 2n/g (see Theorem 4.3), and 2n/g is even if and only if G(N n ) is orientable (seeTheorem 4.2). Furthermore we will mention related topics and open problems.
This article is organized as follows: In Section 1, we recall the notions and elementary
results of the Hamiltonian deformations of a Lagrangian submanifold in a symplectic manifolds, the moment map of Hamiltonian group action and Lagrangian orbits, the Maslov
index and minimal Maslov number, the monotone and cyclic properties for Lagrangiansubmanifolds. In Section 2, we treat Lagrangian submanifolds in a Kähler manifold, especially in an Einstein-Kähler manifold, and describe the definitions and results of the mean
curvature form, the Hamiltonian minimality, the Hamiltonian stability, and the integral formula for Maslov index. In Section 3, we review the fundamental structures and known
results on isoparametric hypersurfaces in spheres. In Section 4, we shall discuss the basic
structures and properties of compact Lagrangian submanifolds in complex hyperquadrics
obtained as the Gauss images of isoparametric hypersurfaces in spheres. In Section 5,
2

we shall mention about a construction of special Lagrangian submanifolds in the tangent
vector bundle of the standard sphere from isoparametric hypersurfaces in the spheres.
1. Lagrangian submanifolds in a symplectic manifold
1.1. Hamiltonian deformations of Lagrangian submanifolds. Let (M, ω) be a connected
symplectic manifold of dimension 2n with a symplectic form ω. A smooth immersion(resp. embedding) φ : L → M of a connected smooth manifold L into M is called a
Lagrangian immersion (resp. Lagrangian embedding) if dim L = n and φ∗ ω = 0. A submanifold of M satisfying only the condition φ∗ ω = 0 is called an isotropic submanifoldof M. We say that L is a immersed (resp. embedded) Lagrangian submanifold in M. If
φ : L → M is a Lagrangian immersion, then the vector bundle homomorphism between
the normal bundle of φ and the cotangent bundle of L defined by(1.1)

φ−1 T M/φ∗ T L ∋ v 7−→ αv := ω(v, ·) ∈ T ∗ L

is an isomorphism.
The Lagrangian deformation of a Lagrangian immersion φ is by definition a oneparameter smooth family of Lagrangian immersions φt : L → M (|t| < ε) with φ0 = φ.
{φt } is a Lagrangian deformation if and only if a 1-form αVt on L is closed for each t, whereVt := ∂t∂ φt ∈ C ∞ (φ−1 T M) is the variational vector field of {φt }. Moreover, if a 1-form αVt
on L is exact for each t, then such a Lagrangian immersion {φt } is called a Hamiltonian
deformation of φ = φ0 . Note that if H 1 (L; R) = {0}, then a Lagrangian deformation and a
Hamiltonian deformation are same.
1.2. Moment maps and Lagrangian submanifolds. Suppose that a connected Lie group
K has a Hamiltonian group action on (M, ω) with the moment map µ : M → k∗ . Let eξ
denotes the fundamental vector field on M corresponding to ξ ∈ k defined byd(eξ) x :=
exp(tξ) · x|t=0
dt
for each x ∈ M. The following results are standard and well-known :Lemma 1.1.
(1) Let L be a connected isotropic submanifold of (M, ω). If L is invariant under theaction of K, then there is an element η ∈ z(k∗ ) such that L ⊂ µ−1
K (η). Herez(k∗ ) := {η ∈ k∗ | (Ad∗ a)(η) = η (∀a ∈ K)}.
When K is compact, we can identify z(k∗ ) with the center c(k) of Lie algebra k.
(2) An orbit L = K · p ⊂ M of K is a Lagrangian orbit if and only if L = K · p ⊂ M is∗a connected component of µ−1
K (η) for some η ∈ z(k )  c(k).
(3) Assume that K and M are compact and connected. Then an orbit L = K · p ⊂ M∗
of K is a Lagrangian orbit if and only if L = µ−1
K (η) for some η ∈ z(k )  c(k).
1.3. Maslov index of monotone and cyclic Lagrangian submanifolds. In this subsection we recall some basic notions and invariants of Lagrangian submanifolds in a symplectic manifold (cf. [26], [28], [31], [32]). The Floer chomology theory of LagrangianIntersection was investigated in [26], [27], [29] and see FOOO ([9]) for further great developments. In Subsection 2.3 we shall mention some useful results on those invariants for
Lagrangian submanifolds in Einstein-Kähler manifolds.
Let (M, ω) be a symplectic manifold and L be a Lagrangian submanifold of M. Letw : (D2 , ∂D2 ) → (M, L) be a smooth map of pairs, where D2 and ∂D2 denote a unit open3

disk of R2 and a unit circle as the boundary of D2 , respectively. We take an identification w−1 (T M)  D2 × R2n . Set Lagr(R2n ) := {Lagrangian vector subspaces of R2n } =O(2n)/U(n). By using the Maslov class µ ∈ H 1 (O(2n)/U(n); R), we define a loop w̃ ofLagr(R2n ) byw̃ : ∂D2 ∋ p 7−→ T w(p) L ∈ ΛR2 .

(1.2)

The Maslov index of L is a homomorphism Iµ,L : π2 (M, L) → Z defined by
Iµ,L ([w]) := µ(w̃) ∈ Z.

(1.3)

Iµ,L is invariant under Lagrangian deformations.
We define the minimal Maslov number ΣL of L by
(1.4)

ΣL := min{Iµ,L (A) | A ∈ π2 (M, L), Iµ,L (A) > 0}

Another homomorphism Iω : π2 (M, L) → R is defined by∫(1.5)
Iω ([w]) :=w∗ ω ∈ R.
D2

for any smooth map w : (D , ∂D ) → (M, L). Iω is invariant under Hamiltonian deformations, but not invariant under Lagrangian deformations in general.
A Lagrangian submanifold L in a symplectic manifold (M, ω) is called monotone if
there is a positive constant λ > 0 such that
2

2

Iµ,L = λ Iω .

(1.6)

Let (M, ω) be a symplectic manifold. The period group of (M, ω) is defined by theadditive subgroup(1.7)Γω := {[ω](A) | A ∈ H2 (M; Z)} ⊂ R.

Note that if M is simply connected, then
(1.8)Γω = {[ω](u) | u : S 2 −→ M smooth} ⊂ R.

A symplectic manifold (M, ω) is prequantizableif Γω is discrete[in ]R, or equivalently there[ ]ωωis a non-zero constant γ such thatis an integral class, i.e.
∈ i(H 2 (M; Z)), whereγγ
i denotes the natural homomorphismi : H 2 (M; Z) → H 2 (M; R) induced by the inclusion[ ]ωZ ⊂ R. It is well-known thatis an integral class if and only if there is a complex lineγ
√ ωbundle E over M with a U(1)-connection ∇ whose curvature form is 2π −1 . If M isγprequantizable, then we can choose a non-negative real number γ = γω such thatΓω = γω Z.

(1.9)

Suppose that a symplectic manifold (M, ω) is prequantizable. A Lagrangian submanifold L of M is called cyclic if the additive subgroup(1.10)

Γω,L := {[ω](B) | B ∈ H2 (M, L; Z)} ⊂ R

is discrete. If L is cyclic, then we can choose a non-negative real number γω,L such that
(1.11)

Γω,L = γω,L Z.

and there is an integer k such that(1.12)γω = k γω,L .

and moreover, there is an positive integer k such that ⊗k (φ−1 E, φ−1 ∇) is trivial.
4Assume that M is simply connected. Then L is cyclic if and only if there is an integer ksuch that ⊗k (φ−1 E, φ−1 ∇) is trivial. Defineγω(1.13)
nL := min{k ∈ Z | k ≥ 1, ⊗k (φ−1 E, φ−1 ∇) is trivial} =
∈ Z.
γω,L
2. Lagrangian submanifolds in Kähler manifolds
2.1. Mean curvature form of Lagrangian submanifolds. Suppose that (M, ω, J, g) is acomplex n-dimensional Kähler manifold with Kähler form ω, complex structure tensor J
and Kähler metric g. Let Ric M denote the Ricci tensor field of (M, g) and ρ M denotes the
Ricci form of a Kähler manifold (M, ω, J, g) defined by ρ M (X, Y) := Ric M (JX, Y). The first
1 M
Chern class c1 (M) of (M, J) is an integral class of H 2 (M; R) represented byρ .
2π
Let φ : L → M be a Lagrangian immersion into a Kähler manifold. Denote by B and
H the second fundamental form and the mean curvature vector field of the immersion φ,respectively. Define a symmetric tensor field S of degree 3 on L by
S (X, Y, Z) := ω(B(X, Y), Z)

for each X, Y, Z ∈ T L

and the mean curvature form αH of φ by
(2.1)

αH (X) := ω(H, X)

for each X ∈ T L.

The mean curvature form αH satisfies the identity ([6]):(2.2)

d αH (X) = φ∗ ρ M .

The condition H = 0, equivalently αH = 0, is the usual minimal submanifold condition. In
that case, we call it a minimal Lagrangian submanifold in a Kähler manifold.
2.2. Hamiltonian minimality and Hamiltonian stability of Lagrangian submanifolds.
The notion of Hamiltonian minimality and Hamiltonian stability for Lagrangian submanifolds in a Kähler manifold was introduced and investigated first by Y.-G. Oh ([23], [24],[25], [28]). For the simplicity we assume that L is compact without boundary. A Lagrangian immersion φ : L → (M, ω, J, g) is called Hamiltonian minimal if for each Hamiltonian deformation φt : L → M with φ = φ0dVol(L, φ∗t g)|t=0 = 0.
dt
The Hamiltonian minimal equation is given by(2.3)

δ αH = 0,

where δ denotes the codiﬀerential operator to the exterior diﬀerentiation d on L with respect
to φ∗ g.
A Hamiltonian minimal Lagrangian immersion φt : L → (M, ω, J, g) is called Hamiltonian stable if for each Hamiltonian deformation φ : L → M with φ = φ0d2
Vol(L, φ∗t g)|t=0 ≥ 0.
dt2
The Lagrangian version of the second variational formula was given in [25]. The null spaceof the Morse index form of the second variational formula for a compact Hamiltonian
minimal Lagrangian submanifold in a Kähler manifold is the vector space of all solutions
to the linearized Hamiltonian minimal equation. We denote its dimension by n(φ) and callit the nullity of a Hamiltonian minimal Lagrangian immersion φ.
(2.4)

5

For each holomorphic Killing vector field X on the ambient Kähler manifold M, thecorresponding 1-form αX := ω(X, ·) on M is closed. If we assume that dim H 1 (M; R) = 0
or M is simply connected, then αX is exact, that is, αX = d f for some f ∈ C ∞ (M).
Thus a holomorphic Killing vector field X is a Hamitonian vector field and X generates
a Hamiltonian deformation preserving the metric and thus the volume of any Lagrangiansubmanifold φ : L → M. Hence φ∗ αX = d( f ◦φ) is a solution to the linearized Hamiltonian
minimal equation. Denote by
nhk (φ) := dim { φ∗ αX | X is a holomorphic Killing vector field on M },which is called the holomorphic Killing nullity of a Hamiltonian minimal Lagrangian immersion φ In general nhk (φ) ≤ n(φ) and a Hamiltonain minimal Lagrangian immersion φis said to be Hamiltonain rigid if nhk (φ) = n(φ).
In the case when (M, ω, J, g) is an Einstein-Kähler manifold with Einstein constant κ,from the second variational formula we know that a compact minimal Lagrangian Lagrangian submanifold immersed in M if and only if the first (positive) eigenvalue λ1 of
the Laplacian on functions is greater than or equal to κ. When a Lagrangian submanifold
embedded in a Kähler manifold M is obtained as a Lagrangian orbit of a compact Lie
subgroup of the automorphism group Aut(M, ω, J, g), we call it a compact homogeneous
Lagrangian submanifold in M.
Proposition 2.1 ([18]). Any compact homogeneous Lagrangian submanifold in a Kähler
manifold is Hamiltonian minimal.
The classification of compact homogeneous Lagrangian submanifolds in a specific Kähler
manifold is an interesting and important problem in the sense of Riemannian and Symplectic geometry:
Problem. Classify all compact homogeneous Lagrangian submanifolds in in a specificKähler manifold such as complex projective spaces, complex Euclidean spaces, complex
space forms, Hermitian symmetric spaces and generalized flag manifolds with an invariant
symplectic form.
2.3. Minimal Maslov number of cyclic Lagrangian submanifolds in Einstein-Kähler
manifolds.
Theorem 2.1 ([28]). Let L be a Lagrangian submanifold in an Einstein-Kähler manifold(M, ω, J, g). Then the real cohomology class [αH ] of the mean curvature form of L in
H 1 (L; R) is globally invariant under every Hamiltonian deformation of L.
Hajime Ono showed the following integral formula of Maslov index Iµ,L of a Lagrangiansubmanifold in a Kähler manifold and it enables us to improve some results of Y. G. Oh([28]).
Theorem 2.2 ([31]). Let L be a Lagrangian submanifold in a Kähler manifold (M, ω, J, g).
For each smooth map of pairs w : (D2 , ∂D2 ) → (M, L), it holds
∫
∫
1
1
∗
(2.5)
Iµ,L ([w]) =w ρM +(w|∂D2 )∗ αH .
π D2
π ∂D2
The following two theorems are applications of his integral formula.
Theorem 2.3 ([31]). Suppose that (M, ω, J, g) is a simply connected Einstein-Kähler manifold with positive Einstein constant and L is a compact Lagrangian submanifold in M.
Then L is monotone if and only if [αH ] = 0 in H 1 (L; R).
6

We see that if L is a compact monotone Hamiltonian minimal Lagrangian submanifold
in M, then L must be minimal, and any compact minimal Lagrangian submanifold in a
simply connected Einstein-Kähler manifold with positive Einstein constant is monotone.
Defineγc1 := min{c1 (M)(A) | A ∈ H2 (M; Z), c1 (M)(A) > 0} ∈ Z,
γc1 ,L := min{c1 (M)(B) | B ∈ H2 (M, L; Z), c1 (M)(B) > 0} ∈ Z.
The following formula (2.6) will be essentially used in the proof of Theorem 4.3.
Theorem 2.4 ([31]). Suppose that (M, ω, J, g) is a simply connected Einstein-Kähler manifold with positive Einstein constant. If L is a compact monotone Lagrangian submanifold
in M, then L is cyclic and it holds the formula(2.6)

nL ΣL = 2 γc1 .

Remark. The number γc1 for each Hermitian symmetric space M of compact type is given
as follows ([2, p.521]): If M = S U(p + q)/S (U(p) × U(q)), then γc1 = p + q. If M =S O(2p)/U(p), then γc1 = 2p − 2. If M = S p(p)/U(p), then γc1 = p + 1. If M =S O(p + 2)/(S O(2) × S O(p)) (p ≥ 2), then γc1 = p. If M = E6 /T 1 · S pin(10), then γc1 = 12.
If M = E7 /T 1 · E6 , then γc1 = 18.
3. Isoparametric hypersurfaces in spheres
3.1. Structure theory. In this subsection we shall explain briefly the fundamental structure of isoparametric hypersurfaces in the unit standard sphere due to Elie Cartan and H.
F. Münzner ([21], [22], cf. G. Thorbergsson [39], T. E. Cecil [4]).
Let N n be a connected oriented hypersurface embedded in the unit standard spheren+1
S (1) with g distinct constant principal curvatures k1 > k2 > · · · > kg and the corresponding multiplicities mα (α = 1, · · · , g). Let x(p) and n(p) denote the position vector
from the origin O and the unit normal vector in S n+1 (1) at p ∈ N n . Let x denote the position vector of N n and n the unit normal vector field to N n in S n+1 (1) compatible with theorientation.
Theorem 3.1 ([21]). Set kα = cot θα (α = 1, · · · , g) with 0 < θ1 < · · · < θg < π. Then thefollowing properties hold:π(3.1)(α = 1, · · · , g),θα = θ1 + (α − 1)g(3.2)
mα = mα+2 indices modulo g,(m1 + m2 ) g(g ≥ 2),

(3.3)
n=
2


 m1
(g = 1).
Thus if g is odd, then m1 = m2 = · · · = mg . Since θ1 + (α − 1)

gπ
π
<π=
, we havegg(g − α + 1)ππ, and particularly 0 < θ1 < .
gg
We define a smooth function V in a tubular neighborhood A of N n by

0 < θ1 <V(q) := cos(g t(q))
for each q ∈ A. Here θ1 − t(q) is equal to the distance from a point q to N n in S n+1 (1).
DefineF̄(q) := rg cos(g t(q)) = rg V(q)7∪for each r > 0 and each q ∈ A. The function F̄ on an open cone r>0 rA ⊂ Rn+2 extends
to a homogeneous polynomial F : Rn+2 → R of degree g, the so called Cartan-Münzner
polynomial, satisfying the diﬀerential equations: ∆ F = c rg−2 ,
(3.4)


 ∥gradF∥2 = g2 r2g−2 ,m2 − m1
and r = ∥x∥2 = (x1 )2 + (x2 )2 + (x3 )2 + · · · + (xn+2 )2 . Moreover2
V = F|S n+1 (1) satisfies the isoparametric function equation in S n+1 (1): ∆¯ V = −g(g + n) V + c = S (V),
(3.5)


¯ 2 = g2 (1 − V 2 ) = T (V), ∥∇V∥
where c := g2

where ∇¯ and ∆¯ denote the covariant diﬀerentiation of the Levi-Civita connection and theLaplace-Beltrami operator of S n+1 (1). In particular, V = F|S n+1 (1) is an isoparametricfunction on S n+1 (1). As 0 < θ1 < πg , we have cos(gθ1 ) , ±1 and thus cos(gθ1 ) is a regularvalue of the function V on S n+1 (1). The level hypersurface V −1 (cos(gθ1 )) is a compact
connected orientable isoparametric hypersurface embedded in S n+1 (1) and N n is an open
subset of V −1 (cos(gθ1 )). Each N± := V −1 (±1) is a compact connected minimal submanifold
embedded in S n+1 (1) of codimension at least 2, which is called a focal submanifold of an
isoparametric hypersurface N n .
Suppose that N n is a compact connected oriented isoparametric hypersurface embedded
in S n+1 (1). From the above argument we can assume that N n = V −1 (cos(gθ1 )) and n(p) =(gradV)x(p)for each p ∈ N n . Then we have∥(gradV)x(p) ∥Lemma 3.1. For each p ∈ N n ,cos θ x(p) + sin θ n(p) ∈ V −1 (cos(gθ1 )) = N n
if and only ifθ =2π(α − 1)2π(α − 1)or 2θ1 +gg

for some α = 1, · · · , g.

The following results are famous and important results of Münzner and Abresch.
Theorem 3.2 ([22]).
(1) g must be 1, 2, 3, 4 or 6.
(2) If g = 6, then m1 = m2 .
Theorem 3.3 ([1]). If g = 6, then m1 = m2 = 1 or 2.
3.2. Minimal isoparametric hypersurfaces in spheres. It is well-known that there exists
only one minimal isoparametric hypersurface N n in each isoparametric family of S n+1 (1).
From [21] we easily compute its principal curvatures as follows (cf. [30, p.265]):
Proposition 3.1.
(1) If g = 1, then k1 = 0.
√

√
m2
m1
(2) If g = 2, then k1 =and k2 =
.
m2 √
√ m1
(3) If g = 3, then k1 = 3, k2 = 0 and k3 = − 3.
8

(4) If g = 4, then√√√√
m1 + m2 + m2
m1 + m2 − m1
κ1 =, κ2 =
,
√
√
m1
m2
√
√
√
√
m1 + m2 − m2
m1 + m2 + m1
κ3 = −, κ4 = −
.
√
√
m1
m2
(5) If g = 6, then√√
κ1 = 2 + 3, κ2 = 1, κ3 = 2 − 3,√√
κ4 = −(2 − 3), κ5 = −1, κ6 = −(2 + 3).
3.3. Homogeneous isoparametic hypersurfaces in spheres. Due to W.-Y. Hsiang and
H. B. Lawson, Jr. ([14]) and R. Takagi and T. Takahashi ([38]), any homogeneous isoparametric hypersurface in a sphere can be obtained as a principal orbit of the isotropy representation of a Riemannian symmetric pair (U, K) of rank 2
Table 1. Homogeneous isoparametric hypersurfaces in spheres
g
1
2

T ype
S 1×
BDII
BDII×
BDII

3
3
3
3
4
4

AI2
a2
AII2
EIV
b2
AIII2

4

BDI2

4

CII2

4
4
6
6

DIII2
EIII
g2
G

(U, K)(S × S O(n + 2), S O(n + 1))
n ≥ 1, [R ⊕ A1 ](S O(p + 2) × S O(n + 2 − p),
S O(p + 1) × S O(n + 1 − p))
1 ≤ p ≤ n − 1, [A1 ⊕ A1 ](S U(3), S O(3)) [A2 ](S U(3) × S U(3), S U(3)) [A2 ](S U(6), S p(3)) [A2 ](E6 , F4 ) [A2 ](S O(5) × S O(5), S O(5)) [B2 ](S U(m + 2), S (U(2) × U(m)))
m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)
(S O(m + 2), S O(2) × S O(m))
m ≥ 3, [B2 ](S p(m + 2), S p(2) × S p(m))
m ≥ 2, [BC2 ](m ≥ 3), [B2 ](m = 2)
(S O(10), U(5)) [BC2 ](E6 , U(1) · S pin(10)) [BC2 ](G2 × G2 , G2 ) [G2 ](G2 , S O(4)) [G2 ]1

dim N nn

m1 , m2n

N n = K/K0
Snnp, n − p

S p × S n−p

3
6
12
24
8
4m − 2

1, 1
2, 2
4, 4
8, 8
2, 2
2,
2m − 3
1,m−2
4,4m − 5
4, 5
6, 9
2, 2
1, 1

S O(3)
Z2 +Z2
S U(3)T2
S p(3)
S p(1)3
F4
S pin(8)S O(5)T2
S (U(2)×U(m))
S (U(1)×U(1)×U(m−2))

2m − 28m − 218
30
12
6

S O(2)×S O(m)Z2 ×S O(m−2)
S p(2)×S p(m)S p(1)×S p(1)×S p(m−2)
U(5)
S U(2)×S U(2)×U(1)U(1)·S pin(10)S 1 ·S pin(6)G2
T2
S O(4)
Z2 +Z23.4. Isoparametric hypersurfaces of OT-FKM type. The Cliﬀord construction of nonhomogeneous isoparametric hypersurfaces in spheres was discovered first by [33], [34] andgeneralized by [8]. It is another important class of isoparametric hypersurfaces in the unit
standard sphere, which are called “isoparametric hypersurfaces of OT-FKM type”.
Let Cl(Rm−1 ) be the Cliﬀord algebra over Euclidean space (Rm−1 , ⟨ , ⟩). A representation of Cl(Rm−1 ) on Rl of degree l is an algebra homomorphism
Cl(Rm−1 ) −→ M(l; R).
9

Note that
Cl(Rm−1 )  Cl0 (Rm ) ⊃ S pin(m).
Then we can choose E1 , · · · , Em−1 ∈ O(l) such that
Ei2 = −I, Ei E j = −E j Ei , i , j .
Denote by h(R2l ) the vector space of symmetric endomorphisms on R2l . Define P0 , P1 , · · · , Pm ∈h(R2l ) by
P0 (u, v) := (u, −v), P1 (u, v) := (v, u), P1+i (u, v) := (Ei v, −Ei u)
Cl(Rm−1 ) has an irreducible representation of degree l if and only if l = δ(m) as in thetable:m
Cl(Rm−1 )
1
R
2
C
3
H
4
H⊕H
5
H(2) = M(2, H)
6C(4) = M(4, C)
7
R(8) = M(8, R)
8R(8) ⊕ R(8) = M(8, R) ⊕ M(8, R)
k+8
M(Cl(Rk−1 ), 16)

δ(m)
1
2
4
4
8
8
8
8
16δ(k)

Any reducible representation of Cl(Rm−1 ) of degree l for l = k δ(m) with k > 1 is a direct
sum of k irreducible representations of Cl(Rm−1 ) on Rδ(m) .
The system (P1 , · · · , Pm ) is called a Cliﬀord system of R2l . Set(3.6)

m1 := m,m2 := l − m − 1 = k δ(m) − m − 1.

Then the polynomial function F : R2l → R defined by
m
∑
(3.7)
F(x) := ⟨x, x⟩2 − 2 ⟨Pi x, x⟩2
i=0

is a Cartan-Münzner polynomial, that is, F satisfies the Cartan-Münzner diﬀerential equations (3.4) for g = 4 and thus F provides a family of isoparametric hypersurfaces with
4 distinct principal curvatures and multiplicities (m1 , m2 ) = (m, l − m − 1) in the (2l − 1)dimensional unit standard sphere S 2l−1 (1) ⊂ R2l = Rl ⊕Rl , which is called an isoparametric
hypersurface of OT-FKM type.
Table 2. Multiplicities of principal curvatures of isoparametric hypersurfaces of OT-FKM typek\δ(m)
1
2
3
4
5
..
.

1
−
−
(1, 1)(1, 2)(1, 3)..
.

2−(2, 1)(2, 3)(2, 5)(2, 7)..
.

4−(3, 4)(3, 8)(3, 12)(3, 16)..
.

4−(4, 3)(4, 7)(4, 11)(4, 15)..
.

8(5, 2)(5, 10)(5, 18)(5, 26)(5, 34)..
.

10

8(6, 1)(6, 9)(6, 17)(6, 25)(6, 33)..
.

8−(7, 8)(7, 16)(7, 24)(7, 32)..
.

8−(8, 7)(8, 15)(8, 23)(8, 31)..
.

16(9, 6)(9, 22)(9, 38)(9, 54)·..
.

32(10, 21)(10, 53)(10, 85)
·
·
..
.

···
···
···
···
···
···
..
.

Remark. An isoparametric hypersurface of OT-FKM type in the sphere is homogeneous if
and only if it is of type AIII2 , BDI2 , CII2 or EIII.
3.5. Classification problem of isoparametic hypersurfaces in spheres. Now all known
examples of isoparametic hypersurfaces in spheres are homogeneous isoparametric hypersurfaces and isoparametric hypersurfaces of OT-FKM type. It is conjectured that they are
ALL isoparametic hypersurfaces in spheres.
Let N n ⊂ S n+1 (1) be an isoparametric hypersurface in the sphere. If g = 1, 2 or 3,
then N n is homogeneous (E. Cartan). If g = 6 and m = 1, then N n it is homogeneous(Dorfmeister and Neher [7], R. Miyaoka [20]). The case g = 6 and m = 2 is investigated by
R. Miyaoka. If g = 4, then the multiplicities (m1 , m2 ) must be the same as those in examples
of homogeneous one and of OT-FKM type (Stolz [37]), and N n must be homogeneous orof OT-FKM type except for the cases (m1 , m2 ) = (4, 5), (3, 4), (6, 9), (7, 8) (Cecil, Chi andJensen [5], Immervoll [15]). The research of the remaining cases is still in progress by
Q.-S. Chi.
4. Lagrangian submanifolds in complex hyperquadrics
obtained from isoparametric hypersurfaces
4.1. Complex hyperquadrics. The complex hyperquadricf 2 (Rn+2 )  S O(n + 2)/(S O(2) × S O(n))
Qn (C)  Gr
is a compact Hermitian symmetric space of rank 2, where
Qn (C) := {[z] ∈ CPn+1 | z20 + z21 + · · · + z2n+1 = 0},
f 2 (Rn+2 ) := {W | oriented 2-dimensional vector subspace of Rn+2 }.
Gr
f 2 (Rn+2 ) is given by
The identification between Qn (C) and Gr
CPn+1 ⊃ Qn (C) ∋ [a +2∧√
f 2 (Rn+2 ) ⊂−1 b] ←→ W = a ∧ b ∈ Gr
Rn+2 .

Here {a, b} is an orthonormal basis of W compatible with its orientation. If n = 2, thenQ2 (C)  S 2 × S 2 . If n ≥ 3, then Qn (C) is irreducible.
n+2fNote that the standard Kähler metric gstd
) induced from theQn (C) on Qn (C)  Gr 2 (Rn+2standard inner product of R is Hermitian symmetric and an Einstein-Kähler metric with
the Einstein constant κ equal to n.
4.2. Gauss map of an oriented hypersurface in a sphere. Let N n ,→ S n+1 (1) ⊂ Rn+2 bean oriented hypersurface immersed or embedded in the (n + 1)-dimensional unit standard
sphere.
Let x and n denote the position vector of points of N n and the unit normal vector fieldof N n in S n+1 (1), respectively. The Gauss map defined by√
f 2 (Rn+2 )G : N n ∋ p 7−→ [x(p) + −1 n(p)] = x(p) ∧ n(p) ∈ Qn (C)  Gr
is always a Lagrangian immersion.
Proposition 4.1 ([17], [18]).
(1) Let N1 , N2 ⊂ S n+1 (1) be oriented hypersurfaces in the unit sphere. Then N1 isparallel to N2 if and only if G(N1 ) = G(N2 ).
11

(2) The Gauss maps for any deformation of an oriented hypersurface N n in S n+1 (1)gives a Hamiltonian deformation of the Gauss map G. Conversely, a small Hamiltonian deformation of the Gauss map G corresponds to a deformation of the oriented hypersurface N n in S n+1 (1).
Let κi (i = 1, · · · , n) denote the principal curvatures of N n ⊂ S n+1 (1). Set κi = cot θi (i =1, · · · , n), where 0 < θi < π. Choose an orthonormal frame {ei } on N n ⊂ S n+1 (1) suchthat the second fundamental form h of N n in S n+1 (1) with respect to n is diagonalized asnh(ei , e j ) = κi δi j and let {θi } be its dual coframe. Then the induced metric G∗ gstd
Qn (C) on N
by the Gauss map G is expressed as
n
∑
(4.1)
G∗ gstd=(1 + κi2 ) θi ⊗ θi .
Qn (C)
i=1

Let H denote the mean curvature vector field of G. Then B. Palmer showed that themean curvature form of the Gauss map G is expressed in terms of the principal curvaturesof N n as follows:
Lemma 4.1 ([35]).

 n 
 

n
√
∑   ∏
(1 + −1κi ) = −d  θi  .
αH = d Im log
i=1

i=1

Remark. The last equality was pointed by Jianquan Ge (Beijing Normal University).
Particularly, if N n ⊂ S n+1 (1) is an oriented hypersurface in S n+1 (1) with constant principal curvatures, then the Gauss map G : N n −→ Qn (C) is a minimal Lagrangian immersion.
4.3. Gauss images of isoparametric hypersurfaces. Suppose that N n ,→ S n+1 (1) ⊂Rn+2 is a compact connected oriented isoparametric hypersurface embedded in the (n + 1)dimensional unit standard sphere. In this subsection we use the same notations as in Subsection 3.1.
By Lemma 3.1, for each p ∈ N n a normal geodesic (a great circle) γ = γ(θ) defined byxθ (p) := cos θ x(p) + sin θ n(p)has intersection with N n at 2g points as
2π(α − 1)2π(α − 1)γ ∩ N n = {xθ (p) | θ =or 2θ1 +for some α = 1, · · · , g}gg
For each xθ (p) ∈ γ∩ N n , let pθ ∈ N n be a point of N n with the position vector xθ (p) = x(pθ ).
2π(α − 1)
If θ =(α = 1, · · · , g), thengG(pθ ) = x(pθ ) ∧ n(pθ ) = x(p) ∧ n(p) = G(p).
2π(α − 1)(α = 1, · · · , g), thenIf θ = 2θ1 +g
G(pθ ) = x(pθ ) ∧ n(pθ ) = x(p) ∧ (−n(p)) = −x(p) ∧ n(p) , G(p).
2π(α − 1)(α =Conversely, if G(p) = G(q) for p, q ∈ N n , then q = pθ for some θ =g1, · · · , g). Indeed, since x(p) ∧ n(p) = x(q) ∧ n(q), we can express asx(q) = cos ψ x(p) + sin ψ n(p),n(q) = − sin ψ x(p) + cos ψ n(p)12for some 0 ≤ ψ < 2π, it follows that ψ =α = 1, · · · , g. If ψ = 2θ1 +

2π(α − 1)2π(α − 1)or 2θ1 +for somegg2π(α − 1), then G(p) , G(q), a contradiction. Hence ψ =g2π(α − 1)for some α = 1, · · · , g.
g
From these observations we can define the free action of a finite cyclic group Zg oforder g on N n . A map
( )
( )
2π
2π
x(p) + sin
n(p) ∈ N n(4.2)
ν : N n ∋ p 7−→ cosgg
is a diﬀeomorphism of N n onto itself of order g. {Id M = ν0 , ν, · · · , νg−1 } is a finite cyclic
group of order g which acts freely on N n . Set Zg := {Id M = ν0 , ν, · · · , νg−1 }.
Proposition 4.2. Let p, q ∈ N n . Then G(p) = G(q) if and only if q = υ(p) for some υ ∈ Zg .
Therefore we obtain
Theorem 4.1 ([17]). The image G(N n ) of the Gauss map G : N n −→ Qn (C) is diﬀeomorphic to the quotient smooth manifold N n /Zg by the free action of Zg , that is, G(N n ) N n /Zg . Hence G(N n ) is a compact connected minimal Lagrangian submanifold embedded
in Qn (C)
Problem. Investigate properties of compact minimal Lagrangian submanifolds embedded
in complex hyperquadrics obtained as Gauss images of isoparametric hypersurfaces in
spheres.
We shall compute the diﬀerential of a diﬀeomorphism ν ∈ Zg at p ∈ N n . Let {ei | i =1, · · · , n} be an orthonormal basis of T p N n such that A(ei ) = kα ei and set kα = cot θα with
0 < θα < πg . Then we have
( )
( )
2π
2π
(dν) p (ei ) = ∂ei x 2πg = (cos− sin
cot θα ) ei .
g
g
If g = 1, then (dν) p (ei ) = ei (i = 1, · · · , n).
If g = 2, then (dν) p (ei ) = −ei (i = 1, · · · , n).
Suppose that g ≥ 3. Then2π2πcos( ) − sin( ) cot θα < 0gg
2π⇐⇒ cot( ) < cot θα
g
2π
π
⇐⇒
> θα = θ1 + (α − 1)
(α = 1, · · · , g)gg
(3 − α)π> θ1 (α = 1, · · · , g)
⇐⇒
g
and
2π
π
(g − 3)π(α = 1) > (α = 2) > θ1 > 0 (α = 3) ≥ · · · ≥ −(α = g).
g
g
g
Hence the number of negative eigenvalues of (dν) p : T x N → T xθ N is equal to m1 + m2 .
Lemma 4.2. Assume that g ≥ 2. Then the diﬀeomorphism ν : N → N preserves theorientation if and only if m1 + m2 is even.
13Remark. The Zg -action on N n ⊂ S n+1 (1) does not preserve the induced metric from
S n+1 (1) if g ≥ 3. The metric (4.1) on N n induced from Qn (C) by G is preserved by theZg -action. It is possible to check it by a direct computation too.
From Lemma 3.1 we know m1 + m2
2n 

=

 2 m1g

if g is even,
if g is odd.

Hence by Lemma 4.2 the orientability of G(N n ) is determined as follows:
Theorem 4.2.
2n(1) Ifis even, then L = G(N n )  N n /Zg is orientable.
g2n(2) Ifis odd, then L = G(N n )  N n /Zg is non-orientable.
g
Now the minimal Maslov number of the Gauss image of an isoparametric hypersurface
in a sphere can be determined as follows :Theorem 4.3 ([19]). L = G(N n ) is a compact monotone and cyclic Lagrangian submanifold embedded in Qn (C) and its minimal Maslov number ΣL is given by
2n 
 m1 + m2 if g is even,
ΣL =
=

2m
g
if g is odd.
1

Proof. By Theorem 2.3 and minimality of G(N n ), G(N n ) is a monotone Lagrangian submanifold in Qn (C). Moreover, by Theorem 2.4 G(N n ) is a cyclic Lagrangian submanifold
in Qn (C) and the formula (2.6). In case M = Qn (C) we know that γc1 = n if n ≥ 2 andγc1 = 2 if n = 1. We shall determine the number nL .
Let Ñ n be the Legendrian lift of N n to the unit tangent sphere bundle US n+1 (1) =V2 (Rn+2 ) :N n −→ G(N n ) = N n /Zg .
-

Ñ n
Zg
?
G(N n )=-

V2 (Rn+2 )|Ln

- V (Rn+2 )2

π S O(2)  U(1)  S 1
π S O(2)?
?
G(N n ) = Ln ⊂ Qn (C)
Lagr.

Then
π : V2 (Rn+2 )|L −→ L = G(N n )is a flat principal fiber bundle with structure group S O(2) and the covering map
π : Ñ n −→ G(N n )with Deck transformation group Zg coincides with its holonomy subbundle with the holonomy group Zg :
}
{(
)
g−1
1
cos t − sin t
Zg =t = 0, 2π , · · · , 2π
sin t cos t
g
g
√
1
g
−
1
}
={e −1t | t = 0, 2π , · · · , 2πgg14Let E denote the flat complex line bundle over G(N n ) associated with the principal fiber
bundleπ : V2 (Rn+2 )|L −→ G(N n )by the standard action of S O(2)  U(1) on C.
f 2 (Rn+2 ) is defined by
The tautological complex line bundle W over Qn (C) = Gr√W x := C(a + −1b)√for each x = [a + −1b] ∈ Qn (C) Then E = W if n ≥ 2 and ⊗2 E = W if n = 1. Indeed,c1 (W)(CP1 ) = 1 if n ≥ 2. Here CP1 := {W ⊂ U | 1-dimensional complex vector subspaces}
and U is a complex 2-dimensional isotropic
vector subspace of Cn+2 , namely, U ⊥ Ū.
√ 2π
For k = 1, 2, · · · ,√g, the generator e −1 g of the holonomy group Zg on E|L induces themultiplicationby e −1 g on ⊗k E|L . Thus the holonomy group of ⊗k E|L is generated by√ 2πk
e −1 g of Zg . Hence ⊗g E|L has trivial holonomy and for k = 1, 2, · · · , g − 1, ⊗k E|L has
non-trivial holonomy. Therefore we obtain nL = g if n ≥ 2 and nL = 2 if n = 1 . Moreover, 2 (m + m )g

1
2


= m1 + m2 if g is even,



g
2


2n 
ΣL =
=


g


2



 g m1 = 2m1 if g is oddg
2πkRemark. Theorems 4.1 and 4.3 are described in [17], [19] without detail of the proof.
Proposition 4.3 ([17]). An isoparametric hypersurface N n in S n+1 (1) is homogeneous if
and only if its Gauss image G(N n ) is a compact homogeneous Lagrangian submanifold in
Qn (C).
In [17], we classified ALL compact homogeneous Lagrangian submanifolds in complex
hyperquadrics Qn (C).
Palmer ([35]) showed that the Gauss map G : N n → Qn (C) is Hamiltonian stable if andonly if g = 1. In [17], [19], we determined completely the (strict) Hamiltonian stability of
the Gauss images G(N n ) of ALL compact homogeneous isoparametric hypersurfaces N n
in spheres S n+1 (1).
Problem. Investigate the similar theory in the case of oriented hypersurfaces in a real
hyperbolic space form and semi-Riemannian space forms. J. Hahn’s work ([10], [11]) will
be useful to provide homogeneous and non-homogeneous examples.
5. Cohomogeneity 1 Lagrangian submanifolds in the cotangent bundle of S n+1 (1)Let T S n+1 (1) and T ∗ S n+1 (1) be the tangent and cotangent vector bundles over the (n+1)dimensional unit standard sphere S n+1 (1). The special orthogonal group S O(n + 2) acts
eﬀectively and transitively on S n+1 (1) ⊂ Rn+2 as the identity component of its isometrygroup and induces the group actions on T S n+1 (1) and T ∗ S n+1 (1) in the natural way. Relative to the standard metric of S n+1 (1), we identify T S n+1 (1) with T ∗ S n+1 (1).
More generally, suppose that N m is an m-dimensional submanifold immersed in the unit
standard sphere S n+1 (1). Let ν∗N denote the conormal bundle of N m in S n+1 (1) and the unit
conormal bundle U(ν∗N ) := {ξ ∈ ν∗N | ∥ξ∥ = 1} of N m in S n+1 (1). It is a classical fact that theconormal bundle ν∗N is a Lagrangian submanifold in the cotangent vector bundle T ∗ S n+1 (1)15

of the unit standard sphere S n+1 (1). Notice that the unit cotangent bundle U(T ∗ S n+1 (1)) is
diﬀeomorphic to the Stiefel manifold(5.1) V2 (Rn+2 ) := { (a, b) | a, b ∈ Rn+2 , ∥a∥ = ∥b∥ = 1, ⟨a, b⟩ = 0 }  S O(n + 2)/S O(n).
f 2 (Rn+2 ) 
Furthermore, the unit cotangent bundle U(T ∗ S n+1 (1)) is a circle bundle over Gr∗ n+1
Qn (C). Hence U(T S (1)) carries the canonical contact structure and then the unit conormal bundle U(ν∗N ) of N is a Legendrian submanifold of U(T ∗ S n+1 (1)). Then the projectionof U(ν∗N ) via p2 gives a Lagrangian submanifold immersed in Qn (C). In the case when
N m = N n+1 is an oriented hypersurface in S n+1 (1), (if we take a connected component ofU(ν∗N ), ) this construction coincides with the above Gauss map construction. We have thefollowing diagram :ν∗N?
U(ν∗N )?
p2 (U(ν∗N ))

Lag.
Leg.

T ∗ S n+1 (1)?
U(T ∗ S n+1 (1)) V2 (Rn+2 )p2 S 1?
- Q (C)nLag.

p1 S n?
S n+1 (1) ⊃ N mimm. submfd.

The unit cotangent bundle U(T ∗ S n+1 (1))  V2 (Rn+2 ) has the standard homogeneousf 2 (Rn+2 )  Qn (C) and(S O(n + 2)-invariant) Einstein-Sasakian manifold structure over Gr
n+2
n+2
the corresponding cone metric on (0, ∞) × V2 (R ) = CV2 (R ) × CU(T ∗ S n+1 (1)) T S n+1 (1) \ {0} is an Ricci-flat Kähler metric (cf. [3]). Moreover, the cone CU(T ∗ S n+1 (1))
over U(T ∗ S n+1 (1)) is a Lagrangian submanifold in CV2 (Rn+2 ) relative to the Kähler cone
metric. Then it is a well-known fact that the following three conditions are equivalent eachother:(1) p2 (U(ν∗N )) is a minimal Lagrangian submanifold in Qn (C) with respect to the standard Hermitian symmetric Einstein-Kähler metric.
(2) U(ν∗N ) is a minimal Legendrian submanifold in U(T ∗ S n+1 (1))  V2 (Rn+2 ) withrespect to the standard homogeneous Einstein-Sasakian metric.
(3) CU(ν∗N ) is a minimal Lagrangian submanifold, i.e. a special Lagrangian submanifold of some phase, in CV2 (Rn+2 ) with respect to the Ricci-flat Kähler cone metric.
Therefore each isoparametric hypersurface N n in S n+1 (1) gives a special Lagrangian submanifold in CV2 (Rn+2 ) equipped with the Ricci-flat Kähler cone metric. In particular we
see that if N n is a homogeneous isoparametric hypersurface in S n+1 (1), then the coneCU(ν∗N ) is a cohomogeneity 1 special Lagrangian submanifold in CV2 (Rn+2 ) with the
Ricci-flat Kähler cone metric.
Stenzel ([36]) constructed a complete Ricci-flat Kähler metric on the (co)tangent vector
bundle of a compact rank one symmetric space G/K invariant under the induced group
action of G, and thus of cohomogeneity 1. The metric is called the Stenzel meteric. The
Stenzel metric on T ∗ S n+1 (1)  T S n+1 (1) is a complete Ricci-flat Kähler metric invariant
under the standard group action of S O(n + 2).
16

S. Karigiannis and M. Min-Oo ([16]) showed that N m is an austere submanifold inS (1) if and only if its conormal bundle ν∗N is a special Lagrangian submanifold in
T ∗ S n+1 (1)  T S n+1 (1) with respect to the Stenzel metric. Here a minimal submanifold
of a Riemannian manifold is called austere if for each normal vector ν, the eigenvalues ofthe shape operator Aν are invariant under multiplication by −1 ([12, p.102]). Note that aminimal isoparametric hypersurface N n in S n+1 (1) is an austere submanifold in S n+1 (1) if
and only if m1 = m2 (see Proposition 3.1).
On the other hand by Lemma 4.1 we see that if an oriented minimal hypersurface N n in
n+1
S (1) is austere, then the Gauss map G : N n → Qn (C) is a minimal Lagrangian immersion. Hence CU(ν∗N n ) is a special Lagrangian submanifold in CV2 (Rn+2 ) with the Ricci-flatKähler cone metric and at the same time its conormal bundle ν∗N n is a special Lagrangiansubmanifold in T ∗ S n+1 (1)  T S n+1 (1) with the Stenzel metric. However we should notice that if N n is an isoparametric hypersurface in S n+1 (1) but not austere, then CU(ν∗N n )is a special Lagrangian submanifold in CV2 (Rn+2 ) with the Ricci-flat Kähler cone metricand but by the result of Karigiannis and Min-Oo its conormal bundle ν∗N n is not always aspecial Lagrangian submanifold in T ∗ S n+1 (1)  T S n+1 (1) with the Stenzel metric. And
also we see that if N n is an austere homogeneous isoparametric hypersurface in S n+1 (1),then its conormal bundle ν∗N n is a cohomogeneity 1 special Lagrangian submanifold in
T ∗ S n+1 (1)  T S n+1 (1) with the Stenzel metric.
Recently Kaname Hashimoto (Osaka City University, Ph.D. student) and Takashi Sakai(Tokyo Metropolitan University/OCAMI) ([13]) investigated a classification of cohomogeneity 1 special Lagrangian submanifolds in T ∗ S n+1 (1)  T S n+1 (1) with the Stenzel
metric deformed from special Lagrangian cones CU(ν∗N n ) in CV2 (Rn+2 ) with the Ricciflat Kähler cone metric in the case when N n is an isoparametric hypersurface in S n+1 (1)with g = 1, 2. Their research is in progress at present.
n+1

Acknowledgements. This article is based on my lectures “Geometry of Lagrangian submanifolds and isoparameric hypersurfaces, I & II”at the 14th International Workshop on“Diﬀerential Geometry and Related Fields”and the 3rd KNUGRG-OCAMI Joint Diﬀerential Geometry Workshop at Kyungpook National University in Korea, on November 4-6,
2010. The author would like to thank Professor Young Jin Suh and Dr. Hyunjin Lee for
their kind hospitality and and excellent organization.

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Osaka City University Advanced Mathematical Institute (OCAMI) & Department of Mathematics, Osaka
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