Isoparametric submanifolds admitting a reflective focal...

.

......

Isoparametric submanifolds admitting

a reflective focal submanifold

in symmetirc spaces of non-compact type

Naoyuki Koike

Tokyo University of Science

[email protected]

Workshop on the Isoparametric Theory

Beijing Normal University

June 3, 2019

[email protected]

Content

1. Introduction

2. Isoparametric submanifold and complex equifocal

submanifold

3. ∞-dimensional isoparametric submanifold

submanifold

4. ∞-dim. anti-Kaehler isoparametric submanifold

5. Outline of the proofs of results

1. Introduction

Lift to Hilbert space

M ⊂ G/K

φ

M := (π ◦ φ)−1(M) ⊂ H0([0, 1], g)

lift G

π

G/K : a simply connected symmetric space of compact type

.Theorem A.1(Terng-Thorbergsson, 1995)..

...... M : equifocal ⇐⇒ M : isoparametric

Homogeneity of ∞-dim. isoparametric submanifolds

• In 1999, Heintze-Liu proved the homogeneity theorem for

isoparametric submanifolds in a Hilbert space.

• In 2002, Christ proved the homogeneity theorem for an

equifocal submanifold M in G/K by applying Heintze-Liu’

theorem to M = (π ◦ φ)−1(M) ⊂ H0([0, 1], g).

In the proof, he used the fact that M is homogeneous by

a Banach Lie group action.

• In 2012, Gorodski-Heintze proved that the homogeneity in

Heintze-Liu’s theorem means the homogeneity by a Banach

Lie group action.


V : (seprable) Hilbert space

M(⊂ V ) : complete proper Fredholm submanifold

.Theorem A.2(Heintze-Liu, 1999)..

......

M : full irreducible isoparametric submanifold

of codimM ≥ 2 in V

=⇒ M : homogeneous (i.e., M = H · p (∃H ⊂ I(V )))

Remark H is given by

H := {F ∈ I(V ) |F (M) = M}.I(V ) is not a Banach Lie group.

Hence H also is not a Banach Lie group in general.

A Banach Lie group of isometries

Ib(V ) :=

F ∈ I(V )

∣∣∣∣∣∣∣∣∣∃ {Ft}t∈[0,1] : a one para transf . gr.

s.t.

• F1 = F

• the Killing vec. fd. ass. to

{Ft}t∈[0,1] is defined on V

idF

t 7→ Ft

XI(V )

X : the ass. vec. field of {Ft}t∈[0,1]

u

t 7→ Ft(u)

Xu I(V )V

A Banach Lie group of isometries

.Fact......... Ib(V ) is a Banach Lie group.

Proof

ϕ : Ib(V ) −→ ob(V ) ⊕ V (Banach space)

F 7→(dAt

dt

∣∣∣∣t=0

,dbt

dt

∣∣∣∣t=0

)(Ft(u) = At(u) + bt (F1 = F ))

D := {(LF (U), (ϕ ◦ L−1F )|LF (U))}F∈Ib(V ) gives

a Banach Lie group str. of Ib(V ), where U is

a suff. small nbd of id.

Remark Xu =dAt

dt

∣∣∣∣t=0

u +dbt

dt

∣∣∣∣t=0

(u ∈ V )

An example of an element of Ib(V )

Example 1 V := l∞ =

{(ai)

∞i=1 |

∞∑i=1

a2i < ∞

}

Ft(u) := At(u) + bt (u ∈ V )(At =

∞⊕k=1

(cos t

k− sin t

k

sin tk

cos tk

) )

Then the Killing vec. fd. X ass. to {Ft}t is given by

Xu = B(u) + b (u ∈ V )(B =

∞⊕k=1

(0 −1

k1k

0

) )

Since B is bounded, X is defined on V .

Hence we have F1 ∈ Ib(V )．

An example of an element of I(V ) \ Ib(V )

Example 2 V := l∞

Ft(u) := At(u) + bt (u ∈ V )(At =

∞⊕k=1

(cos kt − sin kt

sin kt cos kt

) )

Then the Killing vec. fd. X ass. to {Ft}t is given by

Xu = Bu + b (u ∈ V )(B =

∞⊕k=1

(0 −k

k 0

) )

Since B is not bounded, X is not defined on V .

Hence we have F1 /∈ I(V ) \ Ib(V ).

An example of an element of I(V ) \ Ib(V )

u =

1[i+12

] = (1, 1, 2, 2, 3, 3, · · · ) ∈ l∞

B(u) = (−1, 1,−1, 1,−1, 1, · · · ) /∈ l∞

Hence X is not defined at u.


M(⊂ V ) : complete proper Fredholm submanifold

.Theorem A.3(Gorodski-Heintze, 2012)..

......

M : full irreducible isoparametric submanifold

of codimM ≥ 2 in V

=⇒ M = Hb · p (∃Hb ⊂ Ib(V ))

Remark Hb = {F ∈ Ib(V ) |F (M) = M}

Homogeneity of equifocal submanifolds

G/K : simply connected symmetric space

of compact type

.Theorem A.4(Christ)..

......

M : full irreducible equifocal submanifold

of codimM ≥ 2 in G/K

=⇒ M : homogeneous (i.e., M = H ·p (∃H ⊂ I(G/K)))

Complexification and Lift to ∞-dim. anti-Kaheler space

M ⊂ G/Kextrinsic

complexification

MC ⊂ GC/KC

φ

MC := (π ◦ φ)−1(MC) ⊂ H0([0, 1], gC)

lift GC

π

G/K : symmetric space of non-compact type

.Theorem B.1(K, 2005)..

......

M : complex equifocal

⇐⇒ MC : anti-Kaehler isoparametric

Homogeneity of ∞-dim. anti-Kaehler isoparametric

submanifolds

V : ∞-dim. anti-Kaehler space

M(⊂ V ) : complete anti-Kaehler Fredholm submanifold

.Theorem B.2(K,2014)..

......

M : full irr. anti-Kaehler isoparametric submanifold of

codimM ≥ 2 with J-diagonalizable shape op. in V


Remark H is given by

H := {F ∈ I(V ) |F (M) = M}.I(V ) is not a Banach Lie group.

Hence H also is not a Banach Lie group in general.

Homogeneity of ∞-dim. anti-Kaehler isoparametric

submanifolds

Ib(V ) :=

F ∈ I(V )

∣∣∣∣∣∣∣∣∣∃ {Ft}t∈[0,1] : a one para transf . gr.

s.t.

• F1 = F

• the hol. Killing v. fd. ass. to

{Ft}t∈[0,1] is defined on V


......



=⇒ M : homogeneous (i.e., M = Hb · p (∃Hb ⊂ Ib(V )))


Homogeneity of isoparametric submanifolds in sym. sp. of

non-cpt type


(∗C) For any unit normal vec. v of M ,

the nullity spaces of the complex focal radii

along the normal geodesic γv span

(TpM)C ∩ ((KerAv ∩ KerR(v))C)⊥.


......

M : full irreducible curvature-adapted isoparametric

Cω-submanifold of codimM ≥ 2 in G/K s.t. (∗C)=⇒ M : homogeneous (i.e., M = H ·p (∃H ⊂ I(G/K)))

(M : curv.-adapted & (∗C) ⇒ MC : has J-diag. shape op.)


non-cpt type


......

M : full irreducible isoparametric Cω-submanifold

of codimM ≥ 2 in G/K admitting


=⇒ M : a principal orbit of a Hermann type action

Remark (i) Let H be a symmetric subgroup of G. Then the

natural action of H on G/K is called a Hermann type

action.

(ii) Principal orbits of a Hermann type action are

curvature-adapted isoparametric submanifolds.


non-cpt type

.Question..

......

Can we delete the assumption of the real anlayticity

of M in Theorem B.4?

(∗R) For any unit normal vec. v of M , the nullity spaces

of the focal radii along γv span TpM .


......


C∞-submanifold of codimM ≥ 3 in G/K s.t. (∗R)=⇒ M : homogeneous (i.e., M = H ·p (∃H ⊂ I(G/K)))

(M : a principal orbit of the isotropy action of G/K)


non-cpt type

We proved this theorem by constructing a Tits building

associtaed to M and using

Burns-Spatzier’s theorem (1987).

2. Isoparametric submanifold andcomplex equifocal submanifold

Equifocal submanifold

G/K : symmetric space of compact type

M(⊂ G/K) : compact submanifold

.Def(Equifocal submanifold)..

......

M : equifocal submanifold

⇐⇒def

• M is a submanifold with flat section

• The normal holonomy gr. of M is trivial

• For each parallel normal vec. fd. v,

the focal radii along γvp is independent

of p ∈ M

Isoparametric submanifold

(M, g) : complete Riemannian manifold

M(⊂ M) : complete submanifold

.Def(Isoparametric submanifold in Heintze-Liu-Olmos-sense)..

......

M : isoparametric submanifold with flat section

⇐⇒def



• Sufficciently close parallel submanifolds of M

are of CMC w.r.t. the radial direction

In this talk, we call this submanifold

“isoparametic submanifold” for simplicity.

Isoparametric submanifold

Mηv(M)

v

the radial directions

p

the section of M thr. p

Equifocality and isoparametricness

.Proposition 2.1(Heintze-Liu-Olmos,2006)...

......

Assume that M is compact. Then

M : equifocal ⇐⇒ M : isoparametric

Complex focal radius

G/K : symmetirc space of non-compact type

GC/KC : the complexification of G/K

M(⊂ G/K) : Cω-submanifold in G/K

MC(⊂ GC/KC) : the complexification of

M(⊂ G/K)

γv : the normal geodesic of M of direction

v(∈ T⊥p M) (‖v‖ = 1)

γCv : the complexification of γv


.Def(Complex focal radius)..

......

z0 = s0 + t0√−1 : complex focal radius along γv

⇐⇒def

γC(z0) : a focal point of MC along s 7→ γC(sz0)


G/K

γvM

v

Jv

MC

γCv

x

γCv (sz0) = γss0v+st0Jv(1)

orγv γv

xx

γCv (z0)

γCv (z0)

γCv (z0)

GC/KC

Complex equifocal submanifold


M(⊂ G/K) : complete Cω-submanifold

.Def(Complex equifocal submanifoldi)..

......

M : complex equifocal

⇐⇒def



• For each parallel normal vec. fd. v,

the complex focal radii along γvp is

independent of p ∈ M

Complex equifocality and isoparametricness

M : submanifold in a Riemannian manifold.Def(Curvature-adapted)..

......

M : curvature-adapted

⇐⇒def

for any v ∈ T⊥M , [Av, R(v)] = 0

(Av : shape operator, R(v) := R(·, v)v)

.Proposition 2.2(K,2005)...

......

Assume that M is curvature-adapted. Then

M : complex equifocal ⇐⇒ M : isoparametric

3. ∞-dim. isoparametric submanifold

Proper Fredholm submanifold

(V, 〈 , 〉) : (∞-dim. separable) Hilbert space

M(⊂ V ) : immersed submanifold of finite codimension

A : the shape tensor of M

.Def(Proper Fredholm submanifold)..

......

M : proper Fredholm

⇐⇒def

{• exp⊥ |B⊥

1 (M) : proper

• exp⊥∗u : Fredholm operator (∀u ∈ M)

∞-dimensional isoparametric submanifold

M(⊂ V ) : proper Fredholm submanifold

.Def(∞-dim. isoparametric submanifold)..

......

M(⊂ V ) : isoparametric

⇐⇒def

• The normal holonomy group of M is trivial

• For any parallel normal vec. fd. v of M ,

the eignvalues of Avpis independent of

p ∈ M with considered the multiplicites

Principal curvature and curvature distribution

M(⊂ V ) : isoparametric submanifold

{Av | v ∈ T⊥p M} : a commuting family of

symmetric operators

TpM = Ep0 ⊕

(⊕i∈I

Epi

) (Ep

0 := ∩v∈T⊥

p MKerAv

)(

the common eigenspace decomposition

of {Av | v ∈ T⊥p M}

)For each v ∈ T⊥

p M ,

λpi (v) ⇐⇒

defAv|Ep

i= λp

i (v) · id

Then λpi : v 7→ λp

i (v) is linear, that is, λpi ∈ (T⊥

p M)∗.

Principal curvature and curvature distribution

.Def(principal curvature, curvature distribution)..

......

λi ∈ Γ((T⊥M)∗) ⇐⇒def

(λi)p := λpi (p ∈ M)

principal curvature

ni ∈ Γ(T⊥M) ⇐⇒def

λi(·) = 〈ni, ·〉curvature normal

Ei (a subbundle of TM) ⇐⇒def

Ei := qp∈M

Epi

curvature distribution

lpi ⊂ T⊥p M ⇐⇒

deflpi := (λi)

−1p (1)

focal hyperplane

Focal radii and Focal hyperplanes

M ⊂ G/K

φ

M := (π ◦ φ)−1(M) ⊂ H0([0, 1], g)

liftG

π

u ∈ (π ◦ φ)−1(p)

γv : the normal geodesic of M with γ′(0) = v

γvLu

: the normal geodesic of M with γ′(0) = vLu

equifocal

isoparametric

rank two

Focal radii and focal hyperplanes

u = γvLu(0)

γvLu(s1)

γvLu(s2)

γvLu(s3) γvL

u(s4)

γvLu(s5)

γvLu(s6)

T⊥u M(≈ R2)

4. ∞-dim. anti-Kaehler isoparametricsubmanifold

∞-dim. anti-Kaehler space

V : ∞-dim. topological (real) vector space

〈 , 〉 : continuous non-deg. sym. bilinear form of V

J : continuous linear op. of V satisfying

J2 = −id, 〈JX, JY 〉 = −〈X,Y 〉 (∀X,Y ∈ V )

.Def(∞-dim. anti-Kaehler space)..

......

(V, 〈 , 〉, J) : anti-Kaehler space

⇐⇒def

∃ V = V1 ⊕ V+

s.t.

• 〈 , 〉|V−×V− : negative defnite

• 〈 , 〉|V+×V+ : positive definite

• 〈 , 〉|V−×V+ = 0, JV− = V+

• (V, 〈 , 〉V±) : Hilbert space

(〈 , 〉V± := −π∗V−

〈 , 〉 + π∗V+

〈 , 〉)

Anti-Kaehler Fredholm submanifold

(V, 〈 , 〉, J) : ∞-dim. anti-Kaehler space

M(⊂ V ) : anti-Kaehler submanifold

(i.e., J(TM) = TM)

A : the shape tensor of M

.Def(Anti-Kaehler Fredholm submanifold)..

......

M : anti-Kaehler Fredholm

⇐⇒def

∀ v ∈ T⊥M, Av : a compact op. w.r.t. 〈 , 〉V±

Remark

M : anti-Kaehler Fredholm ⇒ exp⊥ : Fredholm map

J-eigenvalues and J-eigenvectors

M(⊂ V ) : anti-Kaehler Fredholm submanifold

.Def(J -eigenvalue)..

......

z = a + b√−1 : J-eigenvalue of Av

⇐⇒def

∃X( 6= 0) ∈ TpM s.t. AvX = aX + bJX

Also X is called J-eigenvector of Av.

Anti-Kaehler isoparametric submanifold

M(⊂ V ) : anti-Kaehler Fredholm submanifold

.Def(∞-dim. anti-Kaehler isoparametric submanifold)..

......

M(⊂ V ) : anti-Kaehler isoparametric

⇐⇒def

• The normal holonomy group of M is trivial

• For any parallel normal vec. fd. v of M ,

the J-eignvalues of Avpis independent of

p ∈ M with considered the multiplicites

AK isoparametric submfd with J-diag. shape op.

M(⊂ V ) : anti-Kaehler isoparametric submanifold

.Def(J -diagonalizable shape operator)..

......

M has J-diagonalizable shape operators

⇐⇒def

For any normal vec. v of M ,

there exists an orthonormal base

consisting of the J-eignvectors of Av

Complex principal curvature, complex curvature distribution

M(⊂ V ) : anti-Kaehler isoparametric submanifold

with J-diagonalizable shape operators

{Av | v ∈ T⊥p M} : a commuting family of

J-diagonalizable operators,

TpM = Ep0 ⊕

(⊕i∈I

Epi

) (Ep

0 := ∩v∈T⊥

p MKerAv

)(

the common J − eigenspace decomposition

of {Av | v ∈ T⊥p M}

)For each v ∈ T⊥

p M ,

λpi (v) ⇐⇒

defAv|Ep

i= Re(λp

i (v))id + Im(λpi (v))Jp

Then λpi : v 7→ λp

i (v) is complex linear, that is,

λpi ∈ (T⊥

p M)∗C .

Complex principal curvature, complex curvature distribution

.Def(complex curvature distribution)..

......

λi ∈ Γ((T⊥M)∗C) ⇐⇒def

(λi)p := λpi (p ∈ M)

complex principal curvature

ni ∈ Γ(T⊥M) ⇐⇒def

λi(·) = 〈ni, ·〉 −√−1〈Jni, ·〉

complex curvature normal

Ei (a subbundle of TM) ⇐⇒def

Ei := qp∈M

Epi

complex curvature distribution

lpi ⊂ T⊥p M ⇐⇒

deflpi := (λi)

−1p (1)

complex focal hyperplane

Complex focal radii and complex focal hyperplanes

M ⊂ G/K

extrinsiccomplexification

MC ⊂ GC/KC

φ

MC := (π ◦ φ)−1(MC) ⊂ H0([0, 1], gC)

lift GC

π

γv : the normal geodesic of MC with γ′(0) = v

γvLu

: the normal geodesic of MC with γ′vLu(0) = vL

u

u ∈ (π ◦ φ)−1(p)

curvature-adapted

s.t. (∗C)

anti-Kaeh. isopara.

with J-diag. shape op.

isoparametric

Complex focal radii and complex focal hyperplanes

l1

l3

l4

T⊥u MC(≈ C2)

l2l1

γCvLu(≈ C)

γCvLu(√−1R)

γCvLu(R)

γCvLu(z1)

T⊥u MC(≈ C2)

5. Outline of the proof of results

Recall Theorem B.2.


......




We proved this theorem by refering the proof of

E. Heintze and X. Liu, Ann. of Math. 149, (1999).

Outline of the proof of Theorem B.2

Proof of Theorem B.2.

{Ei}i∈I∪{0} : complex curvature distributions of M

γ : [0, 1] → M : geodesic in LEip

(LEip : the leaf of Ei through p)

(Step I) We construct a C∞-family {F γt }t∈[0,1] in I(V )

s.t.

{F γt (γ(0)) = γ(t)

(F γt )∗γ(0)|T⊥

γ(0)M = τ⊥

γ|[0,t]

(Step II) We show F γt (M) = M by using the assumption:

“M : full, irreducible and codimM ≥ 2”.


(Step III) We show that M = H ′ · p holds for some

subgroup H ′ of I(V ) satisfying

〈qγ

{F γt }t∈[0,1]〉 ⊂ H ′ ⊂ 〈q

γ{F γ

t }t∈[0,1]〉.Hence we have

M = H · p (H = {F ∈ I(V ) |F (M) = M}).


.Theorem B.3(K.2017)..

......

M : full irr. anti-Kaehler isoparametric Cω-submanifold

of codimM ≥ 2 with J-diagonalizable shape op. in V

=⇒ M : homogeneous (i.e., M = Hb · p (∃ Hb ⊂ Ib(V )))


We proved this theorem by refering the proof of

Gorodski and Heintze, J. Fixed Point Theory Appl. 11 (2012).


w ∈ (Ei)p (i ∈ I)

γw : [0, 1] → LEip : the geodesic in LEi

p s.t. γ′w(0) = w

Fwt := F γw

t (Fwt (u) = Aw

t (u) + bwt )

Xw ∈ X (U) ⇐⇒def

(Xw)u :=d

dtFwt (u)

∣∣∣∣t=0

(u ∈ U)(U :=

{u ∈ V

∣∣∣∣ d

dtFwt (u)

∣∣∣∣t=0

is defined

})Remark U is dense in V .

Γw : U → TpM homogeneous structure

⇐⇒def

Γw(u) :=

(d

dtAw

t (u)

∣∣∣∣t=0

)TpM

(u ∈ U)


Proof of Theorem B.3.

Fwt (u) = Aw

t (u) + bwt (u ∈ V )

(Xw)u = ddt

∣∣∣t=0

Fwt (u) =

(ddt

∣∣∣t=0

Awt

)(u) +

dbwtdt

∣∣∣t=0

Xw is defined on V

⇔d

dt

∣∣∣∣t=0

Awt is defined continuosly on V

⇔ Γw is defined continuosly on V (U = V )

⇔ supu∈U s.t. ‖u‖=1

‖Γw(u)‖ < ∞

(‖ · ‖ : the norm defined by 〈 , 〉V±)


By long deliacte discusion, we can show

supw∈∪i∈I(Ei)p s.t. ‖w‖=1

supu∈U s.t. ‖u‖=1

‖Γw(u)‖ < C < ∞.

Hence we can derive the followings:

Xw

(w ∈ ∪

i∈I(Ei)p

)are defined continuously on V ,

that is, Fwt ∈ Ib(V )

(w ∈ ∪

p∈M∪i∈I

(Ei)p

).

Furthermore, we can show the following:

H ′b · p = M(

H ′b := qp∈M q

w{Fw

t }t∈[0,1] ⊂ Ib(V )).

Recall Theorem B.4.


......


Cω-submanifold of codimM ≥ 2 in G/K s.t. (∗C)=⇒ M : homogeneous (i.e., M = H ·p (∃H ⊂ I(G/K)))

(∗C) For any unit normal vec. v of M ,

the nullity spaces of the complex focal radii

along the normal geodesic γv span

(TpM)C ∩ ((KerAv ∩ KerR(v))C)⊥.


M ⊂ G/Kextrinsic

complexification

MC ⊂ GC/KC

φ

MC := (π ◦ φ)−1(MC) ⊂ H0([0, 1], gC)

lift GC

π


V := H0([0, 1], gC)


Outline of the proof of Theorem B.4.

By the assumption for M ,

MC : full irr. anti-Kaehler isoparametric Cω-submfd of


By Theorem B.3,

MC : homogeneous

(i.e., MC = Hb · p (∃ Hb ⊂ Ib(V ))).

Without loss of generailty, we may assume MC = Hb · 0.


Since H1([0, 1], GC) acts on V isometrically,

we can regard as H1([0, 1], GC) ⊂ I(V ).

By delicate long disccussion, we can show

Hb ⊂ H1([0, 1], GC),

where we use the fact that Hb is a Banach Lie group.

H ′ := 〈{(h(0), h(1)) |h ∈ Hb}〉0 (⊂ GC × GC)

Then we can show

H ′ · (e, e) = π−1(MC).


H ′R := 〈(H ′ ∩ (G × G))0 ∪ ({e} × K)〉0

Then we can show

H ′R · e = π−1(MC) ∩ (G × G).

H ′′R := {g ∈ G | ({g} × K) ∩ H ′

R 6= ∅}

Then we can show

H ′′R · (eK) = M .

Recall Theorem B.5


......

M : full irreducible isoparametric Cω-submanifold

of codimM ≥ 2 in G/K admitting


=⇒ M : a principal orbit of a Hermann type action


Proof

M admits a reflective focal submanifold

⇓M is curvature-adapted and satisfies (∗C)

⇓ Theorem B.4

M is homogeneous

⇓ ∃ reflective f. s.

M is a principal orbit of Hermann type action

Recall Theorem B.6


......


C∞-submanifold of codimM ≥ 3 in G/K s.t. (∗R)=⇒ M : homogeneous (i.e., M = H · p (∃H ⊂ I(G/K))

(∗R) For any unit normal vec. v of M , the nullity spaces

of the focal radii along γv span TpM .

Topological Tits building

∆ = (V,S) : r-dim. simplicial complex

A := {Aλ}λ∈Λ family of subcomplexes of ∆

O : Hausdorff topology of VB := (∆,A,O) is called a topological Tits building

if the following conditions (B1)∼(B6) hold:

(B1) Each (r − 1)-dim. simplex of ∆ is contained in at least

three chambers.

(B2) Each (r − 1)-dim. simplex in a subcomplex Aλ are

contained in exactly two chambers of Aλ.

Topological Tits building

(B3) Any two simplices of ∆ are contained in some Aλ.

(B4) If two subcomplexes Aλ1 and Aλ2 share a chamber,

then there is an isomorphism of Aλ1 onto Aλ2 fixing

Aλ1 ∩ Aλ2 pointwisely.

(B5) Each apartment Aλ is a Coxeter complex.

(B6) For k ∈ {1, · · · , r},Sk := {(x1, · · · , xk+1) ∈ Vk+1 | |x1 · · ·xk+1| ∈ Sk}

is closed in (Vk+1,Ok+1).

If Aλ is finite (resp. infinite), then B is said to be

spherical type (resp. affine type).


Outline of the proof of Theorem B.6.

(Step I) We construct a topological Tits building ass. to M .

Σp : the section of M through p(∈ M)

We can show that ∩p∈M

Σp is a one-point set.

∩p∈M

Σp = {p0}, b := d(p, p0)

Sm−1(b) : the sphere of radius b in Tp0(G/K)

(m = dimG/K)


Then we can construct a topological Tits building

BM = (4M := (VM ,SM), AM ,OM)

satisfying

(i) VM = exp−1p0

(F1 q · · · q Fl) (⊂ Sm−1(b))

(F1, · · · , Fl : focal submanifolds of M)

(ii) |4M | = Sm−1(b)

(iii) AM = {Ap}p∈M , |Ap| = Sm−1(b) ∩ exp−1p0

(Σp)

(iv) OM : the relative topology of Sm−1(b).


Σp

exp−1p0

(Σp)

M

F1

F2F1

F2F1 F2

p

Sm−1(b)

Aplaminate

M

Fi p

expp0(Ap)

lp1lp2lp3

expp0

v

p0

lp1lp2

lp3Tp0(G/K)

G/K


v

v′ApAp′

Sm−1(b)

(v := exp−1p0

(p), v′ := exp−1p0

(p′))


G′ : the topological automorphism group of BM

G′ a semi-simple Lie group by [Burns-Spatizier,1987]

G′0 : the identity component of G′

s : SM → SM ⇐⇒def

s(σ) = −σ (σ ∈ SM)

K′ := {g ∈ G′0 | g ◦ s = s ◦ g}

p′ := To′(G′0/K

′) =ident.

To(G/K)

G′ y VM −→hence

K′ y VM −→extension

K′ y p′


v

ApAp′

Sm−1(b)

σ

k′(σ)

k′ · v


(Step II) We show that K′ y p′ is orbit equivalent

to the s-representation of G/K.

(by using the discussion in Pge 444-445

of [Thorbergsson, 1991])

Hence it follows that

M ′ := exp−1o (M) is a principal orbit of

the s-representation of G/K.

Therefore it follows that

M is a principal orbit of the isotropy action of G/K.

Thank you for your attention!

Affine root system

Affine root system associeted to M

We define the affine root system (in the sense of

Macdonald) associtated to M as follows.

lpi := (λi)−1p (1) (⊂ T⊥

p M) (i ∈ I)，

(T⊥p M)R := SpanR{(ni)p | i ∈ I}

(lpi )R := lpi ∩ (T⊥p M)R

Remark M : full ⇒ T⊥p M = SpanC{(ni)p | i ∈ I}

Affine root system associated to M

{(lpi )R | i ∈ I} is described as follows:

{(lpi )R | i ∈ I} = {(lpa,j)R | a = 1, · · · , k j ∈ Z}(• (lpa,j)R (j ∈ Z) are parallel.

• (lpa,i)R and (lpb,j)R (a 6= b) are not parallel.

).Fact...

......

4M := {((lpa,j)R, (na,0)x) | a = 1, · · · , k, j ∈ Z}is the affine root system (in the sense of Macdonald).

Affine root system associated to M

(lp1,−1)R

(lp1,0)R

(lp1,1)R

(lp1,2)R

(lp2,1)R (lp2,0)R (lp2,−1)R

(lp3,−1)R (lp3,0)R (lp3,1)R

(n1,0)p

(n2,0)p

(n3,0)p

0

Proof of supw1

supw2

‖Γw1w2‖ < ∞

(I) In case of 4M is of type (A), (D), (E),

a 6= b, 〈(na,i)p, (nb,j)p〉 = 0 =⇒ Γ(Ea,i)p(Eb,j)p = 0

(II) In case of 4M is of type (A), (D), (E),

a 6= b, 〈(na,i)p, (nb,j)p〉 6= 0

=⇒ ||Γwa,iwb,j|| ≤ ||wa,i|| · ||wb,j|| · ||(na,i)p||(∀wa,i ∈ (Ea,i)p, ∀wb,j ∈ (Eb,j)p)

(III) In general cases,

Γ(Ea,i)p(Ea,j)p ⊂ (E0)p⊕(Ea,2i−j)p⊕(Ea,2j−i)p⊕(Ea,(i+j)/2)p

Proof of supw1

supw2

‖Γw1w2‖ < ∞

By showing many other facts for Γ, we can show

supw

supu

||Γwu|| < ∞.

Isoparametric submanifolds admitting a reflective focal...

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