Isoparametric submanifolds admitting a reflective focal submanifold - - PowerPoint PPT Presentation

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Isoparametric submanifolds admitting a reflective focal submanifold in symmetirc spaces of non-compact type

Naoyuki Koike

Tokyo University of Science koike@rs.kagu.tus.ac.jp

Workshop on the Isoparametric Theory Beijing Normal University June 3, 2019

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.

Homogeneity of ∞-dim. isoparametric submanifolds

V : (seprable) Hilbert space M(⊂ V ) : complete proper Fredholm submanifold . Theorem A.2(Heintze-Liu, 1999) . . M : full irreducible isoparametric submanifold

• f codim M ≥ 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         

id F

t → Ft

X I(V )

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

Xu I(V ) V

A Banach Lie group of isometries

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

ϕ : Ib(V ) − →

• b(V ) ⊕ V

(Banach space) F → ( dAt dt

• t=0

, dbt dt

• t=0

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

D := {(LF (U), (ϕ ◦ L−1

F )|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

a2

i < ∞

} Ft(u) := At(u) + bt (u ∈ V ) ( At =

∞

⊕

k=1

( cos t

k

− sin t

k

sin t

k

cos t

k

) ) Then the Killing vec. fd. X ass. to {Ft}t is given by Xu = B(u) + b (u ∈ V ) ( B =

∞

⊕

k=1

( − 1

k 1 k

) ) 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

( −k k ) ) 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+1 2

]   = (1, 1, 2, 2, 3, 3, · · · ) ∈ l∞ B(u) = (−1, 1, −1, 1, −1, 1, · · · ) / ∈ l∞ Hence X is not defined at u.

Homogeneity of ∞-dim. isoparametric submanifolds

M(⊂ V ) : complete proper Fredholm submanifold . Theorem A.3(Gorodski-Heintze, 2012) . . M : full irreducible isoparametric submanifold

• f codim M ≥ 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

• f compact type

. Theorem A.4(Christ) . . M : full irreducible equifocal submanifold

• f codim M ≥ 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/K extrinsic complexification M C ⊂ GC/KC

φ

• M C := (π ◦ φ)−1(M C) ⊂ H0([0, 1], gC)

lift

GC π G/K : symmetric space of non-compact type . Theorem B.1(K, 2005) . . M : complex equifocal ⇐ ⇒

• M C : 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

codim M ≥ 2 with J-diagonalizable shape op. 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.

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         

. Theorem B.3(K,2017) . . M : full irr. anti-Kaehler isoparametric submanifold of

codim M ≥ 2 with J-diagonalizable shape op. in V

= ⇒ M : homogeneous (i.e., M = Hb · p (∃ Hb ⊂ Ib(V ))) Remark Hb = {F ∈ Ib(V ) | F (M) = M}

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type

G/K : symmetric space of non-compact 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 ∩ ((Ker Av ∩ Ker R(v))C)⊥. . Theorem B.4(K,2018) . . M : full irreducible curvature-adapted isoparametric Cω-submanifold of codim M ≥ 2 in G/K s.t. (∗C) = ⇒ M : homogeneous (i.e., M = H · p (∃ H ⊂ I(G/K)))

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

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type

. Theorem B.5(K,2018) . . M : full irreducible isoparametric Cω-submanifold

• f codim M ≥ 2 in G/K admitting

a reflective focal submanifold = ⇒ 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.

Homogeneity of isoparametric submanifolds in sym. sp. of non-cpt type

. Question . . Can we delete the assumption of the real anlayticity

• f M in Theorem B.4?

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

• f the focal radii along γv span TpM.

. Theorem B.6(K,2018) . . M : full irreducible curvature-adapted isoparametric C∞-submanifold of codim M ≥ 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)

Homogeneity of isoparametric submanifolds in sym. sp. of 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 and

complex 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

• f 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

        

• M is a submanifold with flat section
• The normal holonomy gr. of M is trivial
• 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

M η

v(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 M C(⊂ GC/KC) : the complexification of M(⊂ G/K) γv : the normal geodesic of M of direction v(∈ T ⊥

p M) (v = 1)

γC

v : the complexification of γv

Complex focal radius

. Def(Complex focal radius) . . z0 = s0 + t0 √−1 : complex focal radius along γv ⇐ ⇒

def

γC(z0) : a focal point of M C along s → γC(sz0)

Complex focal radius

G/K γv M v Jv M C γC

v

x

γC

v (sz0) = γss0v+st0Jv(1)

• r

γv γv x x γC

v (z0)

γC

v (z0)

γC

v (z0)

GC/KC

Complex equifocal submanifold

G/K : symmetric space of non-compact type M(⊂ G/K) : complete Cω-submanifold . Def(Complex equifocal submanifoldi) . . M : complex equifocal ⇐ ⇒

def

              

• M is a submanifold with flat section
• The normal holonomy gr. of M is trivial
• 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 A

vp is 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 = Ep

0 ⊕

( ⊕

i∈I

Ep

i

) ( Ep

0 :=

∩

v∈T ⊥

p M Ker Av

) ( the common eigenspace decomposition

• f {Av | v ∈ T ⊥

p M}

) For each v ∈ T ⊥

p M,

λp

i (v) ⇐

⇒

def

Av|Ep

i = λp

i (v) · id

Then λp

i : v → λp i (v) is linear, that is, λp i ∈ (T ⊥ p M)∗.

Principal curvature and curvature distribution

. Def(principal curvature, curvature distribution) . . λi ∈ Γ((T ⊥M)∗) ⇐ ⇒

def

(λi)p := λp

i

(p ∈ M) principal curvature ni ∈ Γ(T ⊥M) ⇐ ⇒

def

λi(·) = ni, · curvature normal Ei (a subbundle of T M) ⇐ ⇒

def

Ei := ∐

p∈M Ep i

curvature distribution lp

i ⊂ T ⊥ p M ⇐

⇒

def

lp

i := (λi)−1 p (1)

focal hyperplane

Focal radii and Focal hyperplanes

M ⊂ G/K

φ

• M := (π ◦ φ)−1(M)

⊂ H0([0, 1], g) lift

G π u ∈ (π ◦ φ)−1(p)

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

u : the normal geodesic of

M with γ′(0) = vL

u

equifocal isoparametric rank two

Focal radii and focal hyperplanes

u = γvL

u (0)

γvL

u (s1)

γvL

u (s2)

γvL

u (s3)

γvL

u (s4)

γvL

u (s5)

γvL

u (s6)

T ⊥

u

M(≈ R2)

• 4. ∞-dim. anti-Kaehler isoparametric

submanifold

∞-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(T M) = T M) 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(= 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 A

vp is 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 = Ep

0 ⊕

( ⊕

i∈I

Ep

i

) ( Ep

0 :=

∩

v∈T ⊥

p M Ker Av

) ( the common J − eigenspace decomposition

• f {Av | v ∈ T ⊥

p M}

) For each v ∈ T ⊥

p M,

λp

i (v) ⇐

⇒

def

Av|Ep

i = Re(λp

i (v))id + Im(λp i (v))Jp

Then λp

i : v → λp i (v) is complex linear, that is,

λp

i ∈ (T ⊥ p M)∗C.

Complex principal curvature, complex curvature distribution

. Def(complex curvature distribution) . . λi ∈ Γ((T ⊥M)∗C) ⇐ ⇒

def

(λi)p := λp

i

(p ∈ M) complex principal curvature ni ∈ Γ(T ⊥M) ⇐ ⇒

def

λi(·) = ni, · − √ −1Jni, · complex curvature normal Ei (a subbundle of T M) ⇐ ⇒

def

Ei := ∐

p∈M Ep i

complex curvature distribution lp

i ⊂ T ⊥ p M ⇐

⇒

def

lp

i := (λi)−1 p (1)

complex focal hyperplane

Complex focal radii and complex focal hyperplanes

M ⊂ G/K extrinsic complexification M C ⊂ GC/KC

φ

• M C := (π ◦ φ)−1(M C) ⊂ H0([0, 1], gC)

lift

GC π

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

u : the normal geodesic of

M C with γ′

vL

u (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

M C(≈ C2) l2 l1

γC

vL

u (≈ C)

γC

vL

u (√−1R)

γC

vL

u (R)

γC

vL

u (z1)

T ⊥

u

M C(≈ C2)

• 5. Outline of the proof of results

Recall Theorem B.2.

. Theorem B.2(K,2014) . . M : full irr. anti-Kaehler isoparametric submanifold of

codim M ≥ 2 with J-diagonalizable shape op. in V

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

p

(LEi

p

: the leaf of Ei through p) (Step I) We construct a C∞-family {F γ

t }t∈[0,1] in I(V )

s.t. { F γ

t (γ(0)) = γ(t)

(F γ

t )∗γ(0)|T ⊥

γ(0)M = τ ⊥

γ|[0,t]

(Step II) We show F γ

t (M) = M by using the assumption:

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

Outline of the proof of Theorem B.2

(Step III) We show that M = H′ · p holds for some subgroup H′ of I(V ) satisfying

∐

γ {F γ t }t∈[0,1] ⊂ H′ ⊂ ∐ γ {F γ t }t∈[0,1].

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

Outline of the proof of Theorem B.3

. Theorem B.3(K.2017) . . M : full irr. anti-Kaehler isoparametric Cω-submanifold

• f codim M ≥ 2 with J-diagonalizable shape op. in V

= ⇒ M : homogeneous (i.e., M = Hb · p (∃ Hb ⊂ Ib(V ))) Remark Hb = {F ∈ Ib(V ) | F (M) = M} We proved this theorem by refering the proof of

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

Outline of the proof of Theorem B.3

w ∈ (Ei)p (i ∈ I) γw : [0, 1] → LEi

p

: the geodesic in LEi

p

s.t. γ′

w(0) = w

F w

t

:= F γw

t

(F w

t (u) = Aw t (u) + bw t )

Xw ∈ X(U) ⇐ ⇒

def

(Xw)u := d dtF w

t (u)

• t=0

(u ∈ U) ( U := { u ∈ V

• d

dtF w

t (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)

Outline of the proof of Theorem B.3

Proof of Theorem B.3. F w

t (u) = Aw t (u) + bw t

(u ∈ V ) (Xw)u =

d dt

• t=0 F w

t (u) =

(

d dt

• t=0 Aw

t

) (u) + dbw

t

dt

• t=0

Xw is defined on V ⇔ d dt

• t=0

Aw

t is defined continuosly on V

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

u∈U s.t. u=1

Γw(u) < ∞

( · : the norm defined by , V±)

Outline of the proof of Theorem B.3

By long deliacte discusion, we can show sup

w∈∪i∈I(Ei)p s.t. w=1

sup

u∈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, F w

t

∈ Ib(V ) ( w ∈ ∪

p∈M ∪ i∈I(Ei)p

) . Furthermore, we can show the following: H′

b · p = M

( H′

b := ∐p∈M ∐ w {F w t }t∈[0,1] ⊂ Ib(V )

) .

Recall Theorem B.4.

. Theorem B.4(K,2018) . . M : full irreducible curvature-adapted isoparametric Cω-submanifold of codim M ≥ 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 ∩ ((Ker Av ∩ Ker R(v))C)⊥.

Outline of the proof of Theorem B.4

M ⊂ G/K extrinsic complexification M C ⊂ GC/KC

φ

• M C := (π ◦ φ)−1(M C) ⊂ H0([0, 1], gC)

lift

GC π G/K : symmetric space of non-compact type V := H0([0, 1], gC)

Outline of the proof of Theorem B.4

Outline of the proof of Theorem B.4. By the assumption for M,

• M C : full irr. anti-Kaehler isoparametric Cω-submfd of

codim M ≥ 2 with J-diagonalizable shape op. in V

By Theorem B.3,

• M C : homogeneous

(i.e., M C = Hb · p (∃ Hb ⊂ Ib(V ))). Without loss of generailty, we may assume M C = Hb · 0.

Outline of the proof of Theorem B.4

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(M C).

Outline of the proof of Theorem B.4

H′

R := (H′ ∩ (G × G))0 ∪ ({e} × K)0

Then we can show H′

R · e = π−1(M C) ∩ (G × G).

H′′

R := {g ∈ G | ({g} × K) ∩ H′ R = ∅}

Then we can show H′′

R · (eK) = M.

Recall Theorem B.5

. Theorem B.5(K,2018) . . M : full irreducible isoparametric Cω-submanifold

• f codim M ≥ 2 in G/K admitting

a reflective focal submanifold = ⇒ M : a principal orbit of a Hermann type action

Outline of the proof of Theorem B.5

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

. Theorem B.6(K,2018) . . M : full irreducible curvature-adapted isoparametric C∞-submanifold of codim M ≥ 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

• f 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 V B := (∆, 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

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 = dim G/K)

Outline of the proof of Theorem B.6

Then we can construct a topological Tits building BM = (△M := (VM, SM), AM, OM) satisfying                (i) VM = exp−1

p0 (F1 ∐ · · · ∐ Fl) (⊂ Sm−1(b))

(F1, · · · , Fl : focal submanifolds of M) (ii) |△M| = Sm−1(b) (iii) AM = {Ap}p∈M, |Ap| = Sm−1(b) ∩ exp−1

p0 (Σp)

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

Outline of the proof of Theorem B.6

Σp exp−1

p0 (Σp)

M F1 F2 F1 F2F1 F2 p Sm−1(b) Ap laminate M Fi p expp0(Ap) lp

1

lp

2

l p

3

expp0 v p0 ¯ lp

1

¯ lp

2

¯ lp

3

Tp0(G/K) G/K

Outline of the proof of Theorem B.6

v v′ Ap Ap′ Sm−1(b) (v := exp−1

p0 (p), v′ := exp−1 p0 (p′))

Outline of the proof of Theorem B.6

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′ VM − →

hence K′ VM

− →

extension K′ p′

Outline of the proof of Theorem B.6

v Ap Ap′ Sm−1(b) σ k′(σ) k′ · v

Outline of the proof of Theorem B.6

(Step II) We show that K′ p′ is orbit equivalent to the s-representation of G/K. (by using the discussion in Pge 444-445

• f [Thorbergsson, 1991])

Hence it follows that M ′ := exp−1

• (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. lp

i := (λi)−1 p (1) (⊂ T ⊥ p M) (i ∈ I)，

(T ⊥

p M)R := SpanR{(ni)p | i ∈ I}

(lp

i )R := l p i ∩ (T ⊥ p M)R

Remark M : full ⇒ T ⊥

p M = SpanC{(ni)p | i ∈ I}

Affine root system associated to M

{(lp

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

{(l p

i )R | i ∈ I} = {(lp a,j)R | a = 1, · · · , k j ∈ Z}

(

• (lp

a,j)R (j ∈ Z) are parallel.

• (lp

a,i)R and (lp b,j)R (a = b) are not parallel.

) . Fact. . . △M := {((lp

a,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

(lp

1,−1)R

(lp

1,0)R

(lp

1,1)R

(lp

1,2)R

(lp

2,1)R

(l p

2,0)R

(l p

2,−1)R

(lp

3,−1)R

(lp

3,0)R

(lp

3,1)R

(n1,0)p (n2,0)p (n3,0)p

Proof of sup

w1

sup

w2

Γw1w2 < ∞

(I) In case of △M is of type ( A), ( D), ( E), a = b, (na,i)p, (nb,j)p = 0 = ⇒ Γ(Ea,i)p(Eb,j)p = 0 (II) In case of △M is of type ( A), ( D), ( E), a = b, (na,i)p, (nb,j)p = 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 sup

w1

sup

w2

Γw1w2 < ∞

By showing many other facts for Γ, we can show sup

w

sup

u ||Γwu|| < ∞.