.
......
Isoparametric submanifolds admitting
a reflective focal submanifold
in symmetirc spaces of non-compact type
Naoyuki Koike
Tokyo University of Science
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
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.
Homogeneity of ∞-dim. isoparametric submanifolds
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
=⇒ 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
codimM ≥ 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 ∩ ((KerAv ∩ KerR(v))C)⊥.
.Theorem B.4(K,2018)..
......
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.)
Homogeneity of isoparametric submanifolds in sym. sp. of
non-cpt type
.Theorem B.5(K,2018)..
......
M : full irreducible isoparametric Cω-submanifold
of codimM ≥ 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
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 .
.Theorem B.6(K,2018)..
......
M : full irreducible curvature-adapted isoparametric
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)
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 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
• 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
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
Complex focal radius
.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)
Complex focal radius
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
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 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.
.Theorem B.2(K,2014)..
......
M : full irr. anti-Kaehler isoparametric submanifold of
codimM ≥ 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 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”.
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
〈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}).
Outline of the proof of Theorem B.3
.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 )))
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] → 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)
Outline of the proof of Theorem B.3
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±)
Outline of the proof of Theorem B.3
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.
.Theorem B.4(K,2018)..
......
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)))
(∗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)⊥.
Outline of the proof of Theorem B.4
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
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 ,
MC : full irr. anti-Kaehler isoparametric Cω-submfd of
codimM ≥ 2 with J-diagonalizable shape op. in V
By Theorem B.3,
MC : homogeneous
(i.e., MC = Hb · p (∃ Hb ⊂ Ib(V ))).
Without loss of generailty, we may assume MC = 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(MC).
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(MC) ∩ (G × G).
H ′′R := {g ∈ G | ({g} × K) ∩ H ′
R 6= ∅}
Then we can show
H ′′R · (eK) = M .
Recall Theorem B.5
.Theorem B.5(K,2018)..
......
M : full irreducible isoparametric Cω-submanifold
of codimM ≥ 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 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
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)
Outline of the proof of Theorem B.6
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).
Outline of the proof of Theorem B.6
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
Outline of the proof of Theorem B.6
v
v′ApAp′
Sm−1(b)
(v := exp−1p0
(p), v′ := exp−1p0
(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′ y VM −→hence
K′ y VM −→extension
K′ y p′
Outline of the proof of Theorem B.6
v
ApAp′
Sm−1(b)
σ
k′(σ)
k′ · v
Outline of the proof of Theorem B.6
(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|| < ∞.