Weakly reﬂective orbits and tangentiallydegenerate orbits of s-representations.
Takashi Sakai.

Osaka City University Advanced Mathematical Institute
April 1, 2008
at National Taiwan University

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reﬂective
submanifolds and austere submanifolds,
arXiv:math.DG/0612384
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations
with degenerate Gauss mappings, arXiv:math.DG/0803.2391.

1

Deﬁnitions and fundamental results related to weakly
reﬂective submanifolds

2

A classiﬁcation of wealky reﬂective orbits ofs-representations

3

A classiﬁcation of orbits of s-representations withdegenerate Gauss mappings

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reﬂective
submanifolds and austere submanifolds,
arXiv:math.DG/0612384
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations
with degenerate Gauss mappings, arXiv:math.DG/0803.2391.

1

Deﬁnitions and fundamental results related to weakly
reﬂective submanifolds

2

A classiﬁcation of wealky reﬂective orbits ofs-representations

3

A classiﬁcation of orbits of s-representations withdegenerate Gauss mappings

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reﬂective
submanifolds and austere submanifolds,
arXiv:math.DG/0612384
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations
with degenerate Gauss mappings, arXiv:math.DG/0803.2391.

1

Deﬁnitions and fundamental results related to weakly
reﬂective submanifolds

2

A classiﬁcation of wealky reﬂective orbits ofs-representations

3

A classiﬁcation of orbits of s-representations withdegenerate Gauss mappings

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reﬂective
submanifolds and austere submanifolds,
arXiv:math.DG/0612384
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations
with degenerate Gauss mappings, arXiv:math.DG/0803.2391.

1

Deﬁnitions and fundamental results related to weakly
reﬂective submanifolds

2

A classiﬁcation of wealky reﬂective orbits ofs-representations

3

A classiﬁcation of orbits of s-representations withdegenerate Gauss mappings

Deﬁnition of a reﬂective submanifold

Deﬁnition (D. Leung)f be a complete Riemannian manifold. A connected
Let M
component M of the ﬁxed point set of an involutive isometry σ
f is called a reﬂective submanifold.
of M.
A reﬂective submanifold is a complete totally geodesic
submanifold.
For all x ∈ M and all ξ ∈ Tx⊥ M , the reﬂection σ .satisﬁes
σ(x) = x,(dσ)x ξ = −ξ,σ(M ) = M.

Deﬁnition of a weakly reﬂective submanifold
Deﬁnitionf : weakly reﬂective submanifold (WRS)M ⊂Mdef⇐⇒ for each x ∈ M and each ξ ∈ Tx⊥ M ,f which satisﬁes
there exists an isometry σξ of Mσξ (x) = x,(dσξ )x ξ = −ξ,σξ (M ) = M.
.
We call σξ a reﬂection of M with respect to ξ.

.

An example of a weakly reﬂective submanifold
ExampleM = S n−1 (1) × S n−1 (1) = {(u, v) ∈ Rn × Rn | u, v ∈ S n−1 (1)}√is a weakly reﬂective submanifold in S 2n−1 ( 2).

.

x = (e1 , e1 ) ∈ M
ξ = (e1 , −e1 ) ∈ Tx⊥ M√√σξ : S 2n−1 ( 2) −→ S 2n−1 ( 2)(u, v) 7−→ (v, u)Then σξ satisﬁesσξ (x) = x,(dσξ )x ξ = −ξ,σξ (M ) = M.

Deﬁnition of an austere submanifold

Deﬁnition (Harvey-Lawson)f : austere submanifold
M ⊂Mdef⇐⇒ for all ξ ∈ Tx⊥ M , the set of eigenvalues of the shapeoperator Aξ of M is invariant under the multiplication by −1,concerning multiplicities.
.
An austere submanifolds is a minimal submanifold.
A minimal surface is an austere submanifold.

.

Twisted normal bundle (Harvey-Lawson, Borrelli-Gorodski)

M ⊂ S n : submanifold
Φ : N 1 M × S 1 −→ S 2n+1 ⊂ R2n+2(vx , eiθ ) 7−→ (cos θ x, sin θ vx )Aξ does not have 0-eigenvalue =⇒ Φ : Legendrianimmersion
M ⊂ S n : austere =⇒ Φ : minimal

Classes of submanifolds

Proposition

reﬂective ⊂ WRS ⊂ austere ⊂ minimal.

Proposition (Podestà, Ikawa-Tasaki-S.)
Any singular orbit of a cohomogeniety one action on a.
Riemannian manifold is a weakly reﬂective submanifold.

Orbits of cohomogeneity one actions
Proposition
Let G be a connected Lie group acting isometrically on acomplete, connected Riemannian manifold M̃ . Suppose that theaction of G on M̃ is cohomogeneity one with two singularorbits. If there exists a principal orbit which is a weakly. fromreﬂective submanifold in M̃ , then it has a same distance
two singular orbits and two singular orbits are isometric.

.

Orbits of s-representations(G, K) : compact symmetric pair
g=k+m
The linear isotropy representation K y m is called ans-representation. For H ∈ S ⊂ m, the orbit Ad(K)H ⊂ S
through H is called an R-space.
Related topics
an orbit of the s-representation of a symmetric space ofrank 2 ⇐⇒ a homogeneous hypersurface in S n
G : compact, simple Lie algebra.
Ad(G)H ⊂ g ⇐⇒ a Kähler C-spacea symmetric R-space ⇐⇒ a symmetric submanifold in Rn
an s-representation ⇐⇒ a polar representation

Orbits of s-representations(G, K) : compact symmetric pair
g=k+m
The linear isotropy representation K y m is called ans-representation. For H ∈ S ⊂ m, the orbit Ad(K)H ⊂ S
through H is called an R-space.
Related topics
an orbit of the s-representation of a symmetric space ofrank 2 ⇐⇒ a homogeneous hypersurface in S n
G : compact, simple Lie algebra.
Ad(G)H ⊂ g ⇐⇒ a Kähler C-spacea symmetric R-space ⇐⇒ a symmetric submanifold in Rn
an s-representation ⇐⇒ a polar representation

a ⊂ m : maximal abelian subspacemλ = {X ∈ m | [H, [H, X]] = −⟨λ, H⟩2 X (∀ H ∈ a)}
R = {λ ∈ a | mλ ̸= {0}} : restricted root system
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamber
Ad(K)C̄ = m⇐⇒C̄ ∩ S : orbit space

For ∆ ⊂ FC ∆ = {H ∈ C̄ | ⟨α, H⟩ > 0 (α ∈ ∆), ⟨β, H⟩ = 0 (β ∈ F − ∆)}C̄ =[∆⊂FC∆

disjoint uniona ⊂ m : maximal abelian subspacemλ = {X ∈ m | [H, [H, X]] = −⟨λ, H⟩2 X (∀ H ∈ a)}
R = {λ ∈ a | mλ ̸= {0}} : restricted root system
F : fundamental system of RC = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamberAd(K)C̄ = m⇐⇒

C̄ ∩ S : orbit space

For ∆ ⊂ FC ∆ = {H ∈ C̄ | ⟨α, H⟩ > 0 (α ∈ ∆), ⟨β, H⟩ = 0 (β ∈ F − ∆)}C̄ =[∆⊂FC∆

disjoint uniona ⊂ m : maximal abelian subspacemλ = {X ∈ m | [H, [H, X]] = −⟨λ, H⟩2 X (∀ H ∈ a)}
R = {λ ∈ a | mλ ̸= {0}} : restricted root system
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamber
Ad(K)C̄ = m⇐⇒C̄ ∩ S : orbit space

For ∆ ⊂ FC ∆ = {H ∈ C̄ | ⟨α, H⟩ > 0 (α ∈ ∆), ⟨β, H⟩ = 0 (β ∈ F − ∆)}C̄ =[∆⊂FC∆

disjoint unionOrbits of s-representations
Assume H ∈ C ∆ ⊂ C̄H
ZK= {k ∈ K | Ad(K)H = H}¯
= {k ∈ K | Ad(K)¯C ∆ = idC ∆ }H
Ad(K)H ∼= K/ZK

∆1 ⊂ ∆2 ⊂ F⇐⇒

H1 ∈ C ∆ 1 , H 2 ∈ C ∆ 2

C ∆1 ⊂ C̄ ∆2

=⇒

H1
H2ZK
⊃ ZKTheorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such that
Ad(K)H is a minimal submanifold in the hypersphere S.

Orbits of s-representations
Assume H ∈ C ∆ ⊂ C̄HZK= {k ∈ K | Ad(K)H = H}¯= {k ∈ K | Ad(K)¯C ∆ = idC ∆ }H
Ad(K)H ∼= K/ZK

∆1 ⊂ ∆2 ⊂ F⇐⇒

H1 ∈ C ∆ 1 , H 2 ∈ C ∆ 2C ∆1 ⊂ C̄ ∆2

=⇒

H1
H2ZK
⊃ ZK

Theorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such that
Ad(K)H is a minimal submanifold in the hypersphere S.

Austere orbits of s-representations
An orbit Ad(K)H of an irreducible s-representation which is anaustere submanifold in the hypersphere is one of the following
list:.

.

1An orbit through a restricted root

2

R = A2 ; H = 2e1 − e2 − e3 , e1 + e2 − 2e3

3

R = A3 ; H = e1 + e2 − e3 − e44

R = Dn ; H = e15

R = D4 ; H = e1 + e2 + e3 ± e4

6.
7.

2
R = B2 with constant multiplicities; H = e1 + e1√+e2(regular orbit)

α2R = G2 ; H = α1 + √3(regular orbit)

Weakly reﬂective orbits of s-representations

An orbit Ad(K)H of an irreducible s-representation which is aweakly reﬂective submanifold in the hypersphere is one of thefollowing list:

.

.

1

An orbit through a restricted root

2

R = A2 ; H = 2e1 − e2 − e3 , e1 + e2 − 2e3

3

R = A3 ; H = e1 + e2 − e3 − e44

R = Dn ; H = e1

5

R = D4 ; H = e1 + e2 + e3 ± e4

Gauss mappingf : M l −→ S nimmersion

Gauss map

γ:M−→ Gl+1 (Rn+1 )x 7−→ Rf (x) ⊕ Tf (x) (f (M ))Deﬁnition
A submanifold f (M ) ⊂ S n is said to be tangenetiallydegenerate if the Gauss map γ is degenerate.

Ferus inequality
Theorem (Ferus)
M l : compact, connected manifold, f : M −→ S n : immersion.
Then， rank γ < F (l) =⇒ rank γ = 0F (l) = min{k | A(k) + k ≥ l} : Ferus number
A((2k + 1)2c+4d ) = 2c + 8d − 1 : Adams number .
(0 ≤ c ≤ 3, 0 ≤ d)

Problem (Ishikawa-Kimura-Miyaoka)
1

2Is the inequality rank γ < F (l) best possible for theimplication rank γ = 0?
Classify tangentially degenerate immersions f : M l −→ S nwith rank γ = F (l).

Ferus inequality
Theorem (Ferus)
M l : compact, connected manifold, f : M −→ S n : immersion.
Then， rank γ < F (l) =⇒ rank γ = 0F (l) = min{k | A(k) + k ≥ l} : Ferus number
A((2k + 1)2c+4d ) = 2c + 8d − 1 : Adams number .
(0 ≤ c ≤ 3, 0 ≤ d)

Problem (Ishikawa-Kimura-Miyaoka)
12Is the inequality rank γ < F (l) best possible for theimplication rank γ = 0?
Classify tangentially degenerate immersions f : M l −→ S nwith rank γ = F (l).

Tangentially degenerate orbits

TheoremAn orbit Ad(K)H of an irreducible s-representation which istangentially degenerate is one of the following list:.

1H : a long root

2

R = G2 ;H : a short root

Moreover, if λ is such a root, thenker(dγ)λ = mλ .
.

.
These orbits are weakly reﬂective submanifolds.

In general, for the Gauss map γ of a submanifold M ⊂ S nker(dγ)x = {X ∈ Tx (M ) | h(X, Y ) = 0, ∀ Y ∈ Tx (M )}\=ker(Aξ )ξ∈Tx⊥ (M )For the Gauss map γ of the orbit Ad(K)Hker(dγ)H =\H)k∈(ZK0Ad(k)X

mµµ//H

Proposition
For a resticted root λ, the orbit Ad(K)λ is tangentiallydegenerateXmµ has non-zero subspace invariant under ad(zλK ).
⇐⇒µ//λCase of type A2

Case of type B2

Case of type G2

Related problems

Ishikawa-Kimura-Miyaoka’s problem on the Ferus
inequality.
Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Study geometry of special Lagrangian submanifolds
obtained from austere submanifolds via twisted normalbundle.

Related problems

Ishikawa-Kimura-Miyaoka’s problem on the Ferus
inequality.
Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Study geometry of special Lagrangian submanifolds
obtained from austere submanifolds via twisted normal
bundle.

Related problems

Ishikawa-Kimura-Miyaoka’s problem on the Ferus
inequality.
Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Study geometry of special Lagrangian submanifolds
obtained from austere submanifolds via twisted normal
bundle.

Related problems

Ishikawa-Kimura-Miyaoka’s problem on the Ferus
inequality.
Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Study geometry of special Lagrangian submanifolds
obtained from austere submanifolds via twisted normal
bundle.