弱鏡映部分多様体とその周辺の話題.

.
.

酒井 高司
大阪市立大学数学研究所

2008 年 9 月 24 日
日本数学会 2008 年度秋季総合分科会
於 東京工業大学

酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2

Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4Orbits with degenerate Gauss mappings5Special Lagrangian submanifolds in Cn and T ∗ S n酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4Orbits with degenerate Gauss mappings

5

Special Lagrangian submanifolds in Cn and T ∗ S n

酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2

Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4

Orbits with degenerate Gauss mappings

5

Special Lagrangian submanifolds in Cn and T ∗ S n

酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2

Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4

Orbits with degenerate Gauss mappings

5

Special Lagrangian submanifolds in Cn and T ∗ S n酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2

Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4

Orbits with degenerate Gauss mappings

5

Special Lagrangian submanifolds in Cn and T ∗ S n

酒井 高司

弱鏡映部分多様体とその周辺の話題

Table of contents
O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds
and austere submanifolds, to appear in J. Math. Soc. Japan.
O. Ikawa, T. Sakai and H. Tasaki, Orbits of s-representations withdegenerate Gauss mappings, preprint.

.

1

Minimal submanifolds

2

Deﬁnition and fundamental properties of WRS

3Weakly reﬂective orbits and austere orbits of s-representations

4

Orbits with degenerate Gauss mappings

5

Special Lagrangian submanifolds in Cn and T ∗ S n酒井 高司

弱鏡映部分多様体とその周辺の話題

Minimal submanifolds
f, ⟨·, ·⟩)f : M −→ (Mimmersionf) −→ R
Vol : Imm(M, M

volume functionalf, compact support, F (·, 0) = f
F : M × (−ε, ε) −→ Mddt
whereddt∫
Vol(F (M, t)) = −⟨H, V ⟩dµM
M

t=0

H : mean curvature vector ﬁeld∂
V := F∗ ∂t: variational vector ﬁeld
Vol(F (M, t)) = 0 for ∀F

⇐⇒

H≡0

t=0

minimal submanifold
酒井 高司

弱鏡映部分多様体とその周辺の話題

Mean curvature vector fields
f, ⟨·, ·⟩, ∇)eM ⊂ (Mf = Tx M ⊕ T ⊥ M
Tx Mx
For X, Y ∈ Γ(T M ),ξ ∈ Γ(T ⊥ M )e X Y = ∇X Y + h(X, Y )∇(Gauss formula)h : second fundamental forme X ξ = −Aξ (X) + ∇⊥ ξ∇
(Weingarten formula)
XAξ : shape operator
H := trace(h) ∈ Γ(T ⊥ M )⟨h(X, Y ), ξ⟩ = ⟨Y, Aξ (X)⟩酒井 高司

弱鏡映部分多様体とその周辺の話題

Definition of reflective submanifold

Deﬁnition (D. Leung)
A connected component M of the ﬁxed point set of an involutivef, ⟨·, ·⟩) is called a reflective submanifold.
isometry σ of (M.
A reﬂective submanifold is a totally geodesic submanifold.
For all x ∈ M and all ξ ∈ Tx⊥ M , the reﬂection σ satisﬁes.
σ(x) = x,(dσ)x ξ = −ξ,酒井 高司

σ(M ) = M.

弱鏡映部分多様体とその周辺の話題

Definition of reflective submanifold

Deﬁnition (D. Leung)
A connected component M of the ﬁxed point set of an involutivef, ⟨·, ·⟩) is called a reflective submanifold.
isometry σ of (M.
A reﬂective submanifold is a totally geodesic submanifold.
For all x ∈ M and all ξ ∈ Tx⊥ M , the reﬂection σ satisﬁes.
σ(x) = x,(dσ)x ξ = −ξ,酒井 高司

σ(M ) = M.

弱鏡映部分多様体とその周辺の話題

Definition of a weakly reflective submanifold
Deﬁnitionf : weakly reflective submanifold (WRS)M ⊂M
def⇐⇒ 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.

.
酒井 高司

弱鏡映部分多様体とその周辺の話題

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⊥ Mσξ : S2n−1√( 2) −→ S2n−1√( 2)(u, v) 7−→ (v, u)Then σξ satisﬁesσξ (x) = x,(dσξ )x ξ = −ξ,酒井 高司

σξ (M ) = M

弱鏡映部分多様体とその周辺の話題.

Definition of austere submanifold

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

.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Definition of austere submanifold

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

.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Classes of submanifolds
Proposition.
reflective ⊂ WRS ⊂ austere ⊂ minimal.

.

(dσξ )−1x Aξ (dσξ )x = −Aξ

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

弱鏡映部分多様体とその周辺の話題

Classes of submanifolds
Proposition.
reflective ⊂ WRS ⊂ austere ⊂ minimal.

.

(dσξ )−1x Aξ (dσξ )x = −Aξ

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

弱鏡映部分多様体とその周辺の話題Orbits of cohomogeneity one actions
Propositionf, ⟨·, ·⟩) be a cohomogeneity one action with twoLet G y (Mf, thensingular orbits. If ∃ principal orbit which is a WRS in M.

1it has a same distance from two singular orbits,

2

two singular orbits are isometric.

.

.
.

酒井 高司

弱鏡映部分多様体とその周辺の話題Orbits of s-representations(G, K) : compact symmetric pair
g=k+m
Ad
Kym∼
= To (G/K) : s-representation

For H ∈ S ⊂ m,Ad(K)H ⊂ S : R-spacea ⊂ m : maximal abelian subspace
mλ := {X ∈ m | [H, [H, X]] = −⟨λ, H⟩2 X (∀ H ∈ a)}
R := {λ ∈ a | mλ ̸= {0}} : restricted root system∑

m=mλλ∈R+

酒井 高司

弱鏡映部分多様体とその周辺の話題Orbits of s-representations(G, K) : compact symmetric pair
g=k+m
AdKym∼= To (G/K) : s-representation

For H ∈ S ⊂ m,Ad(K)H ⊂ S : R-spacea ⊂ m : maximal abelian subspace
mλ := {X ∈ m | [H, [H, X]] = −⟨λ, H⟩2 X (∀ H ∈ a)}
R := {λ ∈ a | mλ ̸= {0}} : restricted root system∑

m=mλλ∈R+酒井 高司

弱鏡映部分多様体とその周辺の話題Orbits of s-representations
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamberC̄ ∩ S : orbit space of K y S

Ad(K)C̄ = mFor ∆ ⊂ FC{∆=⟨α, H⟩ > 0 (α ∈ ∆),⟨β, H⟩ = 0 (β ∈ F − ∆)H ∈ C̄C̄ ∩ S =∪ (C∆ ∩ S)}

disjoint union∆⊂FTheorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such thatAd(K)H is a minimal submanifold in the hypersphere S.
酒井 高司

弱鏡映部分多様体とその周辺の話題Orbits of s-representations
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamberC̄ ∩ S : orbit space of K y S

Ad(K)C̄ = mFor ∆ ⊂ FC{∆=⟨α, H⟩ > 0 (α ∈ ∆),⟨β, H⟩ = 0 (β ∈ F − ∆)H ∈ C̄C̄ ∩ S =∪ (C∆ ∩ S)}

disjoint union∆⊂FTheorem (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.
酒井 高司

弱鏡映部分多様体とその周辺の話題Orbits of s-representations
F : fundamental system of R
C = {H ∈ a | ⟨α, H⟩ > 0 (∀α ∈ F )} : Weyl chamberC̄ ∩ S : orbit space of K y S

Ad(K)C̄ = mFor ∆ ⊂ FC{∆=⟨α, H⟩ > 0 (α ∈ ∆),⟨β, H⟩ = 0 (β ∈ F − ∆)H ∈ C̄C̄ ∩ S =∪ (C∆ ∩ S)}

disjoint union∆⊂FTheorem (Hirohashi-Song-Takagi-Tasaki)
For a subset ∆ ⊂ F there exists unique H ∈ C ∆ ∩ S such thatAd(K)H is a minimal submanifold in the hypersphere S.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Austere orbits of s-representations (Ikawa-Tasaki-S.)
An orbit Ad(K)H of an irreducible s-representation which is anaustere submanifold in the hypersphere S is one of the following
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 = e15

R = D4 ; H = e1 + e2 + e3 ± e46R = B2 with constant multiplicities;

7α2R = G2 ; H = α1 + √32H = e1 + e1√+e2Moreover, in the cases (1)〜(5), these austere orbits are weakly
reﬂective submanifolds in S.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Gauss mappingf : M l −→ S nimmersion

γ:M−→ Gl+1 (Rn+1 )Gauss map

x 7−→ Rf (x) ⊕ Tf (x) (f (M ))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.

Problem (Ishikawa-Kimura-Miyaoka)
Is the inequality rank γ < F (l) best possible for the implication.
rank γ = 0?
酒井 高司

弱鏡映部分多様体とその周辺の話題

Gauss mappingf : M l −→ S nimmersion

γ:M−→ Gl+1 (Rn+1 )Gauss map

x 7−→ Rf (x) ⊕ Tf (x) (f (M ))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.

Problem (Ishikawa-Kimura-Miyaoka)
Is the inequality rank γ < F (l) best possible for the implication.
rank γ = 0?
酒井 高司

弱鏡映部分多様体とその周辺の話題

Gauss mappingf : M l −→ S nimmersion

γ:M−→ Gl+1 (Rn+1 )Gauss map

x 7−→ Rf (x) ⊕ Tf (x) (f (M ))Theorem (Ferus)
M l : compact, connected manifold, f : M −→ S n : immersion
Then， rank γ < F (l) =⇒ rank γ = 0
F (l) = min{k | A(k) + k ≥ l} : Ferus number
A((2k + 1)2c+4d ) = 2c + 8d − 1 : Adams number.

Problem (Ishikawa-Kimura-Miyaoka)
Is the inequality rank γ < F (l) best possible for the implication.
rank γ = 0?
酒井 高司

弱鏡映部分多様体とその周辺の話題

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

2R = G2 ;H : a short root

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

.

.
These orbits are weakly reﬂective submanifolds.
In this list, we can ﬁnd many orbits which satisfy the equalityof the Ferus inequality.
酒井 高司

弱鏡映部分多様体とその周辺の話題

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

1

H : 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 this list, we can ﬁnd many orbits which satisfy the equalityof the Ferus inequality.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Tangentially degenerate orbits
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 =∩Ad(k)H)k∈(ZK0where∑mλλ∈R+λ//H

H = {k ∈ K | Ad(K)H = H}
ZK酒井 高司

弱鏡映部分多様体とその周辺の話題

Tangentially degenerate orbits
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 =∩Ad(k)H)k∈(ZK0where∑mλλ∈R+λ//H

H = {k ∈ K | Ad(K)H = H}
ZK

酒井 高司

弱鏡映部分多様体とその周辺の話題

Case of type A2

酒井 高司

弱鏡映部分多様体とその周辺の話題

Case of type B2

酒井 高司

弱鏡映部分多様体とその周辺の話題

Case of type G2

酒井 高司

弱鏡映部分多様体とその周辺の話題

Special Lagrangian geometryf, J, ω, Ω) : Calabi-Yau manifold(MDeﬁnitionf : special Lagrangian submanifold of phase θL⊂Mdef⇐⇒ L is calibrated by Re(eiθ Ω) for some θ ∈ R.
{ω|L ≡ 0⇐⇒Im(eiθ Ω|L ) ≡ 0.
Theorem (Harvey-Lawson)
A calibrated submanifold is volume minimizing in its homologyclass.

酒井 高司

弱鏡映部分多様体とその周辺の話題.

Special Lagrangian geometryf, J, ω, Ω) : Calabi-Yau manifold(MDeﬁnitionf : special Lagrangian submanifold of phase θL⊂Mdef⇐⇒ L is calibrated by Re(eiθ Ω) for some θ ∈ R.
{ω|L ≡ 0⇐⇒Im(eiθ Ω|L ) ≡ 0.
Theorem (Harvey-Lawson)
A calibrated submanifold is volume minimizing in its homologyclass.

酒井 高司

弱鏡映部分多様体とその周辺の話題.

Special Lagrangian normal bundles
M ⊂ S n : submanifold
Φ : N 1 M × S 1 −→ S 2n+1 ⊂ R2n+2Legendrian(x, ξ, eiθ ) 7−→ (cos θ x, sin θ ξ)
Proposition (Harvey-Lawson)
M ⊂ S n : austere⇐⇒Φ : minimale modifying Φ and showed that
Borrelli-Gorodski deﬁned a map Φe : Legendrian immersionAξ does not have 0-eigenvalue =⇒ Φ.

酒井 高司

弱鏡映部分多様体とその周辺の話題.

Special Lagrangian normal bundles
M ⊂ S n : submanifold
Φ : N 1 M × S 1 −→ S 2n+1 ⊂ R2n+2Legendrian(x, ξ, eiθ ) 7−→ (cos θ x, sin θ ξ)
Proposition (Harvey-Lawson)
M ⊂ S n : austere⇐⇒Φ : minimale modifying Φ and showed that
Borrelli-Gorodski deﬁned a map Φe : Legendrian immersionAξ does not have 0-eigenvalue =⇒ Φ.

酒井 高司

弱鏡映部分多様体とその周辺の話題.

Stenzel metric on T ∗ S nS n = SO(n + 1)/SO(n) =: G/K{}
T ∗ S n = (x, ξ) ∈ Rn+1 × Rn+1 | ∥x∥ = 1, ⟨x, ξ⟩ = 0{}n∑nn+12Q = (z0 , . . . , zn ) ∈ Cz =1 ∼= GC /K C
ii=0

φ : T ∗ S n −→ Qn ⊂ Cn+1

diﬀeomorphism√ξ(x, ξ) 7−→ x cosh(∥ξ∥) + −1sinh(∥ξ∥)∥ξ∥

G = SO(n + 1) acts on T ∗ S n and Qn with cohomogeneityone.
φ is G-equivariant.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Stenzel metric on T ∗ S n
S n = SO(n + 1)/SO(n) =: G/K{}
T ∗ S n = (x, ξ) ∈ Rn+1 × Rn+1 | ∥x∥ = 1, ⟨x, ξ⟩ = 0{}n∑nn+12Q = (z0 , . . . , zn ) ∈ Cz =1 ∼= GC /K C
ii=0

φ : T ∗ S n −→ Qn ⊂ Cn+1

diﬀeomorphism√ξ(x, ξ) 7−→ x cosh(∥ξ∥) + −1sinh(∥ξ∥)∥ξ∥

G = SO(n + 1) acts on T ∗ S n and Qn with cohomogeneityone.
φ is G-equivariant.
酒井 高司

弱鏡映部分多様体とその周辺の話題

Stenzel metric on T ∗ S n

ωStz =√¯ 2) =−1∂ ∂u(r√−1n∑i,j=0∂2u(r2 )dzi ∧ dz¯j∂zi ∂ z¯jwhere r2 = ∥z∥2 and u is a real function given by
U (τ ) = u(cosh(τ )) where U satisﬁesd(U ′ (τ ))n = cn(sinh τ )n−1dτ(c > 0).

Ω : holomorphic (n, 0)-form on Qn deﬁned byΩ ∧ d(z02 + z12 + · · · + zn2 ) = dz0 ∧ dz1 ∧ . . . ∧ dzn .
(T ∗ S n , J, ωStz , Ω) is a cohomogeneity one Calabi-Yau manifold.
酒井 高司

弱鏡映部分多様体とその周辺の話題Special Lagrangian normal bundles

M ⊂ S n : submanifold
Then L = N ∗ M is a Lagrangian submanifold of T ∗ S n with respect
to the canonical symplectic structure ω0 .
Theorem (Karigiannis-Min-Oo)
L = N ∗ M is a Lagrangian submanifold of T ∗ S n with respect toStenzel metric ωStz .
Moreover, L is a special Lagrangian submanifold of T ∗ S n if andonly if M is an austere submanifold in S n .

酒井 高司

弱鏡映部分多様体とその周辺の話題

Cohomogeneity one special Lagrangian submanifolds
γ(s) ⊂ C : regular curvep+q =n+1

Ψ : I × S p−1 × S q−1 −→ Qn ⊂ Cn()√√(s, x, y) 7−→ γ(s)x1 , . . . , γ(s)xp , 1 − γ(s)2 y1 , . . . , 1 − γ(s)2 yq
L := Ψ(I × S p−1 × S q−1 )=⇒ωStz |L ≡ 0

In addition, if γ satisﬁes()
q−2
Im γ ′ γ p−1 (1 − γ 2 ) 2 = 0,=⇒Im(Ω|L ) ≡ 0

L is a special Lagrangian submanifold of T ∗ S n ∼= Qn w.r.t. ωStz .
酒井 高司

弱鏡映部分多様体とその周辺の話題

Phase space
Case of n = 6, p = 3, q = 421-2-1

12-1-2ﬁgure by 橋本 要
酒井 高司

弱鏡映部分多様体とその周辺の話題

Further problems

Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Ishikawa-Kimura-Miyaoka’s problem for the Ferus inequality.
Study the geometry of special Lagrangian submanifolds
obtained as conormal bundles of austere submanifolds.

酒井 高司

弱鏡映部分多様体とその周辺の話題

Further problems

Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Ishikawa-Kimura-Miyaoka’s problem for the Ferus inequality.
Study the geometry of special Lagrangian submanifolds
obtained as conormal bundles of austere submanifolds.

酒井 高司

弱鏡映部分多様体とその周辺の話題

Further problems

Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Ishikawa-Kimura-Miyaoka’s problem for the Ferus inequality.
Study the geometry of special Lagrangian submanifolds
obtained as conormal bundles of austere submanifolds.

酒井 高司

弱鏡映部分多様体とその周辺の話題

Further problems

Classify all weakly reﬂective submanifolds in Riemanniansymmetric spaces.
Homogeneity of weakly reﬂective submanifolds.
Construct examples of non-homogeneous weakly reﬂective
submanifolds.
Ishikawa-Kimura-Miyaoka’s problem for the Ferus inequality.
Study the geometry of special Lagrangian submanifolds
obtained as conormal bundles of austere submanifolds.

酒井 高司

弱鏡映部分多様体とその周辺の話題