首都大学東京 博士（理学）学位論文（課程博士）

首都大学東京 博士（理学）学位論文（課程博士）

論 文 名 可換な

Hermann

作用の軌道の幾何学的性質（英文）

著 者

審査担当者 主 査 委 員 委 員

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平成 年 月 日

首都大学東京大学院理工学研究科教授会

研究科長

DISSERTATION FOR A DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE

TOKYO METROPOLITAN UNIVERSITY

TITLE ： Geometric properties of orbits of commutative Hermann actions

AUTHOR ：

EXAMINED BY Examiner in chief Examiner

Examiner

QUALIFIED BY THE GRADUATE SCHOOL OF SCIENCE AND ENGINEERING

TOKYO METROPOLITAN UNIVERSITY Dean

Date

HERMANN ACTIONS

SHINJI OHNO

Abstract. In this thesis, we study geometric properties of orbits of commu- tative Hermann actions. A Hermann action is a generalization of isotropy actions of compact symmetric spaces.

Contents

1. Introduction 1

2. Hermann actions and symmetric triads 2

2.1. Symmetric triads 2

2.2. Minimal orbits and austere orbits 5

3. Weakly reflective submanifolds in compact symmetric spaces 19

3.1. Weakly reflective submanifolds 19

3.2. Sufficient conditions for orbits to be weakly reflective 20 4. Biharmonic submanifolds in compact symmetric spaces 29

4.1. Preliminaries 30

4.2. Biharmonic isometric immersions 31

4.3. Characterization theorem 33

4.4. Biharmonic orbits of cohomogeneity one Hermann actions 36

4.5. Classification theorem 40

4.6. Cases of cohomogeneity two or greater 45

References 87

1. Introduction

In Riemannian geometry, often submanifolds appear with special properties.

For example, minimal submanifolds have been studied by many mathematicians.

In spacial cases of minimal submanifolds, there are austere submanifolds and to- tally geodesic submanifolds. Austere submanifolds are associated with special La- grangian submanifolds in the cotangent bundle of the hypersphere. In addition, har- monic maps and biharmonic maps are interesting submanifolds. Geometric proper- ties listed above are described by the local structure of submanifolds. A reflectivity and a weakly reflectivity are geometric properties which require a global structure of submanifolds. Reflective submanifolds and weakly reflective submanifolds are totally geodesic and austere, respectively.

To understand these geometric properties, it is an important problem which constructs an example. One method for constructing examples is a method using

2010Mathematics Subject Classification. Primary 58E20; Secondary 53C43.

1

Lie group actions. W. Hsiang and H. B. Lawson constructed many examples of minimal hypersurfaces in the hypersphere using cohomogeneity one action on the hypersphere. This method can be applied to other geometric properties.

The author have studied Lie group actions on Riemannian symmetric spaces, such as isotropy representations and isotropy actions of compact symmetric spaces.

The second fundamental form of orbits of such actions are expressed by root system.

O. Ikawa ([I]) introduced the notion of symmetric triad as a generalization of the notion of irreducible root system to study orbits of commutative Hermann actions.

O. Ikawa expressed orbit spaces of Hermann actions by using symmetric triads, and gave a characterization of the minimal, austere and totally geodesic orbits of Hermann actions in terms of symmetric triads.

In this thesis, we consider commutative Hermann actions and associated ac- tions on compact Lie groups, and express the minimal, austere, weakly reflective, biharmonic properties of orbits of these actions in terms of symmetric triads.

In Section 2, we review the notion of root systems and symmetric triads. In particular, a minimal point, an austere point and a totally geodesic point are dis- cussed.

In Section 3, we recall the definition of weakly reflective submanifolds, and their fundamental properties, and we gave sufficient conditions for orbits of these actions to be weakly reflective. Using the sufficient conditions, we obtain many examples of weakly reflective submanifolds in compact symmetric spaces.

In Section 4, we give a characterization of biharmonic orbits of commutative Hermann actions and associated actions on Lie groups in terms of symmetric tri- ads. Using the characterization, we give examples of biharmonic submanifolds in compact symmetric spaces which is not necessarily hypersurfaces. The contents of this section is based on joint work with T. Sakai and H. Urakawa.

The author would like to express his deepest gratitude to his advisor, Professor Takashi Sakai. The author is also very grateful to Professor Osamu Ikawa, Professor Hiroyuki Tasaki, Professor Hiroshi Tamaru and Professor Hajime Urakawa for their discussions, valuable comments.

2. Hermann actions and symmetric triads

2.1. Symmetric triads. O. Ikawa ([I]) introduced the notion of symmetric triad as a generalization of the notion of irreducible root system to study orbits of Hermann actions. Ikawa expressed orbit spaces of Hermann actions by using symmetric triads, and gave a characterization of the minimal, austere and totally geodesic orbits of Hermann actions in terms of symmetric triads. We recall the notions of root system and symmetric triad. See [I] for details.

Let (a, ⟨· , ·⟩ ) be a finite dimensional inner product space over R . For each α ∈ a, we define an orthogonal transformation s

α

: a → a by

s

α

(H ) = H − 2 ⟨ α, H ⟩

⟨ α, α ⟩ α (H ∈ a),

namely s

α

is the reflection with respect to the hyperplane { H ∈ a | ⟨ α, H ⟩ = 0 } .

Definition 2.1. A finite subset Σ of a \ { 0 } is a root system of a, if it satisfies the

following three conditions:

(1) Span(Σ) = a.

(2) If α, β ∈ Σ, then s

α

(β) ∈ Σ.

(3) 2 ⟨ α, β ⟩ / ⟨ α, α ⟩ ∈ Z (α, β ∈ Σ).

A root system of a is said to be irreducible if it cannot be decomposed into two disjoint nonempty orthogonal subsets.

Let Σ be a root system of a. The Weyl group W (Σ) of Σ is the finite subgroup of the orthogonal group O(a) of a generated by { s

α

| α ∈ Σ } .

Definition 2.2 ([I] Definition 2.2). A triple ( ˜ Σ, Σ, W ) of finite subsets of a \ { 0 } is a symmetric triad of a, if it satisfies the following six conditions:

(1) ˜ Σ is an irreducible root system of a.

(2) Σ is a root system of a.

(3) ( − 1)W = W, Σ ˜ = Σ ∪ W .

(4) Σ ∩ W is a nonempty subset. If we put l := max {∥ α ∥ | α ∈ Σ ∩ W } , then Σ ∩ W = { α ∈ Σ ˜ | ∥ α ∥ ≤ l } .

(5) For α ∈ W and λ ∈ Σ \ W , 2 ⟨ α, λ ⟩

⟨ α, α ⟩ is odd if and only if s

α

(λ) ∈ W \ Σ.

(6) For α ∈ W and λ ∈ W \ Σ, 2 ⟨ α, λ ⟩

⟨ α, α ⟩ is odd if and only if s

α

(λ) ∈ Σ \ W.

Let ( ˜ Σ, Σ, W ) be a symmetric triad of a. We set

Γ = { H ∈ a | ⟨ λ, H ⟩ ∈ (π/2) Z (λ ∈ Σ) ˜ } , Γ

Σ∩W

= { H ∈ a | ⟨ λ, H ⟩ ∈ (π/2) Z (λ ∈ Σ ∩ W ) } .

A point in Γ is called a totally geodesic point. It is known that Γ = Γ

Σ∩W

. We define an open subset a

r

of a by

a

r

= !

λ∈Σ,α∈W

"

H ∈ a # # # ⟨ λ, H ⟩ ̸∈ π Z , ⟨ α, H ⟩ ̸∈ π 2 + π Z $

.

A point in a

r

is called a regular point, and a point in the complement of a

r

in a is called a singular point. A connected component of a

r

is called a cell. The affine Weyl group W ˜ ( ˜ Σ, Σ, W ) of ( ˜ Σ, Σ, W ) is a subgroup of the affine group of a, which defined by the semidirect product O(a) ! a, generated by

%&

s

λ

, 2nπ

⟨ λ, λ ⟩ λ ' ##

# # λ ∈ Σ, n ∈ Z (

∪

%&

s

α

, (2n + 1)π

⟨ α, α ⟩ α ' ##

# # α ∈ W, n ∈ Z (

. The action of (s

λ

, (2nπ/ ⟨ λ, λ ⟩ )λ) on a is the reflection with respect to the hyperplane { H ∈ a | ⟨ λ, H ⟩ = nπ } , and the action of (s

α

, ((2n + 1)π/ ⟨ α, α ⟩ )α) on a is the reflection with respect to the hyperplane { H ∈ a | ⟨ α, H ⟩ = (n + 1/2)π } . The affine Weyl group ˜ W ( ˜ Σ, Σ, W ) acts transitively on the set of all cells. More precisely, for each cell P , it holds that

a = )

s∈W˜( ˜Σ,Σ,W)

sP .

We take a fundamental system ˜ Π of ˜ Σ. We denote by ˜ Σ

⁺

the set of positive roots in ˜ Σ. Set Σ

⁺

= ˜ Σ

⁺

∩ Σ and W

⁺

= ˜ Σ

⁺

∩ W . Denote by Π the set of simple roots of Σ. We set

W

0

= { α ∈ W

⁺

| α + λ ̸∈ W (λ ∈ Π) } .

From the classification of symmetric triads, we have that W

0

consists of the only one element, denoted by ˜ α. We define an open subset P

0

of a by

(2.1) P

0

= !

H ∈ a " " " ⟨ α, H ˜ ⟩ < π

2 , ⟨ λ, H ⟩ > 0 (λ ∈ Π) # . Then P

0

is a cell. For an nonempty subset ∆ ⊂ Π ∪ { α ˜ } , set

P

₀^∆

=

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

H ∈ P

0

⟨ λ, H ⟩ > 0 (λ ∈ ∆ ∩ Π)

⟨ µ, H ⟩ = 0 (µ ∈ Π \ ∆)

⟨ α, H ˜ ⟩

( < (π/2) (if ˜ α ∈ ∆)

= (π/2) (if ˜ α ̸∈ ∆)

⎫ ⎪

⎪ ⎪

⎬

⎪ ⎪

⎪ ⎭ ,

then

P

0

= ,

∆⊂Π∪{α˜}

P

₀^∆

(disjoint union).

Definition 2.3 ([I] Definition 2.13). Let ( ˜ Σ, Σ, W ) be a symmetric triad of a.

Consider two mappings m and n from ˜ Σ to R

≥0

:= { a ∈ R | a ≥ 0 } which satisfy the following four conditions:

(1) For any λ ∈ Σ, ˜

(1-1) m(λ) = m( − λ), n(λ) = n( − λ), (1-2) m(λ) > 0 if and only if λ ∈ Σ, (1-3) n(λ) > 0 if and only if λ ∈ W .

(2) When λ ∈ Σ, α ∈ W, s ∈ W (Σ), then m(λ) = m(s(λ)), n(α) = n(s(α)).

(3) When λ ∈ Σ, ˜ σ ∈ W ( ˜ Σ), then m(λ) + n(λ) = m(σ(λ)) + n(σ(λ)).

(4) Let λ ∈ Σ ∩ W , α ∈ W . If 2 ⟨ α, λ ⟩ / ⟨ α, α ⟩ is even, then m(λ) = m(s

α

(λ)).

If 2 ⟨ α, λ ⟩ / ⟨ α, α ⟩ is odd, then m(λ) = n(s

α

(λ)).

We call m(λ) and n(α) the multiplicities of λ and α, respectively.

Let ( ˜ Σ, Σ, W ) be a symmetric triad of a with multiplicities m and n. For H ∈ a, we set

m

H

= − -

λ∈Σ⁺

⟨λ,H⟩̸∈πZ

m(λ) cot ⟨ λ, H ⟩ λ + -

α∈W⁺

⟨α,H⟩̸∈(π/2)+πZ

n(α) tan ⟨ α, H ⟩ α.

The vector m

H

is called the mean curvature vector at H. A vector H ∈ a is a minimal point if m

H

= 0.

Proposition 2.4 ([I] Theorem 2.14 ). Let ( ˜ Σ, Σ, W) be a symmetric triad of a with multiplicities. For H ∈ a and σ = (s, X) ∈ W ˜ ( ˜ Σ, Σ, W ), set H

^′

= σH ∈ a, then

m

H^′

= s(m

H

).

Theorem 2.5 ([I] Theorem 2.24 ). For any nonempty subset ∆ ⊂ Π ∪ { α ˜ } , there

exists a unique minimal point H ∈ P

₀^∆

.

A vector H ∈ a is an austere point if the subset of a with multiplicities defined by

{− cot ⟨ λ, H ⟩ λ (multiplicity= m(λ)) | λ ∈ Σ

⁺

, ⟨ λ, H ⟩ ̸∈ π Z}

∪ { tan ⟨ α, H ⟩ α (multiplicity= n(α)) | α ∈ W

⁺

, ⟨ α, H ⟩ ̸∈ (π/2) + π Z}

is invariant with multiplicities under the multiplication by − 1. An austere point is a minimal point.

Proposition 2.6 ([I] Theorem 2.18). A point H ∈ a is austere if and only if the following three conditions holds:

(1) ⟨ λ, H ⟩ ∈ (π/2) Z for any λ ∈ (Σ \ W ) ∪ (W \ Σ).

(2) 2H ∈ Γ

Σ∩W

.

(3) m(λ) = n(λ) for any λ ∈ Σ ∩ W with ⟨ λ, H ⟩ ∈ (π/4) + (π/2) Z .

Ikawa gave the classification of symmetric triad and determined austere points for symmetric triads with multiplicities.

2.2. Minimal orbits and austere orbits. In this section, we consider Hermann actions and associated actions on Lie groups which are hyperpolar actions on com- pact symmetric spaces. A. Kollross ([Kol]) classified the hyperpolar actions on compact irreducible symmetric spaces. By the classification, we can see that a hyperpolar action on a compact symmetric space whose cohomogeneity is two or greater, is orbit-equivalent to some Hermann action.

Let G be a compact, connected, semisimple Lie group, and K

1

, K

2

be closed subgroups of G. For each i = 1, 2, assume that there exists an involutive automor- phism θ

i

of G which satisfies (G

θi

)

0

⊂ K

i

⊂ G

θi

, where G

θi

is the set of fixed points of θ

i

and (G

i

)

0

is the identity component of G

θi

. Then the triple (G, K

1

, K

2

) is called a compact symmetric triad. The pair (G, K

i

) is a compact symmetric pair for i = 1, 2. We denote the Lie algebras of G, K

1

and K

2

by g, k

1

and k

2

, respectively.

The involutive automorphism of g induced from θ

i

will be also denoted by θ

i

. Take an Ad(G)-invariant inner product ⟨· , ·⟩ on g. Then the inner product ⟨· , ·⟩ induces a bi-invariant Riemannian metric on G and G-invariant Riemannian metrics on the coset manifolds M

1

:= G/K

1

and M

2

:= K

2

\ G. We denote these Riemannian metrics on G, M

1

and M

2

by the same symbol ⟨· , ·⟩ . These Riemannian manifolds G, M

1

and M

2

are Riemannian symmetric spaces with respect to ⟨· , ·⟩ . We denote by π

i

the natural projection from G to M

i

(i = 1, 2), and consider the following three Lie group actions:

• (K

2

× K

1

) ! G : (k

2

, k

1

)g = k

2

gk

⁻¹₁

((k

2

, k

1

) ∈ K

2

× K

1

),

• K

2

! M

1

: k

2

π

1

(g) = π

1

(k

2

g) (k

2

∈ K

2

),

• K

1

! M

2

: k

1

π

2

(g) = π

2

(gk

⁻₁¹

) (k

1

∈ K

1

),

for g ∈ G. The three actions have the same orbit space, and in fact, the following diagram is commutative:

G −−−−→

^π2

M

2 π1

⏐ ⏐

" ⏐ ⏐ "

^π˜1

M

1

−−−−→

_π˜

2

K

2

\ G/K

1

,

where ˜ π

i

is the natural projection from M

i

to the orbit space K

2

\ G/K

1

. Ikawa

computed the second fundamental form of orbits of Hermann actions in the case

θ

1

θ

2

= θ

2

θ

1

. We can apply Ikawa’s method to the geometry of orbits of the (K

2

× K

1

)-action. For g ∈ G, we denote the left (resp. right) transformation of G by L

g

(resp. R

g

). The isometry on M

1

(resp. M

2

) induced by L

g

(resp. R

g

) will be also denoted by the same symbol L

g

(resp. R

g

).

For i = 1, 2, we set

m

i

= { X ∈ g | θ

i

(X) = − X } .

Then we have an orthogonal direct sum decomposition of g that is the canonical decomposition:

g = k

i

⊕ m

i

.

The tangent space T

πi(e)

M

i

of M

i

at the origin π

i

(e) is identified with m

i

in a natural way. We define a closed subgroup G

12

of G by

G

12

= { g ∈ G | θ

1

(g) = θ

2

(g) } .

Hence ((G

12

)

0

, K

12

) is a compact symmetric pair, where K

12

is a closed subgroup of (G

12

)

0

defined by

K

12

= { k ∈ (G

12

)

0

| θ

1

(k) = k } . The canonical decomposition of ((G

12

)

0

, K

12

) is given by

g

12

= (k

1

∩ k

2

) ⊕ (m

1

∩ m

2

).

Fix a maximal abelian subspace a in m

1

∩ m

2

. Then exp(a) is a torus subgroup in (G

12

)

0

. Then exp(a), π

1

(exp(a)) and π

2

(exp(a)) are sections of the (K

2

× K

1

)- action, the K

2

-action and the K

1

-action, respectively. To investigate the orbit spaces of the three actions, we consider a equivalent relation ∼ on a defined as follows: For H

1

, H

2

∈ a, H

1

∼ H

2

if K

2

exp(H

1

)K

1

= K

2

exp(H

2

)K

1

. Clearly, we have H

1

∼ H

2

if and only if K

2

π

1

(exp(H

1

)) = K

2

π

1

(exp(H

2

)), and similarly, H

1

∼ H

2

if and only if K

1

π

2

(exp(H

1

)) = K

1

π

2

(exp(H

2

)). Then we have a/ ∼ = K

2

\ G/K

1

. For each subgroup L of G, we define

N

L

(a) = { k ∈ L | Ad(k)a = a } ,

Z

L

(a) = { k ∈ L | Ad(k)H = H (H ∈ a) } . Then Z

L

(a) is a normal subgroup of N

L

(a). We define a group ˜ J by

J ˜ = { ([s], Y ) ∈ N

K2

(a)/Z

K1∩K2

(a) ! a | exp( − Y )s ∈ K

1

} .

The group ˜ J naturally acts on a by the following:

([s], Y )H = Ad(s)H + Y (([s], Y ) ∈ J, H ˜ ∈ a).

Matsuki ([M]) proved that

K

2

\ G/K

1

∼ = a/ J. ˜

Hereafter, we suppose θ

1

θ

2

= θ

2

θ

1

. Then we have an orthogonal direct sum decom- position of g:

g = (k

1

∩ k

2

) ⊕ (m

1

∩ m

2

) ⊕ (k

1

∩ m

2

) ⊕ (m

1

∩ k

2

).

We define subspaces of g as follows:

k

0

= { X ∈ k

1

∩ k

2

| [a, X] = { 0 }} ,

V (k

1

∩ m

2

) = { X ∈ k

1

∩ m

2

| [a, X] = { 0 }} ,

V (m

1

∩ k

2

) = { X ∈ m

1

∩ k

2

| [a, X] = { 0 }} .

For λ ∈ a,

k

λ

= { X ∈ k

1

∩ k

2

| [H, [H, X]] = −⟨ λ, H ⟩

²

X (H ∈ a) } , m

λ

= { X ∈ m

1

∩ m

2

| [H, [H, X]] = −⟨ λ, H ⟩

²

X (H ∈ a) } , V

_λ^⊥

(k

1

∩ m

2

) = { X ∈ k

1

∩ m

2

| [H, [H, X]] = −⟨ λ, H ⟩

²

X (H ∈ a) } , V

_λ^⊥

(m

1

∩ k

2

) = { X ∈ m

1

∩ k

2

| [H, [H, X]] = −⟨ λ, H ⟩

²

X (H ∈ a) } . We set

Σ = { λ ∈ a \ { 0 } | k

λ

̸ = { 0 }} ,

W = { α ∈ a \ { 0 } | V

_α^⊥

(k

1

∩ m

2

) ̸ = { 0 }} , Σ ˜ = Σ ∪ W.

It is known that dim k

λ

= dim m

λ

and dim V

_λ^⊥

(k

1

∩ m

2

) = dim V

_λ^⊥

(m

1

∩ k

2

) for each λ ∈ Σ. Thus we set ˜ m(λ) := dim k

λ

, n(λ) := dim V

_λ^⊥

(k

1

∩ m

2

). Notice that Σ is the root system of the pair ((G

12

)

0

, K

12

), and ˜ Σ is a root system of a (see [I]). We take a basis of a and the lexicographic ordering > on a with respect to the basis. We set

Σ ˜

⁺

= { λ ∈ Σ ˜ | λ > 0 } , Σ

⁺

= Σ ∩ Σ ˜

⁺

, W

⁺

= W ∩ Σ ˜

⁺

. Then we have an orthogonal direct sum decomposition of g:

g = k

0

⊕ !

λ∈Σ⁺

k

λ

⊕ a ⊕ !

λ∈Σ⁺

m

λ

⊕ V (k

1

∩ m

2

) ⊕ !

α∈W⁺

V

_α^⊥

(k

1

∩ m

2

)

⊕ V (m

1

∩ k

2

) ⊕ !

α∈W⁺

V

_α^⊥

(m

1

∩ k

2

).

Furthermore, we have the following lemma.

Lemma 2.7 ([I] Lemmas 4.3 and 4.16). (1) For each λ ∈ Σ

⁺

, there exist or- thonormal bases { S

λ,i

}

^m(λ)i=1

and { T

λ,i

}

^m(λ)i=1

of k

λ

and m

λ

respectively such that for any H ∈ a,

[H, S

λ,i

] = ⟨ λ, H ⟩ T

λ,i

, [H, T

λ,i

] = −⟨ λ, H ⟩ S

λ,i

, [S

λ,i

, T

λ,i

] = λ, Ad(exp H)S

λ,i

= cos ⟨ λ, H ⟩ S

λ,i

+ sin ⟨ λ, H ⟩ T

λ,i

,

Ad(exp H )T

λ,i

= − sin ⟨ λ, H ⟩ S

λ,i

+ cos ⟨ λ, H ⟩ T

λ,i

.

(2) For each α ∈ W

⁺

, there exist orthonormal bases { X

α,j

}

^n(α)j=1

and { Y

α,j

}

^n(α)j=1

of V

_α^⊥

(k

1

∩ m

2

) and V

_α^⊥

(m

1

∩ k

2

) respectively such that for any H ∈ a [H, X

α,j

] = ⟨ α, H ⟩ Y

α,j

, [H, Y

α,j

] = −⟨ α, H ⟩ X

α,j

, [X

α,j

, Y

α,j

] = α,

Ad(exp H)X

α,j

= cos ⟨ α, H ⟩ X

α,j

+ sin ⟨ α, H ⟩ Y

α,j

, Ad(exp H )Y

α,j

= − sin ⟨ α, H ⟩ X

α,j

+ cos ⟨ α, H ⟩ Y

α,j

. Using Lemma 2.7, Ikawa proved the following theorems.

Theorem 2.8 ([I] Lemma 4.22). Let x = exp H for H ∈ a. Then we have:

(1) dL

⁻_x¹

B

H

(dL

x

(T

λ,i

), dL

x

(T

µ,j

)) = cot( ⟨ µ, H ⟩ )[T

λ,i

, S

µ,j

]

^⊥

, (2) dL

⁻¹_x

B

H

(dL

x

(Y

α,i

), dL

x

(Y

β,j

)) = − tan( ⟨ β, H ⟩ )[Y

α,i

, X

β,j

]

^⊥

, (3) B

H

(dL

x

(Y

1

), dL

x

(Y

2

)) = 0,

(4) B

H

(dL

x

(T

λ,i

), dL

x

(Y

2

)) = 0,

(5) B

H

(dL

x

(Y

α,i

), dL

x

(Y

2

)) = 0,

(6) dL

⁻_x¹

B

H

(dL

x

(T

λ,i

), dL

x

(Y

β,j

)) = − tan( ⟨ β, H ⟩ )[T

λ,i

, X

β,j

]

^⊥

, for

λ, µ ∈ Σ

⁺

with ⟨ λ, H ⟩ , ⟨ µ, H ⟩ ̸∈ π Z , 1 ≤ i ≤ m(λ), 1 ≤ j ≤ m(µ), α, β ∈ W

⁺

with ⟨ α, H ⟩ , ⟨ β, H ⟩ ̸∈ π

2 + π Z , 1 ≤ i ≤ n(α), 1 ≤ j ≤ n(β), Y

1

, Y

2

∈ V (m

1

∩ k

2

).

Here X

^⊥

is the normal component, i.e. (Ad(x

⁻¹

)m

2

) ∩ m

1

-component, of a tangent vector X ∈ m

1

.

Theorem 2.9 ([I] Corollaries 4.23, 4.29, 4.24, and [GT] Theorem 5.3). Let g = exp(H) (H ∈ a). Denote the mean curvature vector of K

2

π

1

(g) ⊂ M

1

at π

1

(g) by m

¹_H

. Then we have:

(1)

dL

⁻_g¹

m

¹_H

= − !

λ∈Σ⁺

⟨λ,H⟩̸∈πZ

m(λ) cot ⟨ λ, H ⟩ λ + !

α∈W⁺

⟨α,H⟩̸∈(π/2)+πZ

n(α) tan ⟨ α, H ⟩ α.

(2) The orbit K

2

π

1

(g) ⊂ M

1

is austere if and only if the finite subset of a defined by

{− λ cot ⟨ λ, H ⟩ (multiplicity = m(λ)) | λ ∈ Σ

⁺

, ⟨ λ, H ⟩ ̸∈ π Z}

∪{ α tan ⟨ α, H ⟩ (multiplicity = n(α)) | α ∈ W

⁺

, ⟨ α, H ⟩ ̸∈ (π/2) + π Z}

is invariant under the multiplication by − 1 with multiplicities.

(3) The orbit K

2

π

1

(g) ⊂ M

1

is totally geodesic if and only if ⟨ λ, H ⟩ ∈ (π/2) Z for each λ ∈ Σ ˜

⁺

.

We can apply Theorem 2.9 for orbits K

1

π

2

(g) ⊂ M

2

. Thus, we have the following corollary.

Corollary 2.10 ([I] Corollary 4.30). The orbit K

2

π

1

(g) is minimal (resp. austere, totally geodesic) if and only if K

1

π

2

(g) is minimal (resp. austere, totally geodesic).

Now we consider the second fundamental form of orbits of the (K

2

× K

1

)-action on G. For H ∈ a, we set

Σ

H

= { λ ∈ Σ | ⟨ λ, H ⟩ ∈ π Z} , W

H

= { α ∈ W | ⟨ α, H ⟩ ∈ (π/2) + π Z} , Σ ˜

H

= Σ

H

∪ W

H

, Σ

⁺_H

= Σ

⁺

∩ Σ

H

, W

_H⁺

= W

⁺

∩ W

H

, Σ ˜

⁺_H

= Σ

⁺_H

∪ W

_H⁺

. Let g = exp(H ) (H ∈ a). Then we have

T

g

(K

2

gK

1

) =

"

d

dt exp(tX

2

)g exp( − tX

1

)

# #

# #

_t=0

# #

# # X

1

∈ k

1

, X

2

∈ k

2

$

=dL

g

((Ad(g)

⁻¹

k

2

) + k

1

) (2.2)

=dL

g

⎛

⎝k

0

⊕ V (m

1

∩ k

2

) ⊕ !

λ∈Σ⁺\ΣH

m

λ

⊕ !

α∈W⁺\WH

V

_α^⊥

(m

1

∩ k

2

)

⊕ V (k

1

∩ m

2

) ⊕ !

λ∈Σ⁺

k

λ

⊕ !

α∈W⁺

V

_α^⊥

(k

1

∩ m

2

) '

,

(2.3)

T

_g^⊥

(K

2

gK

1

) = dL

g

((Ad(g)

⁻¹

m

2

) ∩ m

1

) (2.4)

= dL

g

⎛

⎝a ⊕ #

λ∈Σ⁺_H

m

λ

⊕ #

α∈W_H⁺

V

_α^⊥

(m

1

∩ k

2

)

⎞

⎠ . (2.5)

For X = (X

2

, X

1

) ∈ g × g, we define a Killing vector field X

^∗

on G by (X

^∗

)

p

= d

dt exp(tX

2

)p exp( − tX

1

)

&

&

&

&

_t=0

(p ∈ G).

Then

(X

^∗

)

p

= (dL

p

)(Ad(p)

⁻¹

X

2

− X

1

)

holds. If X

2

= 0, then X

^∗

is a left invariant vector field. Denote by ∇ the Levi- Civita connection on G. By Koszul’s formula, we have the following.

Lemma 2.11 ([O] Lemma 3). Let g ∈ G, X = (X

2

, X

1

), Y = (Y

2

, Y

1

) ∈ g × g.

Then we have

( ∇

^X∗

Y

^∗

)

_g

= − 1

2 dL

g

[Ad(g)

⁻¹

X

2

− X

1

, Ad(g)

⁻¹

Y

2

+ Y

1

].

Proof. By Koszul’s formula, we have

2 ⟨∇

^X∗

Y

^∗

, Z ⟩ =X

^∗

⟨ Y

^∗

, Z ⟩ + Y

^∗

⟨ Z, X

^∗

⟩ − Z ⟨ X

^∗

, Y

^∗

⟩

+ ⟨ [X

^∗

, Y

^∗

], Z ⟩ − ⟨ [Y

^∗

, Z], X

^∗

⟩ + ⟨ [Z, X

^∗

], Y

^∗

⟩

for any X = (X

2

, X

1

), Y = (Y

2

, Y

1

) ∈ g × g, Z ∈ g. We compute the right side of the above equation at e. Since ⟨ Y

^∗

, Z ⟩

^h

= ⟨ Ad(h

⁻¹

)Y

2

− Y

1

, Z ⟩ (h ∈ G), we have

(X

^∗

⟨ Y

^∗

, Z ⟩ )

e

= d

dt ⟨ Ad(exp( − t(X

^∗

)

e

))Y

2

− Y

1

, Z ⟩|

^t=0

= ⟨− [(X

^∗

)

e

, Y

2

], Z ⟩ = ⟨− [X

2

− X

1

, Y

2

], Z ⟩ . Similarly, we have

(Y

^∗

⟨ Z, X

^∗

⟩ )

e

= ⟨− [Y

2

− Y

1

, X

2

], Z ⟩ .

Since ⟨ X

^∗

, Y

^∗

⟩

^h

= ⟨ Ad(h

⁻¹

)X

2

− X

1

, Ad(h

⁻¹

)Y

2

− Y

1

⟩ (h ∈ G), we have Z (X

^∗

, Y

^∗

)

e

= d

dt ⟨ Ad(exp( − tZ))X

2

− X

1

, Ad(exp( − tZ))Y

2

− Y

1

⟩|

^t=0

= d

dt ⟨ Ad(exp( − tZ))X

2

, − Y

1

⟩ + ⟨− X

1

, Ad(exp( − tZ))Y

2

⟩|

^t=0

= ⟨ [Z, X

2

], Y

1

⟩ + ⟨ X

1

, [Z, Y

2

] ⟩ = ⟨ Z, [X

2

, Y

1

] ⟩ + ⟨ Z, [Y

2

, X

1

] ⟩

= ⟨ Z, [X

2

, Y

1

] + [Y

2

, X

1

] ⟩ .

Note the sign of the commutator product of X(G) and g × g. Then we have [X

^∗

, Y

^∗

] = − (ad

g×g

(X)Y )

^∗

.

Thus,

⟨ [X

^∗

, Y

^∗

], Z ⟩

^e

= ⟨− ad(X

2

)Y

2

+ ad(X

1

)Y

1

, Z ⟩ . Since Z = (0, − Z)

^∗

we have

⟨ [Y

^∗

, Z], X

^∗

⟩

^e

= ⟨− ad(Y

1

)Z, X

2

− X

1

⟩ = ⟨ Z, ad(Y

1

)(X

2

− X

1

) ⟩ ,

⟨ [Z, X

^∗

], Y

^∗

⟩

^e

= −⟨ Z, ad(X

1

)(Y

2

− Y

1

) ⟩ .

Therefore, we have

2( ∇

^X∗

Y

^∗

)

e

=( − [X

2

− X

1

, Y

2

]) + ( − [Y

2

− Y

1

, X

2

]) − ([X

2

, Y

1

] + [Y

2

, X

1

]) + ( − [X

2

, Y

2

] + [X

1

, Y

1

]) − ([Y

1

, X

2

− X

1

]) + ( − [X

1

, Y

2

− Y

1

])

=[X

2

− X

1

, − Y

2

+ Y

1

] + [X

1

, Y

2

+ Y

1

− Y

2

+ Y

1

] + [X

2

, Y

2

− Y

1

− Y

1

− Y

2

]

=[X

2

− X

1

, − Y

2

+ Y

1

] + 2[X

1

− X

2

, Y

1

]

= − [X

2

− X

1

, Y

2

+ Y

1

].

Hence we obtain

( ∇

^X∗

Y

^∗

)

e

= − 1

2 [X

2

− X

1

, Y

2

+ Y

1

].

(2.6)

Since dL

g

is an isometry, we have

( ∇

^X∗

Y

^∗

)

g

= dL

g

( ∇

dL⁻g¹X^∗

dL

⁻¹_g

Y

^∗

)

e

. Further, we have

(dL

⁻¹_g

X

^∗

)

h

= dL

⁻¹_g

(X

^∗

)

gh

= dL

⁻¹_g

dL

gh

(Ad(gh)

⁻¹

X

2

− X

1

)

= dL

h

(Ad(h)

⁻¹

Ad(g)

⁻¹

X

2

− X

1

)

= (Ad(g)

⁻¹

X

2

, X

1

)

^∗_h

(h ∈ G).

Thus,

dL

⁻_g¹

X

^∗

= (Ad(g)

⁻¹

X

2

, X

1

)

^∗

holds. Summarizing the above, we obtain

( ∇

^X∗

Y

^∗

)

_g

= − 1

2 dL

g

[Ad(g)

⁻¹

X

2

− X

1

, Ad(g)

⁻¹

Y

2

+ Y

1

].

! For H ∈ a, we denote the second fundamental form of the orbit K

2

gK

1

⊂ G by B

H

. By Lemma 2.11, we can express B

H

for H ∈ a.

Theorem 2.12 ([O] Theorem 3). For H ∈ a, we set g = exp(H) and V

1

= !

λ∈Σ⁺\ΣH

m

λ

⊕ !

α∈W⁺\WH

V

_α^⊥

(m

1

∩ k

2

),

V

2

= !

λ∈Σ⁺

k

λ

⊕ !

α∈W⁺

V

_α^⊥

(k

1

∩ m

2

).

Then we have the following:

(1) For X ∈ k

0

, B

H

(dL

g

(X ), Y ) = 0 where Y ∈ T

g

(K

2

gK

1

).

(2) For X ∈ V (k

1

∩ m

2

),

dL

⁻_g¹

B

H

(dL

g

(X ), dL

g

(Y )) =

⎧ ⎨

⎩

0 (Y ∈ k

1

⊕ V (m

1

∩ k

2

))

− 1

2 [X, Y ]

^⊥

(Y ∈ V

1

) . (3) For X ∈ V (m

1

∩ k

2

),

dL

⁻¹_g

B

H

(dL

g

(X), dL

g

(Y )) =

⎧ ⎨

⎩

0 (Y ∈ V (m

1

∩ k

2

) ⊕ V

1

) 1

2 [X, Y ]

^⊥

(Y ∈ V

2

) .

(4) For S

λ,i

(λ ∈ Σ

⁺

, 1 ≤ i ≤ m(λ)),

dL

⁻¹_g

B

H

(dL

g

(S

λ,i

), dL

g

(Y )) =

⎧ ⎨

⎩

0 (Y ∈ V

2

)

− 1

2 [S

λ,i

, Y ]

^⊥

(Y ∈ V

1

) . (5) For X

α,i

(α ∈ W

⁺

, 1 ≤ i ≤ n(α)),

dL

⁻_g¹

B

H

(dL

g

(X

α,i

), dL

g

(Y )) =

⎧ ⎨

⎩

0 (Y ∈ V

2

)

− 1

2 [X

α,i

, Y ]

^⊥

(Y ∈ V

1

) . (6) For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)),

• dL

⁻¹_g

B

H

(dL

g

(T

λ,i

), dL

g

(T

µ,j

)) = cot ⟨ µ, H ⟩ [T

λ,i

, S

µ,j

]

^⊥

where µ ∈ Σ

⁺

\ Σ

H

, 1 ≤ j ≤ m(µ).

• dL

⁻¹_g

B

H

(dL

g

(T

λ,i

), dL

g

(Y

β,j

)) = − tan ⟨ β, H ⟩ [T

λ,i

, X

β,j

]

^⊥

where β ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(β ).

(7) For Y

α,i

(α ∈ W

⁺

\ W

H

, 1 ≤ i ≤ n(α)),

dL

⁻¹_g

B

H

(dL

g

(Y

α,i

), dL

g

(Y

β,j

)) = − tan ⟨ β, H ⟩ [Y

α,i

, X

β,j

]

^⊥

where β ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(β )).

Here, X

^⊥

is the normal component, i.e. the ((Ad(g)

⁻¹

m

2

) ∩ m

1

)-component, of a tangent vector X ∈ g.

Proof. By a simple calculation, we have the following:

• For X ∈ k

0

, dL

g

(X ) = (X, 0)

^∗_g

.

• For X ∈ V (k

1

∩ m

2

), dL

g

(X ) = (0, − X )

^∗_g

.

• For X ∈ V (m

1

∩ k

2

), dL

g

(X ) = (X, 0)

^∗_g

.

• For S

λ,i

(λ ∈ Σ

⁺

, 1 ≤ i ≤ m(λ)), dL

g

(S

λ,i

) = (0, − S

λ,i

)

^∗_g

.

• For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)), dL

g

(T

λ,i

) =

$

− S

λ,i

sin ⟨ λ, H ⟩ , − cot ⟨ λ, H ⟩ S

λ,i

%

∗ g

.

• For X

α,i

(α ∈ W

⁺

, 1 ≤ i ≤ n(α)), dL

g

(X

α,i

) = (0, − X

α,i

)

^∗_g

.

• For Y

α,i

(α ∈ W

⁺

\ W

H

, 1 ≤ i ≤ n(α)), dL

g

(Y

α,i

) =

$ Y

α,i

cos ⟨ α, H ⟩ , tan ⟨ α, H ⟩ X

α,i

%

∗ g

. Then, applying Lemma 2.11, we have follows.

For (1), let X ∈ k

0

. Then we can calculate as follows:

• For Y ∈ k

0

,

B

H

(dL

g

(X), dL

g

(Y )) = &

∇

(Ad(g)⁻¹X,0)^∗

(Ad(g)

⁻¹

Y, 0)

^∗

'

⊥ g

= − 1

2 dL

g

([X, Y ])

^⊥

= 0 since [X, Y ] ∈ k

0

is a tangent vector.

• For Y ∈ V (k

1

∩ m

2

),

B

H

(dL

g

(X), dL

g

(Y )) = &

∇

(Ad(g)⁻¹X,0)^∗

(0, − Y )

^∗

'

⊥ g

= − 1

2 dL

g

([X, − Y ])

^⊥

= 0

since [X, Y ] ∈ k

1

is a tangent vector.

• For Y ∈ V (m

1

∩ k

2

),

B

H

(dL

g

(X), dL

g

(Y )) = !

∇

(Ad(g)⁻¹X,0)^∗

(Ad(g)

⁻¹

Y, 0)

^∗

"

⊥ g

= − 1

2 dL

g

([X, Y ])

^⊥

= 0 since [X, Y ] ∈ V (m

1

∩ k

2

) is a tangent vector.

• For S

λ,i

(λ ∈ Σ

⁺

, 1 ≤ i ≤ m(λ)), B

H

(dL

g

(X ), dL

g

(S

λ,i

)) = !

∇

^{(Ad(g)−1X,0)∗}

(0, − S

λ,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − S

λ,i

])

^⊥

= 0 since [X, − S

λ,i

] ∈ k

1

is a tangent vector.

• For X

α,j

(α ∈ W

⁺

, 1 ≤ j ≤ n(α)), B

H

(dL

g

(X), dL

g

(X

α,j

)) = !

∇

^{(Ad(g)−1X,0)∗}

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − X

α,j

])

^⊥

= 0 since [X, X

α,j

] ∈ k

1

is a tangent vector.

• For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)),

B

H

(dL

g

(X ), dL

g

(T

λ,i

)) =

#

∇

(Ad(g)⁻¹X,0)^∗

(Ad(g)

⁻¹

− S

λ,i

sin ⟨ λ, H ⟩ , − cot ⟨ λ, H ⟩ S

λ,i

)

^∗

$

⊥

g

= − 1

2 dL

g

([X, − 2 cot ⟨ λ, H ⟩ S

λ,i

− T

λ,i

])

^⊥

= 0 since [X, S

λ,i

] ∈ k

1

and [X, T

λ,i

] ∈ m

λ

are tangent vectors.

• For Y

α,j

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)),

B

H

(dL

g

(X), dL

g

(Y

α,j

)) =

#

∇

(Ad(g)⁻¹X,0)^∗

(Ad(g)

⁻¹

Y

α,j

cos ⟨ α, H ⟩ , tan ⟨ α, H ⟩ X

α,j

)

^∗

$

⊥

g

= − 1

2 dL

g

([X, 2 tan ⟨ α, H ⟩ X

α,j

− Y

α,j

])

^⊥

= 0

since [X, X

α,j

] ∈ k

1

and [X, Y

α,j

] ∈ V

_α^⊥

(m

1

∩ k

2

) are tangent vectors.

For (2), let X ∈ V (k

1

∩ m

2

). Then we can calculate as follows:

• For Y ∈ V (k

1

∩ m

2

),

B

H

(dL

g

(X ), dL

g

(Y )) = !

∇

0,−X)^∗

(0, − Y )

^∗

"

⊥ g

= − 1

2 dL

g

([X, − Y ])

^⊥

= 0 since [X, Y ] ∈ V (m

1

∩ k

2

) is a tangent vector.

• For Y ∈ V (m

1

∩ k

2

),

B

H

(dL

g

(X ), dL

g

(Y )) = !

∇

(0,−X)^∗

(Ad(g)

⁻¹

Y, 0)

^∗

"

⊥ g

= − 1

2 dL

g

([X, Y ])

^⊥

.

Then, [X, Y ] ∈ a and ⟨ [X, Y ], H

^′

⟩ = ⟨ X, [Y, H

^′

] ⟩ for all H

^′

∈ a, thus

[X, Y ] = 0. Hence B

H

(dL

g

(X ), dL

g

(Y )) = 0.

• For S

λ,i

(λ ∈ Σ

⁺

, 1 ≤ i ≤ m(λ)), B

H

(dL

g

(X), dL

g

(S

λ,i

)) = !

∇

^(0,−X)^∗

(0, − S

λ,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − S

λ,i

])

^⊥

= 0 since [X, S

λ,i

] ∈ k

1

is a tangent vector.

• For X

α,j

(α ∈ W

⁺

, 1 ≤ j ≤ n(α)), B

H

(dL

g

(X ), dL

g

(X

α,j

)) = !

∇

(0,−X)^∗

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − X

α,j

])

^⊥

= 0 since [X, X

α,j

] ∈ k

1

is a tangent vector.

• For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)),

B

H

(dL

g

(X ), dL

g

(T

λ,i

)) =

#

∇

^(0,−X)^∗

(Ad(g)

⁻¹

− S

λ,i

sin ⟨ λ, H ⟩ , − cot ⟨ λ, H ⟩ S

λ,i

)

^∗

$

⊥

g

= − 1

2 dL

g

([X, − 2 cot ⟨ λ, H ⟩ S

λ,i

+ T

λ,i

])

^⊥

= − 1

2 dL

g

([X, +T

λ,i

])

^⊥

since [X, S

λ,i

] ∈ k

1

is a tangent vector.

• For Y

α,j

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)),

B

H

(dL

g

(X ), dL

g

(Y

α,j

)) =

#

∇

(0,−X)^∗

(Ad(g)

⁻¹

Y

α,j

cos ⟨ α, H ⟩ , tan ⟨ α, H ⟩ X

α,j

)

^∗

$

⊥

g

= − 1

2 dL

g

([X, 2 tan ⟨ α, H ⟩ X

α,j

+ Y

α,j

])

^⊥

= − 1

2 dL

g

([X, Y

α,j

])

^⊥

since [X, X

α,j

] ∈ k

1

is a tangent vector.

For (3), let X ∈ V (m

1

∩ k

2

). Then we can calculate as follows:

• For Y ∈ V (m

1

∩ k

2

),

B

H

(dL

g

(X ), dL

g

(Y )) = !

∇

^(X,0)∗

(Ad(g)

⁻¹

Y, 0)

^∗

"

⊥ g

= − 1

2 dL

g

([X, Y ])

^⊥

= 0.

since [X, Y ] ∈ k

0

is a tangent vector.

• For S

λ,i

(λ ∈ Σ

⁺

, 1 ≤ i ≤ m(λ)), B

H

(dL

g

(X ), dL

g

(S

λ,i

)) = !

∇

(X,0)^∗

(0, − S

λ,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − S

λ,i

])

^⊥

= 1

2 dL

g

([X, S

λ,i

])

^⊥

.

• For X

α,j

(α ∈ W

⁺

, 1 ≤ j ≤ n(α)), B

H

(dL

g

(X ), dL

g

(X

α,j

)) = !

∇

(X,0)^∗

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([X, − X

α,j

])

^⊥

= 1

2 dL

g

([X, X

α,j

])

^⊥

.

• For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)),

B

H

(dL

g

(X ), dL

g

(T

λ,i

)) = B

H

(dL

g

(T

λ,i

), dL

g

(X))

=

#

∇

_(Ad(g)−1 −Sλ,i

sin⟨λ,H⟩,−cot⟨λ,H⟩Sλ,i)^∗

(X, 0)

^∗

$

⊥ g

= − 1

2 dL

g

([T

λ,i

, X])

^⊥

= 0 since [X, T

λ,i

] ∈ k

1

is a tangent vector.

• For Y

α,j

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)),

B

H

(dL

g

(X ), dL

g

(Y

α,j

)) = B

H

(dL

g

(Y

α,j

), dL

g

(X))

=

#

∇

_(Ad(g)−1 Yα,j

cos⟨α,H⟩,tan⟨α,H⟩Xα,j)^∗

(X, 0)

^∗

$

⊥ g

= − 1

2 dL

g

([Y

α,j

, X])

^⊥

= 0 since [X, Y

α,j

] ∈ k

1

is a tangent vector.

For (4), let λ ∈ Σ

⁺

and 1 ≤ i ≤ m(λ). Then we can calculate as follows:

• For S

µ,j

(µ ∈ Σ

⁺

, 1 ≤ j ≤ m(µ)), B

H

(dL

g

(S

λ,i

), dL

g

(S

µ,j

)) = !

∇

^(0,−Sµ,j)^∗

(0, − S

λ,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([S

λ,i

, − S

µ,j

])

^⊥

= 0 since [S

λ,i

, S

µ,j

] ∈ k

1

is a tangent vector.

• For X

α,j

(α ∈ W

⁺

, 1 ≤ j ≤ n(α)), B

H

(dL

g

(S

λ,i

), dL

g

(X

α,j

)) = !

∇

^(0,−Sλ,i)^∗

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([S

λ,i

, − X

α,j

])

^⊥

= 0 since [S

λ,i

, X

α,j

] ∈ k

1

is a tangent vector.

• For T

µ,j

(µ ∈ Σ

⁺

\ Σ

H

, 1 ≤ j ≤ m(µ)),

B

H

(dL

g

(S

λ,i

), dL

g

(T

µ,j

)) = B

H

(dL

g

(T

µ,j

), dL

g

(S

λ,i

))

=

#

∇

_(Ad(g)−1 −Sµ,j

sin⟨µ,H⟩,−cot⟨µ,H⟩Sµ,j)^∗

(0, − S

λ,i

)

^∗

$

⊥ g

= − 1

2 dL

g

([T

µ,j

, − S

λ,i

])

^⊥

= − 1

2 dL

g

([S

λ,i

, T

µ,j

])

^⊥

.

• For Y

α,j

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)),

B

H

(dL

g

(S

λ,i

), dL

g

(Y

α,j

)) = B

H

(dL

g

(Y

α,j

), dL

g

(S

λ,i

))

=

!

∇

_(Ad(g)−1 Yα,j

cos⟨α,H⟩,tan⟨α,H⟩Xα,j)^∗

(0, − S

λ,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([Y

α,j

, − S

λ,i

])

^⊥

= − 1

2 dL

g

([S

λ,i

, Y

α,j

])

^⊥

.

For (5), let α ∈ W

⁺

and 1 ≤ j ≤ n(α). Then we can calculate as follows:

• For X

β,i

(β ∈ W

⁺

, 1 ≤ i ≤ n(β )), B

H

(dL

g

(X

α,j

), dL

g

(X

β,i

), ) = #

∇

^(0,−Xα,j)^∗

(0, − X

β,i

)

^∗

$

⊥ g

= − 1

2 dL

g

([X

α,j

, − X

β,i

])

^⊥

= 0 since [X

α,j

, X

β,i

] ∈ k

1

is a tangent vector.

• For T

λ,i

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)),

B

H

(dL

g

(X

α,j

), dL

g

(T

λ,i

)) = B

H

(dL

g

(T

λ,i

), dL

g

(X

α,j

))

=

!

∇

_(Ad(g)−1 −Sλ,i

sin⟨λ,H⟩,−cot⟨λ,H⟩Sλ,i)^∗

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([T

λ,i

, − X

α,j

])

^⊥

= − 1

2 dL

g

([X

α,j

, T

µ,j

])

^⊥

.

• For Y

β,i

(β ∈ W

⁺

\ W

H

, 1 ≤ i ≤ n(β)),

B

H

(dL

g

(X

α,j

), dL

g

(Y

β,i

)) = B

H

(dL

g

(Y

β,i

), dL

g

(X

α,j

))

=

!

∇

_(Ad(g)−1 Yβ,i

cos⟨β,H⟩,tan⟨β,H⟩Xβ,i)^∗

(0, − X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([Y

β,i

, − X

α,j

])

^⊥

= − 1

2 dL

g

([X

α,j

, Y

β,i

])

^⊥

.

For (6), let λ ∈ Σ

⁺

\ Σ

H

and 1 ≤ i ≤ m(λ). Then we can calculate as follows:

• For T

µ,j

(µ ∈ Σ

⁺

\ Σ

H

, 1 ≤ j ≤ m(µ)), B

H

(dL

g

(T

λ,i

), dL

g

(T

µ,j

))

=

!

∇

_(Ad(g)−1 −Sλ,i

sin⟨λ,H⟩,−cot⟨λ,H⟩Sλ,i)^∗

(Ad(g)

⁻¹

− S

µ,j

sin ⟨ µ, H ⟩ , − cot ⟨ µ, H ⟩ S

µ,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([T

λ,i

, − 2 cot ⟨ µ, H ⟩ S

µ,j

+ T

µ,j

])

^⊥

= cot ⟨ µ, H ⟩ dL

g

[T

λ,i

, S

µ,j

]

^⊥

.

• For Y

α,j

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)), B

H

(dL

g

(T

λ,i

), dL

g

(Y

α,j

))

=

!

∇

_(Ad(g)−1 −Sλ,i

sin⟨λ,H⟩,−cot⟨λ,H⟩Sλ,i)^∗

(Ad(g)

⁻¹

Y

α,j

cos ⟨ α, H ⟩ , tan ⟨ α, H ⟩ X

α,j

)

^∗

"

⊥ g

= − 1

2 dL

g

([T

λ,i

, 2 tan ⟨ α, H ⟩ X

α,j

+ Y

α,j

])

^⊥

= − tan ⟨ α, H ⟩ dL

g

([T

λ,i

, X

α,j

])

^⊥

.

For (7), let α, β ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α) and 1 ≤ i ≤ n(β). Then we have B

H

(dL

g

(Y

α,j

), dL

g

(Y

β,i

))

=

!

∇

_(Ad(g)−1 Yα,j

cos⟨α,H⟩,tan⟨α,H⟩Xα,j)^∗

(Ad(g)

⁻¹

Y

β,i

cos ⟨ β, H ⟩ , tan ⟨ β, H ⟩ X

β,i

)

^∗

"

⊥ g

= − 1

2 dL

g

([Y

α,j

, 2 tan ⟨ β , H ⟩ X

β,i

+ Y

β,i

])

^⊥

= tan ⟨ β, H ⟩ dL

g

([Y

α,j

, X

β,i

])

^⊥

.

Then, we have the consequence. !

We denote the mean curvature vector of the orbit K

2

gK

1

at g by m

H

. By Theorem 2.12, we can see that the following corollary.

Corollary 2.13 ([O] Corollary 2). For H ∈ a, dL

⁻¹_g

m

H

= − #

λ∈Σ⁺\ΣH

m(λ) cot ⟨ λ, H ⟩ λ + #

α∈W⁺\WH

n(α) tan ⟨ α, H ⟩ α.

Moreover, dL

⁻_g¹

m

H

= dL

⁻_g¹

m

¹_H

holds. Hence, an orbit K

2

gK

1

⊂ G is minimal if and only if K

2

π

1

(g) ⊂ M

1

is minimal.

Proof. By Theorem 2.12, we have

dL

⁻_g¹

B

H

(dL

g

(X ), dL

g

(X)) = 0 (X ∈ k

1

),

dL

⁻_g¹

B

H

(dL

g

(T

λ,i

), dL

g

(T

λ,i

)) = − cot ⟨ λ, H ⟩ λ (λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)), dL

⁻_g¹

B

H

(dL

g

(Y

α,j

), dL

g

(Y

α,j

)) = tan ⟨ α, H ⟩ α (α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)).

Thus we have

dL

⁻¹_g

m

H

= − #

λ∈Σ⁺\ΣH

m(λ) cot ⟨ λ, H ⟩ λ + #

α∈W⁺\WH

n(α) tan ⟨ α, H ⟩ α.

Moreover, by (1) of Theorem 2.9, we obtain dL

⁻_g¹

m

H

= dL

⁻_g¹

m

¹_H

. ! Next, we consider austere orbits of the (K

2

× K

1

)-action on G. By using ( ˜ Σ, Σ, W ), Ikawa gave a necessary and sufficient condition for an orbit of the K

2

- action to be an austere submanifold. Similarly, in the (K

2

× K

1

)-action, we also have a necessary and sufficient condition for an orbit to be an austere submanifold.

We investigate the set of eigenvalues of the shape operator A

^dLgξ

of K

1

gK

2

⊂ G for each normal vector dL

g

ξ ∈ T

_g^⊥

K

2

gK

1

∼ = dL

g

((Ad(g)

⁻¹

m

2

) ∩ m

1

). For each g ∈ G, we denote the isotropy subgroup of the (K

2

× K

1

)-action on G at g by (K

2

× K

1

)

g

. Notice that (K

2

× K

1

)

g

is isomorphic to the isotropy subgroup (K

1

)

_π2(g)

of the K

1

-action at π

2

(g). The isotropy subgroup (K

2

× K

1

)

g

acts on the normal space T

_g^⊥

(K

2

gK

1

) by the differential of the (K

2

× K

1

)-action. Then we have

d(k

2

, k

1

)

g

(dL

g

(ξ)) = d

dt k

2

g exp(tξ)k

⁻₁¹

$ $

$ $

_t=0

= dL

g

(Ad(k

1

)ξ).

Therefore, the representation of (K

2

× K

1

)

g

is equivalent to the adjoint represen-

tation of (K

1

)

π2(g)

on (Ad(g)

⁻¹

m

2

) ∩ m

1

. Since Lie((K

1

)

π2(g)

) = k

1

∩ (Ad(g)

⁻¹

k

2

),

the Lie algebra Lie((K

1

)

π2(g)

) ⊕ ((Ad(g)

⁻¹

m

2

) ∩ m

1

) is an orthogonal symmetric Lie

algebra with respect to θ

1

. Moreover, when g ∈ exp(a), a is a maximal abelian sub-

space of ((Ad(g)

⁻¹

m

2

) ∩ m

1

). Thus, a is a section of the representation of (K

1

)

π2(g)

on (Ad(g)

⁻¹

m

2

) ∩ m

1

. Therefore, we have

!

(k2,k1)∈(K2×K1)g

d(k

2

, k

1

)

g

dL

g

(a) = T

_g^⊥

K

2

gK

1

. (2.7)

Thus, without loss of generality we can assume ξ ∈ a. Hence, by Theorem 2.12 we have

A

^dLgξ

(dL

g

(S

λ,i

), dL

g

(T

λ,i

)) (2.8)

= (dL

g

(S

λ,i

), dL

g

(T

λ,i

))

"

0 − (1/2) ⟨ λ, ξ ⟩

− (1/2) ⟨ λ, ξ ⟩ − cot ⟨ λ, H ⟩⟨ λ, ξ ⟩

#

(λ ∈ Σ

⁺

\ Σ

H

, 1 ≤ i ≤ m(λ)), A

^dLgξ

(dL

g

(X

α,j

), dL

g

(Y

α,j

))

(2.9)

= (dL

g

(X

α,j

), dL

g

(Y

α,j

))

"

0 − (1/2) ⟨ α, ξ ⟩

− (1/2) ⟨ α, ξ ⟩ tan ⟨ α, H ⟩⟨ α, ξ ⟩

#

(α ∈ W

⁺

\ W

H

, 1 ≤ j ≤ n(α)), for X ∈ k

0

⊕ V (k

1

∩ m

2

) ⊕ V (m

1

∩ k

2

) ⊕ $

λ∈Σ⁺H

k

λ

⊕ $

α∈WH⁺

V

_α^⊥

(k

1

∩ m

2

), A

^dLgξ

dL

g

(X ) = 0.

(2.10)

Therefore, the set of eigenvalues of A

^dLgξ

is given by

%

− cos ⟨ λ, H ⟩ ± 1

2 sin ⟨ λ, H ⟩ ⟨ λ, ξ ⟩ (multiplicity = m(λ))

&

&

&

& λ ∈ Σ

⁺

\ Σ

H

' (2.11)

∪

% sin ⟨ α, H ⟩ ± 1

2 cos ⟨ α, H ⟩ ⟨ α, ξ ⟩ (multiplicity = n(α))

&

&

&

& α ∈ W

⁺

\ W

H

'

∪{ 0 (multiplicity = l) } where l = dim(k

0

⊕ $

λ∈Σ⁺_H

k

λ

⊕ V (k

1

∩ m

2

) ⊕ $

α∈W_H⁺

V

_α^⊥

(k

1

∩ m

2

) ⊕ V (m

1

∩ k

2

)).

Proposition 2.14 ([IST2] p.459). Let E be a finite subset of a finite dimensional vector space a with an inner product ⟨ , ⟩ . Then, (i) and (ii) are equivalent.

(i) For any ξ ∈ a, the set {⟨ a, ξ ⟩ | a ∈ E } with multiplicity is invariant under the multiplication by − 1.

(ii) The set E is invariant under the multiplication by − 1.

Thus, we have the following corollary.

Corollary 2.15 ([O] Corollary 3). Let g = exp(H ) (H ∈ a). Then the orbit K

2

gK

1

⊂ G is austere if and only if the finite subset of a defined by

%

− cos ⟨ λ, H ⟩ ± 1

2 sin ⟨ λ, H ⟩ λ (multiplicity = m(λ))

&

&

&

& λ ∈ Σ

⁺

\ Σ

H

'

∪

% sin ⟨ α, H ⟩ ± 1

2 cos ⟨ α, H ⟩ α (multiplicity = n(α))

&

&

&

& α ∈ W

⁺

\ W

H

'

is invariant under the multiplication by − 1.

It is easy to prove that the following proposition.

Proposition 2.16 ([O] Proposition 5). For each H ∈ a,

E = {− λ cot ⟨ λ, H ⟩ (multiplicity = m(λ)) | λ ∈ Σ

⁺

\ Σ

H

}

∪{ α tan ⟨ α, H ⟩ (multiplicity = n(α)) | α ∈ W

⁺

\ W

H

} is invariant under the multiplication by − 1 with multiplicities if and only if

E

^′

=

!

− cos ⟨ λ, H ⟩ ± 1

2 sin ⟨ λ, H ⟩ λ (multiplicity = m(λ))

"

"

"

" λ ∈ Σ

⁺

\ Σ

H

#

∪

! sin ⟨ α, H ⟩ ± 1

2 cos ⟨ α, H ⟩ α (multiplicity = n(α))

"

"

"

" α ∈ W

⁺

\ W

H

#

is invariant under the multiplication by − 1 with multiplicities.

Proof. The equation E = − E holds if and only if (i) and (ii) hold, where (i) ⟨ λ, H ⟩ ∈ (π/4) Z (λ ∈ Σ ˜

⁺

\ Σ ˜

H

),

(ii) if ⟨ λ, H ⟩ ∈ (π/4) + (π/2) Z , then m(λ) = n(λ).

When E = − E holds, for each λ ∈ Σ ˜

⁺

\ Σ ˜

H

, if ⟨ λ, H ⟩ ∈ (π/2) Z , then it holds either one of the following:

• λ ∈ Σ

H

and

sin ⟨ λ, H ⟩ + 1

2 cos ⟨ λ, H ⟩ = − sin ⟨ λ, H ⟩ − 1 2 cos ⟨ λ, H ⟩ .

• λ ∈ W

H

and

− cos ⟨ λ, H ⟩ + 1

2 sin ⟨ λ, H ⟩ = cos ⟨ λ, H ⟩ − 1 2 sin ⟨ λ, H ⟩ .

Further, if ⟨ λ, H ⟩ ∈ (π/4) + (π/2) Z , then it holds either one of the following:

• m(λ) = n(λ) and cos ⟨ λ, H ⟩ + 1

2 sin ⟨ λ, H ⟩ = sin ⟨ λ, H ⟩ + 1

2 cos ⟨ λ, H ⟩ , and cos ⟨ λ, H ⟩ − 1

2 sin ⟨ λ, H ⟩ = sin ⟨ λ, H ⟩ − 1 2 cos ⟨ λ, H ⟩ .

• m(λ) = n(λ) and cos ⟨ λ, H ⟩ + 1

2 sin ⟨ λ, H ⟩ = sin ⟨ λ, H ⟩ − 1

2 cos ⟨ λ, H ⟩ , and cos ⟨ λ, H ⟩ − 1

2 sin ⟨ λ, H ⟩ = sin ⟨ λ, H ⟩ + 1 2 cos ⟨ λ, H ⟩ . This implies that E

^′

= − E

^′

. The converse is shown by the same way. ! Corollary 2.17 ([O] Corollary 4). Let g = exp(H ) (H ∈ a). The orbit K

2

gK

1

⊂ G is austere if and only if K

2

π

1

(g) ⊂ M

1

is austere.

Remark 2.18. There is no correspondence in totally geodesic orbits. For example,

when θ

1

and θ

2

cannot be transformed each other by an inner automorphism of g,

K

2

eK

1

⊂ G is not totally geodesic, but K

2

π

1

(e) ⊂ M

1

is totally geodesic (see (4)

and (5) in Theorem 2.12).

3. Weakly reflective submanifolds in compact symmetric spaces Ikawa, Sakai, and Tasaki ([IST2]) proposed the notion of weakly reflective sub- manifold as a generalization of the notion of reflective submanifold ([Le]). In [IST2], they detected a certain global symmetry of several austere submanifolds in a hy- persphere, and classified austere orbits and weakly reflective orbits of the linear isotropy representation of irreducible symmetric spaces. They gave a necessary and sufficient condition for orbits of the linear isotropy representation of irreducible symmetric spaces to be an austere submanifold (further, weakly reflective subman- ifold) in the hypersphere in terms of root systems. We would like to generalize this fact to compact Riemannian symmetric spaces. However, it is known that austere orbits of the isotropy action of compact symmetric spaces are reflective submanifolds. Therefore, we consider Hermann actions, which are a generalization of isotropy actions of compact symmetric spaces. Ikawa ([I]) classified austere orbits of commutative Hermann actions. However, weakly reflective orbits have not been classified yet. In this section, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads.

3.1. Weakly reflective submanifolds. We recall the definitions of reflective sub- manifold and weakly reflective submanifold. Let ( ˜ M , ⟨ , ⟩ ) be a complete Riemann- ian manifold.

Definition 3.1 ([Le]). Let M be a submanifold of ˜ M . Then M is a reflective submanifold of ˜ M if there exists an involutive isometry σ

M

of ˜ M such that M is a connected component of the fixed point set of σ

M

. Then, we call σ

M

the reflection of M .

Definition 3.2 ([IST2]). Let M be a submanifold of ˜ M . For each normal vector ξ ∈ T

_x^⊥

M at each point x ∈ M , if there exists an isometry σ

ξ

on ˜ M which satisfies σ

ξ

(x) = x, σ

ξ

(M ) = M and (dσ

ξ

)

x

(ξ) = − ξ, then we call M a weakly reflective submanifold and σ

ξ

a reflection of M with respect to ξ.

If M is a reflective submanifold of ˜ M , then σ

M

is a reflection of M with respect to each normal vector ξ ∈ T

_x^⊥

M at each point x ∈ M . Thus, a reflective submanifold of ˜ M is a weakly reflective submanifold of ˜ M . Notice that a reflective submanifold is totally geodesic, but a weakly reflective submanifold is not necessarily totally geodesic.

Definition 3.3 ([HL]). Let M be a submanifold of ˜ M . We denote the shape operator of M by A. M is called an austere submanifold if for each normal vector ξ ∈ T

_x^⊥

M , the set of eigenvalues with their multiplicities of A

^ξ

is invariant under the multiplication by − 1.

It is clear that an austere submanifold is a minimal submanifold. Ikawa, Sakai and Tasaki proved that a weakly reflective submanifold is an austere submanifold.

Lemma 3.4 ([IST2] p. 439). Let G be a Lie group acting isometrically on a Riemannian manifold M ˜ . For x ∈ M ˜ , we consider the orbit Gx. If for each ξ ∈ T

_x^⊥

Gx, there exists a reflection of Gx at x with respect to ξ, then Gx is a weakly reflective submanifold of M ˜ .

Proposition 3.5 ([IST2] Proposition 2.7). Any singular orbit of a cohomogeneity

one action on a Riemannian manifold is a weakly reflective submanifold.