Ikawa, Osamu

Sakai, Takashi

Tasaki, Hiroyuki

Osaka City University Repository

Ikawa, O., Sakai, T. and Tasaki, H.
Osaka J. Math.
38 (2001), 923–930
ORBITS OF HERMANN ACTIONS
OSAMU IKAWA, TAKASHI SAKAI and HIROYUKI TASAKI
(Received February 23, 2000)
1. Introduction
In this paper we consider the parallel translations of the normal bundles of the
orbits of Hermann actions on compact symmetric spaces and represent such parallel
translations by the group actions (Theorem 2.1). Using this we can show that their
mean curvature vectors are parallel (Corollary 2.8), moreover those of hyperpolar ac-
tions are parallel (Corollary 2.13).
We first review some definitons and previous results concerning isometric group
actions on compact symmetric spaces. Let ( 1) and ( 2) be compact symmetric
pairs. Then 2 acts isometrically on / 1, which is a compact symmetric space. This
action of 2 on / 1 is called a Hermann action.
The Hermann actions are examples of hyperpolar actions, which is defined in the
following. Let be a Lie group acting isometrically on a Riemannian manifold . A
closed submanifold of is called a section, if all orbits of the action of meet
perpendicularly. The action of on is said to be hyperpolar, if there exists a sec-
tion which is flat with respect to the induced Riemannian metric. The codimension of
the orbit of highest dimension is called the cohomogeneity. The isometric actions on
compact symmetric spaces of cohomogeneity one are another examples of hyperpolar
actions. Recently Kollross [8] proved that the hyperpolar actions on compact symmet-
ric spaces are Hermann actions or cohomogeneity one actions.
We next review previous results concerning geometry of orbits of isometric group
actions on symmetric spaces. The linear isotropy representations of symmetric pairs
have sections which are maximal Abelian subspaces, so they are hyperpolar ac-
tions. All of their orbits have parallel mean curvature vectors, which was proved by
Kitagawa-Ohnita [6]. Ohnita [9] considered the parallel translations of the normal bun-
dles of the orbits of the linear isotropy actions on compact symmetric spaces and rep-
resent such parallel translations by the group actions. One can prove the result of
Kitagawa-Ohnita mentioned above by this. Heintze-Olmos [1] also considered such
parallel translations and described the normal holonomy groups of the orbits. For com-
pact symmetric space / , Hirohashi-Song-Takagi-Tasaki [4] and Hirohashi-Ikawa-
Tasaki [3] considered some geometric properties of orbits of the linear isotropy action
on ( / ) and the isotropy action on / .
924 O. IKAWA, T. SAKAI AND H. TASAKI
2. Orbits of Hermann actions
Let θ1 and θ2 be two involutive automorphisms of a compact connected Lie group
furnished with a biinvariant Riemannian metric 〈 〉. We denote by θ ( = 1 2)
the closed subgroup consisting of all fixed points of θ in . For a closed subgroup
( = 1 2) of which lies between θ and the identity component of θ , ( 1)
and ( 2) are Riemannian symmetric pairs. We consider the Hermann action 2 on
a compact symmetric space 1 = / 1 with the induced Riemannian metric from the
biinvariant Riemannian metric 〈 〉 on . We denote by g, k1 and k2 the Lie algebras
of , 1 and 2, respectively. The involutive automorphisms θ1 and θ2 of induce
involutive automorphisms of g, also denoted by θ1 and θ2, respectively. Since θ1 and
θ2 are involutive, we have
g = k1 + m1 = k2 + m2
where we put
m = { ∈ g | θ ( ) = − } ( = 1 2)
We can identify m1 with ( 1) in a natural manner. For ∈ m1, we consider the
2-orbit 2 Exp ⊂ 1, where Exp is the exponential mapping from m1 into 1.
The tangent space of 2 Exp at Exp is given by
Exp ( 2 Exp ) = (exp )∗(Ad(exp(− ))k2)m1
where (Ad(exp(− ))k2)m1 is the m1-component of Ad(exp(− ))k2. We define a
closed subgroup
2
[ 1] in 2 by
2
[ 1] = { ∈ 2 | exp(− ) exp ∈ 1}
Then we have the following diffeomorphism from 2 Exp onto 2/ 2 [ 1]:
2 Exp → 2/ 2 [ 1]; Exp 7→ 2 [ 1]
We denote by nk2 [k1] the Lie algebra of 2 [ 1]. Then we have
nk2 [k1] = { ∈ k2 | Ad(exp(− )) ∈ k1}
We denote by (nk2 [k1])⊥ the orthogonal complement of nk2 [k1] in k2. We can iden-
tify (nk2 [k1])⊥ with ( 2/ 2 ( 1)) in a natural manner. The above diffeomorphism
2 Exp ∼= 2/ 2 [ 1] induces a linear isomorphism from Exp ( 2 Exp ) onto
(nk2 [k1])⊥.
Theorem 2.1. Let be in (nk2 [k1])⊥. We define a curve ( ) in 2 Exp by
( ) = exp Exp
ORBITS OF HERMANN ACTIONS 925
Let ξ be in (Ad(exp(− ))k2)⊥m1 . We define a normal vector field ξ( ) of 2 Exp
along ( ) by
ξ( ) = (exp )∗(exp )∗ξ
Then ξ( ) is parallel with respect to the normal connection.
In order to show the theorem we prove the lemmas below.
Lemma 2.2. We denote by ∇⊥ the normal connection of 2 Exp ⊂ 1. We
define a curve ( ) in by
( ) = exp exp
Then
∇⊥
˙
ξ( ) = ( )∗[(Ad(exp(− )) )k1 ξ]⊥
Proof. Let π be the natural projection from onto 1 = / 1. We consider the
principal fiber bundle ( 1 1 π). The canonical decomposition g = k1 + m1 induces
an invariant connection on ( 1 1 π). It is known that the Levi-Civita connection
of 1 is reduced to the invariant connection. The tangent bundle = ( 1) of 1
is the vector bundle associated with ( 1 1 π) with standard fiber m1. We denote
by ( ) the vector space of -valued -forms on 1, and by Ad( ) the vector
space of tensorial -forms ˜ξ of type Ad( 1) on , that is, ˜ξ satisfies the following
conditions.
(1) ∗ ˜ξ = Ad( −1) ˜ξ ∈ 1
(2) ˜ξ( 1 . . . ) = 0 when 1 is vertical 1 . . . ∈
It is well known that the linear mapping given by
( ) → Ad( ) ; ξ 7→ ˜ξ
˜ξ( 1 . . . ) = −1∗ (ξ(π∗ 1 . . . π∗ ))
is an isomorphism. We denote by ∇ the covariant derivative on ( 1). When in
0( ) = X( 1) corresponds to ˜ξ in 0Ad( 1) by this correspondence, ∇ in 1( )
corresponds to ˜ξ ◦ in 1Ad( 1) (see [7] Chapter II), where we denote by ( ) the
horizontal component of in X( ). By this relationship, we get the following expres-
sion. Let be in X( 1), in m1 and in k1. Let α( ) be a curve in such that
α(0) = and α˙(0) = + . Then
∇
∗
= ∗
( ∣∣∣∣
=0
α( )−1∗ −1∗ π( α( )) + [ −1∗ π( )]
)
For fixed , we define α( ) by α( ) = ( )−1 ( + ). Then α(0) = and α˙(0) =
Ad(exp(− )) . Hence the lemma follows immediately.
926 O. IKAWA, T. SAKAI AND H. TASAKI
Hence in order to show the theorem it is sufficient to prove
[(Ad(exp(− ))(nk2 [k1])⊥)k1 (Ad(exp(− ))k2)⊥m1 ] ⊂ (Ad(exp(− ))k2)m1
The following lemma is trivial.
Lemma 2.3. Ad(exp )n−k1 [k2] = nk2 [k1]
Lemma 2.4. (Ad(exp(− ))k2)⊥m1 = { ∈ m1 | Ad(exp ) ∈ m2} In particular
(Ad(exp(− ))k2)⊥m1 is a Lie triple system in m1.
Proof.
(Ad(exp(− ))k2)⊥m1 = { ∈ m1 | 〈 (Ad(exp(− ))k2)m1〉 = 0}
= { ∈ m1 | 〈 Ad(exp(− ))k2)〉 = 0}
= { ∈ m1 | 〈Ad(exp ) k2〉 = 0}
= { ∈ m1 | Ad(exp ) ∈ m2}
Hence the lemma is proved.
The following lemma immediately follows from the lemma above.
Lemma 2.5. [(Ad(exp(− ))k2)⊥m1 (Ad(exp(− ))k2)⊥m1 ] ⊂ n−k1 [k2]
Lemma 2.6. (Ad(exp(− ))(nk2 [k1])⊥)k1 ⊂ (n−k1 [k2])⊥
Proof.
〈(Ad(exp(− ))(nk2 [k1])⊥)k1 n−k1 [k2]〉
= 〈Ad(exp(− ))(nk2 [k1])⊥ n−k1 [k2]〉 (by n−k1 [k2] ⊂ k1)
= 〈(nk2 [k1])⊥ Ad(exp )n−k1 [k2]〉
= 〈(nk2 [k1])⊥ nk2 [k1]〉 (by Lemma 2.3)
= {0}
Lemma 2.7. [(n−k1 [k2])⊥ (Ad(exp(− ))k2)⊥m1 ] ⊂ (Ad(exp(− ))k2)m1
Proof.
〈[(n−k1 [k2])⊥ (Ad(exp(− ))k2)⊥m1 ] (Ad(exp(− ))k2)⊥m1〉
= 〈(n−k1 [k2])⊥ [(Ad(exp(− ))k2)⊥m1 (Ad(exp(− ))k2)⊥m1 ]〉
ORBITS OF HERMANN ACTIONS 927
⊂ 〈(n−k1 [k2])⊥ n−k1 [k2]〉 (by Lemma 2.5)
= {0}
Hence the lemma is proved.
By Lemmas 2.6 and 2.7 we have
[(Ad(exp(− ))(nk2 [k1])⊥)k1 (Ad(exp(− ))k2)⊥m1 ] ⊂ (Ad(exp(− ))k2)m1
This completes the proof of Thoerem 2.1.
Corollary 2.8. The mean curvature vector of 2 Exp ⊂ 1 is parallel with
respect to the normal connection.
Proof. We denote by the mean curvature vector of 2 Exp ⊂ 1. Since
exp Exp = (exp )∗ Exp ( ∈ (nk2 [k1])⊥), we have ∇⊥ = 0 by Theorem 2.1.
Hence (∇⊥ )Exp = 0. Therefore ∇⊥ vanishes everywhere by the homogeneity of
2 Exp .
The decomposition
k2 = nk2 [k1]⊕ (nk2 [k1])⊥
defines an invariant connection ∇ of 2 Exp . We denote by α the second funda-
mental form of 2 Exp ⊂ 1 We define ∇ α by
(∇ α)( ) = ∇⊥(α( ))− α(∇ )− α( ∇ )
Corollary 2.9. ∇ α = 0.
Proof. Let and be in (nk2 [k1])⊥. The vector fields (exp )∗(exp )∗
and (exp )∗(exp )∗ of 2 Exp along a curve exp Exp are ∇ -parallel.
Thus we get
(∇(exp )∗ α)( )
= ∇⊥(exp )∗ (α((exp )∗(exp )∗ (exp )∗(exp )∗ ))
= ∇⊥(exp )∗ ((exp )∗(α((exp )∗ (exp )∗ ))
= 0 (Theorem 2.1)
Hence we have (∇ α)Exp = 0. By homogeneity we have ∇ α = 0.
By Corollary 2.9, for any vector fields and of 2 Exp we have
(∇⊥(α( ))) ∈ span{α( ) | ∈ ( 2 Exp )}
928 O. IKAWA, T. SAKAI AND H. TASAKI
In another words we get:
Corollary 2.10. The degree of 2 Exp ⊂ 1 is at most 2.
We consider the normal holonomy representation of 2 Exp ⊂ 1. By Lemma
2.4, we can define an action ρ of
2
[ 1] on (Ad(exp(− ))k2)⊥m1 by
ρ( ) = Ad(exp(− ) exp )
for ∈
2
[ 1] ∈ (Ad(exp(− ))k2)⊥m1 . This action is equivalent to the differential
representation of
2
[ 1] on ⊥Exp ( 2 Exp ).
Corollary 2.11. The normal holonomy representation of 2 Exp ⊂ 1
is equivalent to the (effectively made) action of a subgroup of
2
[ 1] on
(Ad(exp(− ))k2)⊥m1 .
Proof. Every geodesic ( ) of 2/ 2 [ 1] through the origin Exp with re-
spect to the normal homogeneous Riemannian metric is given by
( ) = exp Exp for some ∈ (nk2 [k1])⊥
By Theorem 2.1, the parallel translation along ( ) with respect to the normal con-
nection is given by (exp )∗. Now any curve in 2 Exp can be approximated by
broken geodesics with respect to the normal homogeneous Riemannian metric. It fol-
lows that the normal holonomy representation is equivalent to the action of / on
(Ad(exp(− ))k2)⊥m1 where
= { ∈
2
[ 1] | ρ( ) = 1 on (Ad(exp(− ))k2)⊥m1}
and where is a subgroup of
1
[ 2] with ⊂ .
We shall prove that the mean curvature vector of any orbit of any hyperpolar ac-
tion is parallel with respect to the normal connection. In order to do this we review a
result of Hsiang.
Let be a compact, connected Lie group of isometries of a Riemannian manifold
. We denote by the isotropy subgroup at ∈ . Two orbits ( ) and ( ) are
said to be of the same type if there exists ∈ such that = −1. Each orbit
of has a well defined volume by the induced metric as a submanifold. The volume
of an orbit 0 = ( 0) is said to be extremal among nearby orbits of the same type if∣∣∣∣
=
vol( ) = 0
for all smooth families , | | < ε, of -orbits of the same type on .
ORBITS OF HERMANN ACTIONS 929
Theorem 2.12 (Wu-Yi Hsiang [5]). Let be a compact connected Lie group of
isometries of a Riemannian manifold . Then any orbit of whose volume is ex-
tremal among nearby orbits of the same type is a minimal submanifold of . In par-
ticular if there exists a neighborhood of 0 in which there are no other orbits of the
same type, then 0 is a minimal submanifold of .
Corollary 2.13. The mean curvature vector of any orbit of any hyperpolar ac-
tion on a compact symmetric space is parallel with respect to the normal connection.
Proof. Kollross [8] showed that hyperpolar actions on compact symmetric spaces
are either cohomogeneity one actions or ω-equivalent to Hermann actions. Here two
actions are ω-equivalent if all of their orbits coincide. We have already proved that
the mean curvature vector of any orbit of any Hermann action is parallel in Corollaly
2.8, so it is sufficient to consider the case of cohomogeneity one actions.
We assume that the cohomogeneity of the action is equal to one. When a compact,
connected Lie group acts a Riemannian manifold isometlically, an orbit ( )
through ∈ is said to be of the principal type if for any ∈ there exists
∈ such that ⊂ −1. Since the codimensions of orbits of the principal
type are equal to one, they have parallel mean curvature vectors. It is known that the
set of all orbits of the principal type is open and dense in the orbit space. Since the
cohomogeneity is equal to one, the set of orbits which are not of the principal type
does not include any nonempty open set. Hence we have the conclusion by Theorem
2.12.
References
[1] E. Heintze and C. Olmos: Normal holonomy groups and -representations, Indiana Univ. Math.
J. 41 (1992), 869–874.
[2] S. Helgason: Differential geometry, Lie groups, and symmetric spaces, Academic Press, New
York, San Francisco, London, 1978.
[3] D. Hirohashi, O. Ikawa and H. Tasaki: Orbits of isotropy groups of compact symmetric spaces,
Tokyo J. Math., to appear.
[4] D. Hirohashi, H. Song, R. Takagi and H. Tasaki: Minimal orbits of the isotropy groups of
symmetric spaces of compact type, Differential Geom. Appl. 13 (2000), 167–177.
[5] W.Y. Hsiang: On the compact homogeneous minimal submanifolds, Proc. Nat. Acad. Sci.
U.S.A. 56 (1966), 5–6.
[6] Y. Kitagawa and Y. Ohnita: On the mean curvature of -spaces, Math. Ann. 262 (1983), 239–
243.
[7] S. Kobayashi and K. Nomizu: Foundations of differential geometry, Vol. I, Interscience Pub-
lishers, New York, London, 1963.
[8] A. Kollross: A classification of hyperpolar and cohomogeneity one actions, preprint.
[9] Y. Ohnita: The degrees of the standard imbeddings of -spaces, Toˆhoku Math. J. 35 (1983),
499–502.
930 O. IKAWA, T. SAKAI AND H. TASAKI
O. Ikawa
Department of General Education
Fukushima National College of Technology
Iwaki, Fukushima
970-8034
Japan
e-mail: ikawa@fukushima-nct.ac.jp
T. Sakai
Institute of Mathematics
University of Tsukuba
Tsukuba, Ibaraki
305-8571
Japan
e-mail: tsakai@math.tsukuba.ac.jp
H. Tasaki
Institute of Mathematics
University of Tsukuba
Tsukuba Ibaraki
305-8571
Japan
e-mail: tasaki@math.tsukuba.ac.jp


Orbits of Hermann actions

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Orbits of Hermann actions

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