数理解析研究所講究録
第 1929 巻 2014 年 13-1813GEOMETRY OF POLAR ACTIONS ON COMPLEX
HYPERBOLIC SPACESAKIRA KUBO
ABSTRACT. Polar actions on complex hyperbolic spaces without singularorbits have been classified by Berndt and D\’iaz-Ramos. In this paper, we$wi]]$ focus on geometry of their $orbit_{b},$ aIld lne1ltion classifications of theminimal ones and Ricci soliton ones.

1. INTRODUCTION

A submanifold of a complex hyperbolic space
is said to be homogeneous. Note that by an isometricif it is an orbit of an isometric action onaction we always mean an action of a connected closed subgroup of the isomehave studied very actively andtry group. Homogeneous submanifolds ofdeeply (refer to, for instance, [1, 6, 11, 12, 15] and references therein). They
are of particular interest from the viewpoint not only of submanifold geometry, but also of studies of homogeneous submanifolds in other symmetricofspaces of noncompact type.
are polar,
In this paper, we consider the case where isometric actions onand study the geometry of the orbits of such actions. An isometric action
on a Riemannian manifold $M$ is said to be polar if there exists a connectedcomplete submanifold
of $M$ such that meets each orbit of the action andis perpendicular to each orbit at every intersection point. Note that such a
submanifold is called a section of the action, and it is always a totally geodesic
submanifold (for instance, see [3, Theorem 3.2.1]). For more details of polar
actions, for instance, refer to [2, 4, 5, 9, 13, 14] and references therein.
induce homogeWe moreover restrict to the case where polar actions on
neous polar foliations, that is, have no singular orbits. In this case, the actions
have been classified by Berndt and D\’iaz-Ramos in [4]. They have proved that
there exist exactly $2n-1$ actions which induce nontrivial homogeneous polar
up to orbit equivalence. Recall that a homogeneous foliation
foliations ofofor the leaf coincides
is said to be trivial if the leaves are points in. Moreover, the author have classified orbits of such polar actions
withup to isometric congruence, and have given explicit expressions of the orbits in[15]. The subject of this paper is to review the above results on geometry of theorbits, and also to announce our recent work on Ricci solitons (Subsection 3.2).
$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\Sigma$$\Sigma$$\Sigma$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$

Key words and phrases. homogeneous submanifolds, complex hyperbolic spaces, polaractions.
2010 Mathematics Subject Classification. Primary $53C40$ , Secondary $53C30,$ $53C35,$ $53C25.$
The was supported in part by Grant-in-Aid for JSPS Fellows (26 6060).
$\cdot$14
Remark 1.1. Our results in this paper include the some known results in thein [1, 6, 12]. See Remark 3.3 andcase of cohomogeneity one actions on
3.12 for more details.
$\mathbb{C}H^{n}$This paper is organized as follows. In Section 2, we recall the solvable model, and review the classifications of homogeof a complex hyperbolic space
neous polar foliations ofand the orbits of such actinos. In Section 3, weproperties. Firstly, we mention
mention the curvatureof the orbits inthe minimality of the orbit and the classification of the minimal orbits, which
are obtained in [15]. Secondly, we announce our recent result on homogeneous. In particular, we classify the orbits which
Ricci soliton submanifolds ofare Ricci solitons.
$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$2. PRELIMINARIES
2.1. The solvable models of complex hyperbolic spaces. In this subsection, we prepare the solvable models of complex hyperbolic spaces
with$n\geq 2$ , which we need in the following sections. We refer mainly to [8], [11].
$\mathbb{C}H^{n}$Definition 2.1. We call a triple(1)$\mathfrak{s}$$:=span_{\mathbb{R}}\{A_{0}, X_{1}, Y_{1}, .

.

.

$(\mathfrak{s}, \langle, \rangle, J)$

the solvable model ofifis a Lie algebra whose bracket$\mathbb{C}H^{n}$, X_{n-1}, Y_{n-1}, Z_{0}\}$relations are defined by$[A_{0}, X_{i}]=(1/2)X_{i},$(2)(3)$\langle,$$J$$[A_{0}, Y_{i}]=(1/2)Y_{i},$$[A_{0}, Z_{0}]=Z_{0},$$[X_{i}, Y_{i}]=Z_{0},$is an inner product on such that the above basis is orthonormal,
is a complex structure on defined by$\rangle$$\mathfrak{s}$$\mathfrak{s}$$J(A_{0})=Z_{0}, J(Z_{0})=-A_{0}, J(X_{i})=Y_{i}, J(Y_{i})=-X_{i}.$Let be the simply-connected Lie group with Lie algebra . One knows theexpression of the complex hyperbolic space as a homogeneous space, that is,$S$$\mathfrak{s}$$\mathbb{C}H^{n}=SU(1, n)/S(U(1)\cross U(n))$.

The Lie group coincides with the solvable part of the Iwasawa decomposition
of $SU(1, n)$ . Furthermore, we can naturally identify
with the Lie group $S.$
More precisely, we have the following.
$S$$\mathbb{C}H^{n}$

Proposition 2.2. Denote by the same symbols
andthe induced leftinvariant Riemannian metric and the complex structure on , respectively.
Then,
is holomorphically isometric towith constant holomorphic$-1.$
sectional curvature$\langle,$$\rangle$$J$$S$$\mathbb{C}H^{n}$$(S, \langle, \rangle, J)$

2.2. Classifications of polar actions on complex hyperbolic spaces andtheir orbits. In this subsection, we recall the classification of polar actions on
without singular orbits by Berndt and D\’iaz-Ramos. And then, we recall
the classification of the orbits of such actions by the author.
First of all, we introduce Lie subgroupsof , which play essential rolesin the study of polar actions on
without singular orbits. For $\varphi\in[0, \pi/2],$$\mathbb{C}H^{n}$$S_{b}(\varphi)$$S$$\mathbb{C}H^{n}$

let us define$\xi_{0}:=\cos(\varphi)X_{1}+\sin(\varphi)A_{0}.$We mean always $by\ominus the$ orthogonal complement with respect to$\langle,$$\rangle.$

15Definition 2.3. For $b\in\{1, . . . , n\}$ and $\varphi\in[0, \pi/2]$ , we define(1) if $\varphi\in[0, \pi/2)$ , then set$\mathfrak{s}_{b}(\varphi):=\mathfrak{s}\ominus span_{\mathbb{R}}\{\xi_{0}, X_{2}, .

.

.

, X_{b}\},$$\mathfrak{s}_{b}(\pi/2) :=\mathfrak{s}\ominus span_{\mathbb{R}}\{X_{1}, X_{2}, .

.

.

$\mathfrak{s}_{b}(\varphi)$

as follows:

where $b\in\{1, . . . , n-1\},$(2) if $\varphi=\pi/2$ , then set, X_{b}\},$

where $b\in\{1, . . . , n\}.$One can easily check that
Note that$\mathfrak{s}_{b}(\pi/2)$

is a Lie subalgebra of of codimension
is nilpotent, whereas
is solvable but is not nilpotent for$\mathfrak{s}_{b}(\varphi)$$b.$$\mathfrak{s}$$\mathfrak{s}_{b}(\varphi)$$\varphi\in[0, \pi/2),.$.
Denote by
the connected Lie subgroup of with Lie algebra
induces a homogeneous
Then, one can show that the action ofonpolar foliation of cohomogeneity , and its section is a totally geodesic real
. Furthermore, we have the following classification result.
hyperbolic space$S$$S_{b}(\varphi)$$\mathfrak{s}_{b}(\varphi)$$\mathbb{C}H^{n}$$S_{b}(\varphi)$$b$$\mathbb{R}H^{b}$induces a nontrivial homoTheorem 2.4 ([4]). An isometric action onequivalent to one of thegeneous polar foliation oforbit
onlyanditisififfollowing:, where $b\in\{1, . . . , n-1\},$(1) the action of(2) the action of $S_{b}(\pi/2)$ , where $b\in\{1, . . . , n\}.$$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$$S_{b}(0)$We next mention the classification of the orbits of polar actions on
without singular orbits, up to isometric congruence. Denote by the origin of$\mathbb{C}H^{n}$$0$$\mathbb{C}H^{n}.$without singular
Theorem 2.5 ([15]). Every orbit of polar actions onorbits is isometrically congruent to one of the following:(1) the orbit, where $b\in\{1, . . . , n-1\}$ and $\varphi\in[0, \pi/2$ ),(2) the orbit $S_{b}(\pi/2).0$ , where $b\in\{1, . . . , n\}.$$\mathbb{C}H^{n}$$S_{b}(\varphi).0$, the orbit $S_{b}(0).p$ is isometrically congruent to theRemark 2.6. Fororbit
for some $\varphi\in[0, \pi/2$ ) ([15, Proposition 4.5]). In paticular, we notethat is explicitly given by$p\in \mathbb{C}H^{n}$$S_{b}(\varphi).0$$\varphi$$\varphi=\arcsin(\tanh(t_{0}/2))$,

whereis the distance between the origin and the orbit $S_{b}(0).p$ . On theother hand, the action of $S_{b}(\pi/2)$ has congruency of orbits ([15, Theorem 5.1]).
Namely, for each $b\in\{1, . . . , n\}$ , every orbit of $S_{b}(\pi/2)$ is isometrically congruent
to the orbit $S_{b}(\pi/2).0$ . Refer to [16] for more details of the congruency of orbits.
$0$$t_{0}$

3. CURVATURE PROPERTIESIn this section, we study the curvature properties of the orbit
Let us recall thatcan be identified with the Lie group . One thus canequipped withidentify the submanifoldwith the Lie subgroupthe induced left-invariant metric. Throughout this section, therefore, we shall.
express the curvatures in terms of the metric Lie algebra$S_{b}(\varphi).0.$$S$$\mathbb{C}H^{n}$$S_{b}(\varphi).0$$S_{b}(\varphi)$$\mathfrak{s}_{b}(\varphi)$163.1. The minimality. In this subsection, we mention the minimality of theorbit$S_{b}(\varphi).0$

obtained in [15].

Proposition 3.1 ([15]). The mean curvature vector$\mathcal{H}$of$S_{b}(\varphi)$

is given by$\mathcal{H}=(1/2)(2n-b+1)\sin(\varphi)\xi_{0},$Hence, the orbit$S_{b}(\varphi).0$

is minimal if and only if $\varphi=0.$

Recall that $\varphi=\arcsin(\tanh(t_{0}/2))$ , where denotes the distance between
and $S_{b}(0).p$ . It hence follows from the monotonicity of this function that
if and only if $t_{0}=0$ . Altogether, we have the following result.
$t_{0}$$0$$\varphi=0$

has the unique minimal
Theorem 3.2. For $\varphi\in[0, \pi/2$ ), the action of$S_{b}(\pi/2)$$S_{b}(0).0$, whereas the action ofhas no minimal orbits.
orbit$S_{b}(O)$

Remark 3.3. In the case of cohomogeneity one, namely, $b=1$ , the actions of$S_{1}(\pi/2)$ and $S_{1}(O)$ on
are well-known. Especially, Theorem 3.2 has been
proved in [1] (also refer to [6]).
(1) The action of $S_{1}(\pi/2)$ induces the so-called solvable foliation, and the orbit$S_{1}(0).0$ is a unique minimal orbit. In fact, $S_{1}(0).0$ is known as the homogeneous ruled minimal hypersurface, and the other orbits are equidistant
hypersurfaces to $S_{1}(0).0.$(2) The action of $S_{1}(\pi/2)$ , which is the nilpotent part of the Iwasawa decomposition of $SU(1, n)$ , induces the so-called horosphere foliation, and every. In this case, the action of $S_{1}(\pi/2)$
orbit is known as a horosphere ofhas congruency of orbits, and has no minimal orbits.
$\mathbb{C}H^{n}$$\mathbb{C}H^{n}$

The mean curvature vector is said to be parallel if$\nabla_{X}^{\perp}\mathcal{H}=0$

denotes the normal connection of
the following by direct calculations.
$X\in \mathfrak{s}_{b}(\varphi)$, where$\nabla^{\perp}$

Proposition 3.4. The mean curvature vector$\mathcal{H}$of$S_{b}(\varphi)$$S_{b}(\varphi)$holds for any. One can obtainis always parallel.

Remark 3.5. We note that the proposition above can be shown by the generalis polar,theory of polar actions. As we mentioned before, the action ofis a principal orbit of the action. Therefore, it follows fromand the orbit
is parallel[3, Corollary 3.2.5] that the mean curvature vector field on
with respect to$S_{b}(\varphi)$$S_{b}(\varphi).0$$S_{b}(\varphi).0$$\nabla^{\perp}.$3.2. Ricci solitons. In this subsection, we announce our recent result on $ho-$
mogeneous Ricci soliton submanifolds of complex hyperbolic spaces. In
which are Ricci solitons.
particular, we classify the orbitsFirst of all, let us recall the notion of Ricci solitons.
$\mathbb{C}H^{n}$$S_{b}(\varphi)$

Definition 3.6. A Riemannian manifold $(M, g)$ is called a Ricci soliton if its
Ricci curvature $ric$ satisfies$ric=cg+\mathcal{L}_{X}g$

for someand some vector field $X\in X(M)$ , where
Lie derivative.
$c\in \mathbb{R}$$\mathcal{L}_{X}$denotes the usual

It is easy to see that the notion of Ricci solitons is a generalization of Einsteinmanifolds. We now also recall the notion of algebraic Ricci solitons, which
essentially have been introduced by Lauret (see [17, 1817Definition 3.7. A metric Lie algebra
if its Ricci operator
satisfies$(\mathfrak{g}, \langle, \rangle)$

is called an algebraic Ricci soliton$Ric:\mathfrak{g}arrow \mathfrak{g}$(3.1)$Ric=c\cdot id_{\mathfrak{g}}+D$

for someand somederivations on$c\in \mathbb{R}$, where Der ( ) denotes the Lie algebra of$D\in Der(\mathfrak{g})$$\mathfrak{g}$$\mathfrak{g}.$Remark 3.8. Let $(G, g)$ be a simply-connected Lie group with a left-invariant
metric, andbe the corresponding metric Lie algebra. Then, one knowsthat $(G, g)$ is a Ricci soliton if
is an algebraic Ricci soliton. In addition,
if is completely solvable, which means that the eigenvalues of any
are allreal, then the converse also holds. See [18] for more details.
$(\mathfrak{g}, \langle, \rangle)$$(\mathfrak{g}, \langle, \rangle)$$G$$ad_{X}$

By direct calculations, one can see thatis completely solvable. Therefore, we have only to study whether
algebraicis anRicci soliton for ourgoal.
Firstly, we consider the case of, that is, the nilpotent case. It is easy
to show that a direct sum as metric Lie algebras of an algebraic Ricci soliton
and an abelian Lie algebra is also an algebraic Ricci soliton. Hence, one has
the following.
$S_{b}(\varphi)$$\mathfrak{s}_{b}(\varphi)$$\mathfrak{s}_{b}(\pi/2)$

Proposition 3.9. For each $b\in\{1, . . . , n\}$ , the metric Lie algebraan algebraic Ricci soliton.

$\mathfrak{s}_{b}(\pi/2)$

is

In the other cases, the structure theorem for algebraic Ricci solitons [18,Theorem. 4.8] yields the following.

Proposition 3.10. For $b\in\{1, . . . , n-1\}$ and $\varphi\in[0, \pi/2$ ), the metric Liealgberais an algebraic Ricci soliton if and only if $b=n-1$ and $\varphi=0.$$\mathfrak{s}_{b}(\varphi)$

Altogether, we have the following result.
Theorem 3.11. The orbit of(1) $b=n-1$ and $\varphi=0$ , or(2) $\varphi=\pi/2.$$S_{b}(\varphi).0$in$\mathbb{C}H^{n}$

is a Ricci soliton if and only if

Remark 3.12. In the case of cohomogeneity one, Theorem 3.11 has been
proved in [12]. Note that the proofs in [12], which are based on the explicit
formulas for the Ricci operators, are direct and elementary.

REFERENCES[1] J. Berndt, Homogeneous hypersurfaces in hyperbolic spaces, Math. Z., 229 (1998), no. 4,589-600.
[2] J. Berndt, Polar actions on symmetric spaces, in: Proceedings of the Fifteenth International Workshop on Diﬀ. Geom., 15 (2011), 1-10.
[3] J. Berndt, S. Console & C. Olmos, Submanifolds and holonomy, Chapman & Hall/CRC
Research Notes in Mathematics, 434. Chapman & Hall/CRC, Boca Raton, FL, 2003.
[4] J. Berndt & J. C. Diaz-Ramos, Homogeneous polar foliations of complex hyperbolicspaces, Comm. Anal. Geom., 20 (2012), no. 3, 435-454.
[5] J. Berndt, J. C. D\’iaz-Ramos & H. Tamaru, Hyperpolar homogeneous foliations on symmetric spaces of noncompact type, J. Diﬀerential Geom., 86 (2010), no. 2, 191-235.
[6] J. Berndt & H. Tamaru, Homogeneous codimension one foliations on noncompact symmetric spaces, J. Diﬀerential Geom., 63 (2003), no. 1, 1-40.

18[7] J. Berndt & H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of. Amer. Math. Soc., 359 (2007), no. 7, 3425-3438.
rank one,[8] J. Berndt, F. Tricerri & L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci
harmonic spaces, Lecture Notes in Mathematics, 1598. Springer- Verlag, Berlin, 1995.
[9] J. C. D\’iaz-Ramos, Polar actions in complex space forms, in: Proceedings of the Sixteenth
ational on Diﬀ. Geom., 16 (2012), 71-90.
[10] J. C. D\’iaz-Ramos, M. Dom\’inguez-V\’azquez & A. Kollross, Polar actions on complex
hyperbolic spaces. preprint, arXiv:120S.2823v2.
[11] T. Hamada, Y. Hoshikawa & H. Tamaru, Curvature properties of Lie hypersurfaces inthe complex hyperbolic space, J. Geom., 103 (2012), no. 2, 247-261.
[12] T. Hashinaga, A. Kubo & H. Tamaru, Homogeneous Ricci soliton hypersurfaces in thecomplex hyperbolic spaces, preprint, arXiv:1305.612Sv1.
[13] E. Heintze, X. Liu & C. Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, in: Integrable systems, geometry, and topology, 151-190, AMS/IP Stud.
Adv. Math., 36, Amer. Math. Soc., Providence, RI, 2006.
[14] E. Heintze, R. S. Palais, C. L. Terng & G. Thorbergsson, Hyperpolar actions on symmetric spaces, Geometry, topology, & physics, 214-245, Conf. Proc. Lecture Notes Geom.
Topology, IV, Int. Press, Cambridge, MA, 1995.
[15] A. Kubo, Geometry of homogeneous polar foliations of complex hyperbolic spaces, Hiroshima Math. J., to appear.
[16] A. Kubo & I. Tamaru, A suﬃcient condition for congruency of orbits of Lie grollps andsome applications, Geom. Dedicata, 167 (2013), 233-238.
[17] J. Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann., 319 (2001), 715-733.
[18] J. Lauret, Ricci soliton solvmanifolds, J. reine angew. Math., 650 (2011), 1-21.
$\mathcal{I}rans$$Inte\gamma\gamma\iota$$\ddagger$

DEPARTMENT OF MATHEMATICS, HIROSHIMA UNIVERSITY, HIGASHI-HIROSHIMA 739-8526,
JAPAN
$E$
-mail address: akira-kubo@hiroshima-u.ac.jp