数理解析研究所講究録
第 1929 巻 2014 年 1-12

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GROUP ACTIONS ON SYMMETRIC SPACES RELATED TO
LEFT-INVARIANT GEOMETRIC STRUCTURESHIROSHI TAMARU
ABSTRACT. In this paper, we summarize how the theory and results of group
actions on symmetric spaces can be applied to the study of left-invariant geometric structures on Lie groups. We also present a list of problems on group
actions, which naturally arise from this framework.

1. INTRODUCTION
Isometric actions on Riemannian symmetric spaces of noncompact type, such as
cohomogeneity one actions and (hyper)polar actions, have been studied actively
in these decades (see [1-7, 19] and references therein). The theory and results ofthese actions have recently been applied to the study of left-invariant geometricstructures on Lie groups ([10-12, 17, 18, 38 The aim of this survey paper is topresent our framework and recent results. We also propose several problems ongroup actions, which naturally arise from our framework. The answers of these
problems would be interesting, not only from the viewpoint of group actions andsubmanifold geometry, but also for possible applications to the further studies on
left-invariant geometric structures.
Left-invariant geometric structures on Lie groups, such as (pseudo-)Riemannian
metrics, symplectic structures, and (generalized) complex structures, have provided a lot of interesting examples, and have been studied very actively (forleft-invariant metrics, we refer to [8, 9, 13-16, 20, 22-27, 29-36, 39-44]). One of
the central problems is the following existence and nonexistence problem.

Problem 1.1. For a given Lie group, determine whether it admits “nice” leftinvariant geometric structures or not.

Note that, for a given Lie group and a given left-invariant structure, one candirectly study the properties of them. For example, the following can be studied
in the Lie algebra level
curvatures of a left-invariant metric, the integrability
condition of a left-invariant almost complex structure, and so on. However, thisdoes not mean that the above mentioned problem is easy. One of the diﬃculties
comes from the fact that a Lie group admits so many left-invariant geometricstructures. For left-invariant metrics, one knows the following.
–

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HIROSHI TAMARU

Fact 1.2. Let$G$

be a Lie group of dimension$\overline{\mathfrak{M}}:=$$\cong$$n$.

Then there are identifications{left-invariant Riemannian metrics on }{inner products on $:=Lie(G)$ }$G$$\mathfrak{g}$$\cong GL(n, \mathbb{R})/O(n)$$\overline{\mathfrak{M}}_{(p,q)}:=$,{left-invariant metrics on$\cong GL(n, \mathbb{R})/O(p, q)$$G$with signature $(p, q)$ }.

Therefore, for studying the existence and nonexistence of (nice” left-invariant
metrics, such as Einstein or Ricci soliton, one has to study all points on theabove spaces. For left-invariant metrics, of course the Einstein equation is alinear equation, but it contains $(1/2)n(n+1)$ variables, which is in general very
hard to be solved. In order to avoid this diﬃculty, we have proposed an approachfrom the following viewpoint.

are symmetric spaces (the former isandFact 1.3. The above spaces
noncompact Riemannian, but the latter is pseudo-Riemannian). Furthermore,
there are naturalof the automorphism groups of the Lie algebras on thesespaces.
$\overline{\mathfrak{M}}$$\overline{\mathfrak{M}}_{(p,q)}$$action\mathcal{S}$This connects naturally the studies of left-invariant geometric structures andof group actions on symmetric spaces.

Remark 1.4. We only mention left-invariant metrics in this paper, but believethat similar frameworks also work for other left-invariant geometric structures.
In many cases, the set of some geometric structures forms a symmetric space.
Isometric actions on Riemannian symmetric spaces of compact type are certainly interesting topics, and have been studied by many authors. We would liketo say that, isometric actions on Riemannian symmetric spaces of noncompact
type, and group actions on pseudo-Riemannian symmetric spaces, are also interesting topics. They are of course interesting from the viewpoint of group actions
and submanifold geometry, and also interesting because of possible applications
to the studies on left-invariant geometric structures.
Acknowledgements. The author would like to thank Takashi Sakai for a kind
invitation and for giving me this opportunity. He also thanks to Takayuki Okuda
and Akira Kubo for reading the manuscript carefully. This work was supported
by JSPS KAKENHI Grant Numbers 24654012, 26287012.

2. ISOMETRIC ACTIONS ON RIEMANNIAN SYMMETRIC SPACES OFNONCOMPACT TYPE (1 )Let us consider isometric actions of $H$ on Riemannian symmetric spaces $M$. We
of noncompact type. In this section, we consider the orbit spaces$H\backslash M$

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GROUPS ACTIONS RELATED TO LEFT-INVARIANT GEOMETRIC STRUCTURES

mention some known results, and their applications to the study of left-invariant
Riemannian metrics on Lie groups.

2.1. Results on cohomogeneity one actions. First of all, we recall somefundamental notions on group actions, and review a result on cohomogeneity one
actions.
Definition 2.1. Consider an isometric action of a Lie group $H$ on a Riemannian
manifold $M$ . Then, orbits of maximal dimension are said to be regular, and other
orbits singular. The codimension of a regular orbit is called the cohomogeneityof the action.

For cohomogeneity one actions on symmetric spaces of noncompact type, possible topological types of the orbit spaces have been studied.
Theorem 2.2 (Berndt-Br\"uck ([1])). Let $M$ be a Riemannian symmetric spaceof noncompact type, and consider a cohomogeneity one action of $H$ on $M$ with
$H$ being connected.
Then, the orbit spaceis homeomorphic to either or$\mathbb{R}$$H\backslash M$$[0, +\infty)$.

As an example, we here draw a picture of this situation for cohomogeneity one
actions on the real hyperbolic plane$\mathbb{R}H^{2}=SL(2, \mathbb{R})/SO(2)$ .
Example 2.3. Consider the Iwasawa decomposition $SL(2, \mathbb{R})=KAN$ , where$K=SO(2)$ ,$A=\{(\begin{array}{ll}a 00 -a\end{array})|a>0\},$$N=\{(\begin{array}{ll}1 b0 1\end{array})|b\in \mathbb{R}\}.$

Then, the actions of $K,$ , and $N$ on
are of cohomogeneity one (and in fact
they exhaust all, up to orbit equivalence). The orbits and the orbit spaces ofthese actions are as in Figure 1. More precisely,(K) The action of $K$ has a fixed point, the center . Other orbits are circles
centered at . In this case, the orbit space is homeomorphic to $[0, +\infty$ ).
(A) The orbit of through the center is a geodesic, and other orbits are its
equidistant lines. The orbit space is homeomorphic to(N) The orbits of $N$ are horocycles. In this case, all orbits are congruent toeach other, and the orbit space is homeomorphic to$A$$\mathbb{R}H^{2}$

$0$

$0$

$A$

$0$

$\mathbb{R}.$$\mathbb{R}.$2.2. Results on left-invariant metrics. We now consider , the set of allleft-invariant Riemannian metrics on a given Lie group . Assume thatissimply-connected, for simplicity.
$\overline{\mathfrak{M}}$$G$

Remark 2.4. Denote by the Lie algebra ofof . Then one has identifications$\mathfrak{g}$

$G$

. Let

$n:=\dim \mathfrak{g}$$G$, and fix a basis$\mathfrak{g}$$\mathfrak{g}\cong \mathbb{R}^{n}, GL(\mathfrak{g})\cong GL(n, \mathbb{R}) , \overline{\mathfrak{M}}\cong GL(n, \mathbb{R})/O(n)$.

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HIROSHI TAMARU

type (A)type (K)type (N)FIGURE 1. The orbits and the orbit spaces of actions on$\mathbb{R}H^{2}$equipped with the canonicalThen, it is well-known that-invariant
Riemannian metric is a Riemannian symmetric space, which is noncompact. In
this paper, this Riemannian metric is always assumed to be equipped.
$\overline{\mathfrak{M}}$$GL(n, \mathbb{R})$We consider the group actions on$\mathbb{R}^{\cross}:=\{c.

id:$\overline{\mathfrak{M}}$given by\mathfrak{g}arrow \mathfrak{g}|c\in \mathbb{R}_{\neq 0}\},$$Aut(\mathfrak{g}):=\{\varphi\in GL(\mathfrak{g})|\varphi =[\varphi(\cdot) , \varphi$

Definition 2.5. The orbit space of the action of$\mathbb{R}^{\cross}Aut(\mathfrak{g})$on$\overline{\mathfrak{M}},$$\mathfrak{P}\mathfrak{M}:=\mathbb{R}^{\cross}Aut(\mathfrak{g})\backslash \overline{\mathfrak{M}},$is called the moduli$\mathcal{S}pace$of left-invariant Riemannian metrics on $G.$

gives rise to isometry up to scaling of leftRemark 2.6. The action of
Therefore,
all Riemannian geometric properties ofinvariant Riemannian metrics.
left-invariant metrics are preserved by this action. Thus, in order to examine theexistence and nonexistence of a “nice” metric, one has only to study the moduli
space$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\mathfrak{P}\mathfrak{M}.$We here give one easy example of a description of the moduli spaceis denoted byThroughout this paper, the canonical basis of$\mathfrak{P}\mathfrak{M}.$$\{e_{1}, \cdots, e_{n}\}.$$\mathfrak{g}=\mathbb{R}^{n}$

Proposition 2.7 (Hashinaga-Tamaru-Terada ([12])). Consider the Lie algebra$:=$
the canonicalwith $[e_{1}, e_{2}]=e_{2}$ (and others are zero). Denote byinner product on
Then one(1) the action of
onis of cohomogeneity one,(2) the moduli spacecan be expressed as$(\mathbb{R}^{3}, [, ])$$\langle,$$\rangle_{0}$$\mathfrak{g}$$ha\mathcal{S}$$\mathfrak{g}=\mathbb{R}^{3}$$\overline{\mathfrak{M}}$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\mathfrak{P}\mathfrak{M}$$\mathfrak{P}^{\mathfrak{M}=}\{\mathbb{R}^{\cross}Aut(\mathfrak{g}).((\begin{array}{lll} 1 1 \lambda 1\end{array}).\langle, \rangle_{0})|\lambda\in \mathbb{R}\}.$, one needs direct matrix
Remark 2.8. In order to give an expression of
calculations. On the other hand, one knows possible topological types of the$\mathfrak{P}\mathfrak{M}$5
GROUPS ACTIONS RELATED TO LEFT-INVARIANT GEOMETRIC STRUCTURES

orbit spaces in Theorem 2.2. This general theory of cohomogeneity one actions
does not give an expression of, but gives us a certification.
$\mathfrak{P}\mathfrak{M}$This expression of the moduli spaceprovides the following Milnor-type
theorem. The reason of this naming is that the basis mentioned below is a kindof a generalization of “Milnor frames”, obtained by Milnor ([29]).
$\mathfrak{P}\mathfrak{M}$Proposition 2.9 ([12]). Let
be the Lie algebra in Proposition 2.7,
andbe an arbitrary inner product on . Then, there exist$k>0,$
and an orthonormal
with respect tosuch that the nontrivial
bracket relations are given by$\mathfrak{g}$$\langle,$$:=(\mathbb{R}^{3}, [, ])$$\rangle$$\lambda\in \mathbb{R},$$\mathfrak{g}$$ba\mathcal{S}is\{x_{1}, x_{2}, x_{3}\}$$k\langle,$$\rangle$$[x_{1}, x_{2}]=x_{2}-\lambda x_{3}.$When we apply this Milnor-type theorem, one can assume $k=1$ without lossof generality, since $k>0$ is just a scaling factor. In terms of the basis
one can directly calculate the curvatures, which proves the following.
Corollary 2.10 ([12]). The Lie algebra in Proposition 2.7 does not admit leftinvariant Einstein metrics, and furthermore, it
not admit left-invariant metrics of negative Ricci curvature.
$\{x_{1}, x_{2}, x_{3}\},$$doe\mathcal{S}$The above Lie algebra is just an easy example. We have studied some other
Lie algebras as well, including the following ones.
Remark 2.11. Milnor-type theorems have been obtained for(1) all three-dimensional solvable Lie algebras ([11]),(2) all four-dimensional nilpotent Lie algebras ([10]),(3) some higher-dimensional Lie algebras ([10, 12, 38

It is remarkable that the action ofoncan be of cohomogeneityone, even if the dimension of is high. In fact, for any $n\geq 3$ , there exists a Lie
algebra of dimension whose corresponding action is of cohomogeneity one.
$\overline{\mathfrak{M}}$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\mathfrak{g}$$\mathfrak{g}$$n$2.3. Problems. In the last of this section, we propose some problems concerningwith the orbit spaces of isometric actions on symmetric spaces of noncompact
type.

Problem 2.12. For Riemannian symmetric spaces of noncompact type, classify
possible topological type of orbit spaces of some particular classes of actions (forexamples, cohomogeneity two, (hyper)polar, and so on).
The answers of the above problem would be useful to obtain Milnor-type theorems for more complicated Lie algebras. On the Lie algebra side, the following

problems would be natural.

Problem 2.13. Classify Lie algebras so that the actions ofonhave some particular properties (for examples, cohomogeneity one or two, (hyper)polar, and so on).

$\overline{\mathfrak{M}}$$\mathfrak{g}$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$6HIROSHI TAMARU

Note that Taketomi ([37]) has constructed Lie algebras so that the actions ofare hyperpolar, having a singular orbit and higher cohomogeneity. In
this way, the study of left-invariant metrics would also contribute to the study ofisometric actions on symmetric spaces.
$\mathfrak{g}$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$3. ISOMETRIC ACTIONS ON RIEMANNIAN SYMMETRIC SPACES OFNONCOMPACT TYPE (2)In this section, we continue to consider isometric actions on Riemannian symmetric spaces of noncompact type. We here focus on the geometry of the orbits,
in particular, some “distinguished” orbits.

3.1. Results on cohomogeneity one actions. First of all, we recall a result
on cohomogeneity one actions. The following is not explicitly written in [4], but
can be seen from the classification result (we also refer to [7]).
Theorem 3.1 (Berndt-Tamaru ([4])). Let $M$ be an irreducible Riemannian symmetric space of noncompact type, and consider a cohomogeneity one action of $H$
on $M$ with $H$ being connected. Then, this action satisfies one of the following:(K) There exists a unique singular orbit.
(A) All orbits are regular, and there exists a unique minimal orbit.
are isometrically con(N) All orbits are regular, and furthermore, allgruent to each other.
$orbil\mathcal{S}$One already knows examples of these actions in Figure 1. Recall that the.
names (K) , (A) , and (N) come from the Iwasawa decomposition of$SL(2, \mathbb{R})$Remark 3.2. For a cohomogeneity one action of type (K) or (A) , there exists a
unique “distinguished” orbit, that is, the unique singular orbit for type (K) , or
the unique minimal orbit for type (A) . For the case of type (N) , it looks that
there does not exist a distinguished orbit.

3.2. Results on left-invariant metrics. We now see how the above pictures ofcohomogeneity one actions are related to the study of left-invariant Riemannian
metrics. We are interested in the following class of left-invariant metrics.
Definition 3.3. Let
the Ricci operator. Then,and a derivation$(\mathfrak{g}, \langle, \rangle)$$\langle,$

be a metric Lie algebra, and denote byis said to be algebraic Ricci soliton if there exist
such that$Ric:\mathfrak{g}arrow \mathfrak{g}$$\rangle$$D\in Der(\mathfrak{g})$$c\in \mathbb{R}$$Ric=c\cdot id+D.$

It is easy to see that “Einstein” implies “algebraic Ricci soliton” Recall thatis a linear map satisfying
a derivation $D$ :$\mathfrak{g}arrow \mathfrak{g}$$D =[D(\cdot), ]+ D$

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GROUPS ACTIONS RELATED TO LEFT-INVARIANT GEOMETRIC STRUCTURES

Remark 3.4. If
is algebraic Ricci soliton, then it gives rise to a Ricci
soliton metric $([23, 25$ More precisely,is algebraic Ricci soliton, thenthe corresponding simply-connected Lie group with the induced left-invariant
metricis Ricci soliton, in the sense that there exist
and $X\in X(G)$such that$(\mathfrak{g}, \langle, \rangle)$$if (\mathfrak{g}, \langle, \rangle)$$(G, \langle, \rangle)$$c\in \mathbb{R}$$ric=c\langle, \rangle+\mathfrak{L}_{X}\langle, \rangle.$

Here, $ric$ denotes the Ricci $(0,2)$ -tensor, and$\mathfrak{L}_{X}$the Lie derivative along $X.$

We study algebraic Ricci soliton metrics, from the view point of the actions of. In the following case, there is a very nice correspondence.

$\mathbb{R}^{\cross}Aut(\mathfrak{g})$Theorem 3.5 (Hashinaga-Tamaru ([11])). Let be a three-dimensional solvable
Lie algebra. Then, an inner producton is algebraic Ricci soliton if and onlyis a minimal submanifold.
if the orbit$\mathfrak{g}$$\langle,$$\rangle$$\mathfrak{g}$$\mathbb{R}^{\cross}Aut(\mathfrak{g}).\langle,$$\rangle$

Recall that the ambient space
is equipped with the natural-invariant metric, and is a noncompact Riemannian symmetric space.
$\overline{\mathfrak{M}}=GL(3, \mathbb{R})/O(3)$$GL(3, \mathbb{R})$Remark 3.6. A key point of Theorem 3.5 is that, for a three-dimensional solvable Lie algebra , the action of
is of cohomogeneity at most one (thatis, transitive or cohomogeneity one). For the cohomogeneity one cases, algebraic
Ricci soliton metrics are exactly corresponding to the “distinguished” orbits described in Remark 3.2.
$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\mathfrak{g}$We checked several Lie algebras whether it has the similar property or not.
The answer is aﬃrmative for some cases, but not for some cases. For examples,$\mathfrak{g}$Theorem 3.7 (Hashinaga ([10])). For a four-dimensional nilpotent Lie algebra, an inner productis algebraic Ricci soliton if and only if
is aminimal submanifold. On the other hand, there exists a four-dimensional solvable
Lie algebra such that the both implications do not hold.
$\langle,$$\rangle$$\mathbb{R}^{\cross}Aut(\mathfrak{g}).\langle,$$\mathfrak{g}$$\rangle$$\mathfrak{g}$Despite of the above result, we still expect that (algebraic) Ricci soliton metrics
are corresponding to “distinguished”. . One naive certification isthat, ifis bi-invariant, thenis totally geodesic.
$\langle,$$\rangle$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\langle,$$\rangle$$\mathbb{R}^{\cross}Aut(\mathfrak{g}).\langle,$$\langle,$$\rangle$$\rangle$

3.3. Problems. In the last of this section, we propose some problems on isometric actions, which naturally arise from our framework.
Problem 3.8. For Riemannian symmetric spaces of noncompact type, study
the geometry of orbits of some particular actions. For examples, cohomogeneity one actions (with $H$ not necessarily connected), cohomogeneity two actions,(hyper)polar actions, and so on.
The next problem is about a characterization of particular left-invariant metrics
in terms of the corresponding submanifolds.

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HIROSHI TAMARU

Problem 3.9. Examine whether properties of can be understood by the corresponding group actions. For examples, the following properties can be charac(algebraic) Ricci soliton, flat, positive or negative
terized by submanifolds?
curvatures, and so on.
$\mathfrak{g}$–4. NONISOMETRIC ACTIONS ON$R$-SPACESWe here consider group actions, not necessarily isometric, on -spaces. In thissection, we mention one simple example, and some possible problems. We refer
to [28, 21] and references therein for related and deeper results.
$R$

4.1. Results on group actions. We here see one example ofgroup action on an -space.

$a$(nonisometric)$R$

Definition 4.1. Let be a semisimple Lie group with trivial center, and be aparabolic subgroup of . Then the coset space $M$ $:=L/Q$ is called an -space.
$Q$

$L$

$R$

$L$

An -space is also called a real flag manifold. An example on which we concentrate in this section is the real projective space.
$R$

Example 4.2. The real projective space
expresslon$\mathbb{R}P^{n}$is an-space. In fact, one has an$R$$\mathbb{R}P^{n-1}=SL(n, \mathbb{R})/\{(00* *** ***)|\det=1\}.$

Remark 4.3. It is known that an -space can be realized as an orbit of thecan beisotropy representation of a Riemannian symmetric space. In fact,
realized as an orbit of the isotropy representation of $SL(n, \mathbb{R})/SO(n)$ .
$R$$\mathbb{R}P^{n-1}$

We consider the natural action of $SO(p, q)$ on
choice will be mentioned in the next section.

$\mathbb{R}P^{p+q-1}$. The reason of this

Proposition 4.4 (Kubo-Onda-Taketomi-Tamaru ([18])). Consider $SO(p, q)$ with
$q\geq 1$ . Letwith signature $(p, q)$ ,be the canonical inner product onwhich invariant under the action of $SO(p, q)$ . Then, the action of $SO(p, q)$ on
has exactly three orbits, namely,$\mathbb{R}^{p+q}$$p,$$\langle,$$\rangle_{0}$$i\mathcal{S}$$\mathbb{R}P^{p+q-1}$$\mathcal{O}^{+}:=\{[v]\in \mathbb{R}P^{p+q-1}|\langle v, v\rangle_{0}>0\},$$\mathcal{O}^{0}:=\{[v]\in \mathbb{R}P^{p+q-1}|\langle v, v\rangle_{0}=0\},$$\mathcal{O}^{-}:=\{[v]\in \mathbb{R}P^{p+q-1}|\langle v, v\rangle_{0}<0\}.$are open orbits. Since $SO(p, q)$ preservesandNote thatto see that these orbits are diﬀerent to each other.
$\mathcal{O}^{+}$$\mathcal{O}^{-}$$\langle,$$\rangle_{0}$, it is easy

9
GROUPS ACTIONS RELATED TO LEFT-INVARIANT GEOMETRIC STRUCTURES, that is,
Remark 4.5. Note that $SO(p, q)$ is a symmetric subgroup of$(SL(n, \mathbb{R}), SO(p, q))$ is a symmetric pair. The above mentioned action would beinteresting from this view point, since it can be considered as an analogy of a
“Hermann action”$SL(n,\mathbb{R})$4.2. Problems. In the last of this section, we propose some problems on group
actions on -spaces. The following problems must be related to representation$R$theory.

Problem 4.6. Let $M=L/Q$ be an -space, and $H$ be a Lie subgroup of $L.$
Consider the action of $H$ on $M=L/Q.$$R$(1) Among such actions, find interesting examples.
(2) Study what happens if $(L, H)$ is a symmetric pair.
(3) Examine when it has an open orbit.

Next problem is about submanifold geometry. At the moment, the author
knows no interesting examples, but it would be natural.
Problem 4.7. Consider the same situation as Problem 4.6. Then, orbits $H.p$
can be inhomogeneous with respect to Isom (M) . However, study whether theorbits still have some “nice” properties (as Riemannian submanifold) or not.

5. ISOMETRIC ACTIONS ON PSEUDO RIEMANNIAN SYMMETRIC SPACES
In this section, we consider isometric actions on pseudo-Riemannian symmetricspaces. This is related to the topic of the previous section, and also has applications to the study of left-invariant pseudo-Riemannian metrics on Lie groups.

5.1. Results on group actions. We here mention the following only one example of an isometric action.
Proposition 5.1 (Kubo-Onda-Taketomi-Tamaru ([18])). The action of the following
on $GL(p+q, \mathbb{R})/O(p, q)$ has exactly three orbits:$Q$$Q:=\{ (00* * \cdot*\cdot *)\in GL(p+q, \mathbb{R})\}.$

One can certificate this from Proposition 4.4. In fact, the orbit space of thisaction coincides with the orbit space of the action of $O(p, q)$ on$GL(p+q, \mathbb{R})/Q=\mathbb{R}P^{p+q-1}$

10
HIROSHI TAMARU, the set of all left5.2. Results on left-invariant metrics. We now recall
invariant metrics on a Lie group with signature $(p, q)$ . Denote by the Liealgebra of , and consider the groupas in the Riemannian case.
$\tilde{\mathfrak{M}}_{p,q}$$G$$\mathfrak{g}$$G$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$Definition 5.2. The orbit space of the action ofon$\mathbb{R}^{\cross}Aut(\mathfrak{g})$$\overline{\mathfrak{M}}_{p,q},$$\mathfrak{P}\mathfrak{M}_{p,q}:=\mathbb{R}^{\cross}Aut(\mathfrak{g})\backslash \overline{\mathfrak{M}}_{p,q},$is called the moduli space of left-invariant metrics on$G$with signature $(p, q)$ .

We pic up the following two Lie algebras. Recall that
denotes the
canonical basis of , and we only write nonzero bracket relations.
$\{e_{1}, .

.

.

, e_{n}\}$$\mathbb{R}^{n}$Definition 5.3. We define the following two Lie algebras:(1)
withis called the Heisenberg Lie algebra.
(2)
with the following bracket relations is called the solvable
Lie algebra of:$[e_{1}, e_{2}]=e_{3}$$\mathfrak{h}^{3}=(\mathbb{R}^{3}, [, ])$$\mathfrak{g}_{\mathbb{R}H^{n}}=(\mathbb{R}^{n}, [, ])$$\mathbb{R}H^{n}$$[e_{1}, e_{2}]=e_{2}$, . . . , $[e_{1}, e_{n}]=e_{n}.$Note thatis the real hyperbolic space. The simply-connected Lie groupwith Lie algebra
acts simply-transitively on$\mathbb{R}H^{n}$$\mathbb{R}H^{n}.$$G_{\mathbb{R}H^{n}}$$\mathfrak{g}_{\mathbb{R}H^{n}}$. Then, for all
Theorem 5.4 ([18]). Letorwith $dimg=$$p+q$ , there exists exactly three left-invariant metrics onwith signature $(p, q)$ ,up to $isometr1/and$ scaling.
$:=\mathfrak{h}^{3}$$\mathfrak{g}_{\mathbb{R}H^{n}}$$\mathfrak{g}$$p,$ $q\in \mathbb{N}$$\mathfrak{g}$The proof is based on the fact that, for the Lie algebras mentioned above,is a parabolic subgroup. In fact, it coincides with in Proposition 5.1.
$\mathfrak{g}$$Q$$\mathbb{R}^{\cross}Aut(\mathfrak{g})$Remark 5.5. For the case of , the above result has been known by Rahmani([36]), but the method is diﬀerent. The properties of these three metrics havealso been studied by Onda ([35]).
$\mathfrak{h}^{3}$, we have proved that these three left-invariant
Remark 5.6. For the case of
metrics have constant sectional curvatures ([18]). Note that the Lorentzian caseof these results have been known by Nomizu ([34]), but the method is diﬀerent.
Our argument simplifies the proof of his results, and also extends it to the caseof generic signatures.
$\mathfrak{g}_{\mathbb{R}H^{n}}$5.3. Problems. As mentioned above, the study of isometric actions on $GL(p+$has applications to left-invariant pseudo-Riemannian metrics. This$q,$ $\mathbb{R})/O(p, q)$

motivates to study the following problems.
Problem 5.7. Study isometric actions of $H$ on pseudo-Riemannian symmetricspaces $M$ . First of all, study the case when $H$ is a parabolic subgroup (this casecorresponds to the “symmetric actions”’ on $R$-spaces, mentioned in Problem 4.6).

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GROUPS ACTIONS RELATED TO LEFT-INVARIANT GEOMETRIC STRUCTURES

Problem 5.8. Construct “nice” isometric actions on a pseudo-Riemannian symmetric space $M=U/K$ . For example, let $K’$ be a maximal compact subgroupof , and consider the Riemannian symmetric space $M’=U/K’$ of noncompact
type. If the action of $H$ on $M’$ is nice in some sense, then so is the action of $H$
on $M$ ?
$U$

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DEPARTMENT OF MATHEMATICS, HIROSHIMA $UNIVERSITy_{\}}$ HIGASHI-HIROSHIMA 739-8526,
JAPAN-mail address: tamaru@math.sci.hiroshima-u.ac.jp$E$