A MAXIMAL ELEMENT OF A MODULI SPACE OF
RIEMANNIAN METRICSYUICHIRO TAKETOMI

Abstract. For a given smooth manifold, we consider the moduli space ofRiemannian metrics up to isometry and scaling. One can deﬁne a preorder on
the moduli space by the size of isometry groups. We call a Riemannian metricthat attains a maximal element with respect to the preorder a maximal metric.
Maximal metrics give nice examples of self-similar solutions for various metricevolution equations such as the Ricci ﬂow. In this paper, we construct many
examples of maximal metrics on Euclidean spaces.

1. Introduction
Let X be a connected smooth manifold. Denote by M(X) the set of all smooth
Riemannian metrics on X. Deﬁne an equivalent relation ∼ on M(X) as follows:
there exists λ > 0 such that (X, λg) and (X, h) are isometric with each other.
Denote by M(X)/∼ the quotient space with respect to the equivalent relation∼. Note that the moduli space M(X)/∼ can be understood as the orbit space(R>0 × Diﬀ(X))\M(X).
The moduli space M(X)/∼ has often been considered to understand “nice”
Riemannian metrics. For examples, the normalized total scalar curvatureZ
2−1
scalg dVg
S̃ : M(X) → R, g 7→ vol(X, g) nX

has been studied actively for a compact manifold X. Here, vol(X, g) is the
Riemannian volume, scalg is the scalar curvature, and dVg is the Riemannianvolume element. For examples, Einstein metrics on a compact manifold X arecharacterized by critical points of S̃ : M(X) → R. Since the normalized total
scalar curvature S̃ is invariant under the action of R>0 ×Diﬀ(X) on M(X), S̃ canbe regarded as the function on the moduli space M(X)/∼ . Another important
example is the Ricci ﬂow(1.1)∂
gt = −2Ricgt ,∂tThis work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. This work was partly
supported by Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/ResearchCenter on Mathematics and Theoretical Physics JPMXP0619217849.
1

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YUICHIRO TAKETOMI

where Ricg is the Ricci tensor for a Riemannian metric g. Since the Ricci tensor Ricg is invariant under the scaling (i.e. Riccg = Ricg ) and diﬀeomorphisms(i.e. Ricφ∗ g = φ∗ Ricg , where ∗ means the pullback), the Ricci ﬂow equation canbe regarded as a ﬂow on the moduli space M(X)/∼ . Then, for examples, selfsimilar solutions of the Ricci ﬂow are considered as stationary solutions of thecorresponding ﬂow on M(X)/∼ .
Many “nice” metrics (e.g. Einstein metrics, Ricci soliton) can be understood as
distinguished points on the moduli space M(X)/∼ with respect to some curvature
condition. In this paper, we introduce a class of nice Riemannian metrics, which
is deﬁned as a special point on the moduli spaces M(X)/∼ with respect to the sizeof isometry groups. For [g], [h] ∈ M(X)/∼ , denote by [g] ≺ [h] if Isom(X, g) ⊂Isom(X, h′ ) for some h′ ∈ [h]. Then ≺ deﬁnes an order on M(X)/∼ . Notethat ≺ is not a partial order but a preorder. That is, ≺ satisﬁes reﬂexivity andtransitivity, however, does not satisfy asymmetry.
Deﬁnition 1.1. We call a metric g ∈ M(X) maximal if the equivalent class[g] ∈ M(X)/∼ is a maximal element with respect to the preorder ≺ (i.e. [g] ≺ [h]implies [g] = [h]).
Roughly speaking, the order ≺ deﬁnes a barometer of the excellence of Riemannian metrics via the size of isometry groups, and maximal metrics are “maximally nice” metrics with respect to this barometer.
Note that g ∈ M(X) is a maximal metric if and only if g satisﬁes the following
condition :(1.2)

Isom(X, g) ⊂ Isom(X, h) ⇒ [g] = [h] (h ∈ M(X)).

An important example of maximal metrics is an isotropy irreducible metric.
For more details, see Subsection 2.1.
One of the nice motivation to study maximal metrics is that g give examplesof self-similar solutions for various metric evolution equations∂
gt = R(gt ) (gt ∈ M(X)),∂twhere R is a map from M(X) to the set of all symmetric (0, 2)-tensors S(X). We
show that homogeneous maximal metrics are soliton for various metric evolution
equations. For examples,
Proposition 1.2. A homogeneous maximal metric g is a Ricci soliton.
For more details, see Subsection 2.2.
Remark 1.3. Homogeneous Ricci solitons have been studied actively. Recently, aproof of the Alekseevskii conjecture has been announced by Lafuente and Böhm([3]). Also, Jablonski has shown that the Alekseevskii conjecture is equivalent to
the generalized Alekseevskii conjecture which asserts that only Euclidean spaces
can admit homogeneous expanding Ricci solitons ([10]). Note that a shrinking

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3

homogeneous Ricci soliton manifold is the Riemannian product of a compact Einstein manifold and a Euclidean space ([20, 22]), and a steady homogeneous Ricci
soliton is the Riemannian product of a compact ﬂat manifold and a Euclidean
space ([10, 22]). These and the generalized Alekseevskii conjecture conclude thata noncompact homogeneous Ricci soliton irreducible Riemannian manifold is diffeomorphic to a Euclidean space. Therefore, essential homogeneous maximal
metrics on noncompact manifolds can only exist on Euclidean spaces.
For the compact case, one can show that maximal metrics on compact manifolds must be isotropy irreducible. We will give a proof in forthcoming paper.
Another important property of maximal metrics is that they have maximal
isometry groups in the sense that Isom(X, h, i) ⊂ Isom(X, h, i′ ) implies Isom(X, h, i) =Isom(X, h, i′ ) for all Riemannian metric h, i′ on X. For a maximal metric g ∈M(X), if the number of connected components of Isom(X, g) is ﬁnite then g has
a maximal isometry group. In particular,
Proposition 1.4. A homogeneous maximal metric g has a maximal isometrygroup.
For more details, see Subsection 2.4.
A similar notion of maximal metric has been introduced by Jablonski and
Gordon for left-invariant metrics, which is called maximal symmetry ([6]). The
relationship between maximal metrics and maximal symmetry metrics will be
discussed in Subsection 2.3 and Subsection 2.4, and is summarized in Figure 1
and Figure 2.
A goal of this paper is to construct various examples of maximal metrics on
Euclidean spaces which are not isotropy irreducible. Our strategy to constructexamples is to study a moduli space of left-invariant metrics on a Lie group G,
which is given as the orbit space of the action of R>0 Aut(g) on the set of all innerproducts m(g) on g = Lie(G). As a preparation, in Section 3, we study some
general theory of isolated orbits for an isometric action on an Hadamard space.
In Section 4, we show that
Theorem 1.5. Let G be a simply connected Lie group, and h, i be a left-invariant
metric on G. If the orbit R>0 Aut(g).h, i ⊂ m(g) is an isolated orbit, then the leftinvariant metric h, i is maximal. The converse holds if G is unimodular completely
solvable.
In Section 5, we construct examples of maximal metrics on Euclidean spaces
which are not isotropy irreducible. By applying Theorem 1.5, one has
Theorem 1.6. For w = (w2 , w3 , . . . , wn ) ∈ Rn−1 , deﬁne a metric gw on Rn with
the Cartesian coordinate system (x1 , x2 , . . . , xn ) by
gw := (dx1 )2 + e−2w2 x1 (dx2 )2 + · · · + e−2wn x1 (dxn )2 .
Then gw is a maximal metric for all w ∈ Rn−1 . If wi 6= wj for some i, j ∈{2, 3, . . . , n}, then gw is not isotropy irreducible.

4

YUICHIRO TAKETOMIThe Riemannian metric gw is isometric to a left-invariant metric on a certain
solvable Lie group. For more details, see Subsection 5.2. Note that the symmetricgroup Sn−1 acts on Rn−1 naturally. For w, w′ ∈ Rn−1 , gw and gw′ are isometricwith each other if and only if there exists some permutation σ ∈ Sn−1 such that
σ.w = w′ . Hence Theorem 1.6 gives continuously many examples of maximal
metrics.
The other examples are constructed by considering nilmanifolds attached withgraphs. By applying Theorem 1.5, we show that
Theorem 1.7. If given an edge-transitive graph G with p vertices and q edges,
one can construct maximal metrics on Rp+q . If q 6= 0 then the metrics are notisotropy irreducible.
A precise assertion of Theorem 1.7 will be given in Theorem 5.9. Note that,for graphs G and G ′ , the corresponding metrics are isometric with each other if
and only if G and G ′ are isomorphic as graphs. Hence, Theorem 1.7 also gives
inﬁnitely many nontrivial examples of maximal metrics. For more details, see
Subsection 5.3.
2. Maximal metrics
In this section, we give some general theory on maximal metrics. Recall that,in Section 1, we introduce the preordered set (M(X)/∼ , ≺). Here, M(X)/∼ is themoduli space of Riemannian metrics on X up to isometry and scaling, and ≺ isthe preorder with respect to the size of the isometry groups. A maximal metricg ∈ M(X) is the one whose equivalent class [g] ∈ M(X)/∼ is a maximal elementwith respect to ≺.
2.1. isotropy irreducible metrics and maximal metrics
Firstly, we explain that isotropy irreducible metrics are maximal metrics. Recall
that, for p ∈ X in a Riemannian manifold (X, g), the action of the stabilizer
Isom(X, g)p := {φ ∈ Isom(X, g) | φ(p) = p} on Tp X by diﬀerential is called theisotropy representation of (X, g) at p.
Deﬁnition 2.1. A Riemannian metric g ∈ M(X) is called an isotropy irreducible
metric if the isotropy representation at each point p ∈ X is an irreducible representation. A Riemannian manifold (X, g) with an isotropy irreducible metric g
is called an isotropy irreducible space.
One can see that a complete connected isotropy irreducible space is homogeneous. For examples, see [26]. Strongly isotropy irreducible spaces which are somespecial class of isotropy irreducible spaces have been classiﬁed independently by
Manturov ([17, 18, 19]), Wolf ([28, 29]) and Krämer ([12]). Conclusively, isotropy
irreducible spaces have been classiﬁed by Wang-Ziller ([26]).

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Proposition 2.2. A complete connected Riemannian manifold (X, g) is an isotropy
irreducible space if and only if (X, g) satisﬁes the following:(2.1){h ∈ M(X) | Isom(X, g) ⊂ Isom(X, h)} = R>0 g.

Proof. We prove “only if part”. Assume that (X, g) is isotropy irreducible. Recall
that an isotropy irreducible space (X, g) is homogeneous. Hence the assertion
follows from the Schur’s Lemma applied to the isotropy representation.
We prove “if part”. Firstly we show that if (X, g) satisﬁes the property (2.1),
then (X, g) must be homogeneous. If (X, g) is inhomogeneous, since Isom(X, g)action on the complete connected Riemannian manifold X is proper, there exists
a non-constant Isom(X, g)-invariant smooth function f on X. Then one has ef g
is a smooth Isom(X, g)-invariant Riemannian metric which is not contained in
R>0 g. This concludes that (X, g) does not satisﬁes the property (2.1).
Hence we have only to consider the case (X, g) is homogeneous. Take anyp ∈ X. Then the set of Isom(X, g)-invariant metrics on X is naturally identiﬁed
with the set of inner products which are invariant under the isotropy representation on a tangent space Tp X. Assume that (X, g) is not isotropy irreducible.
Then there exists a subspace V ⊊ Tp X which is invariant under the isotropy
representation. Denote by g1 and g2 the restriction of g to V and the normal
space V ⊥ , respectively. Then ag1 + bg2 is an Isom(X, g)-invariant metrics for alla, b > 0. This implies that, for examples, 2g1 + 3g2 is an Isom(X, g)-invariant
metrics which is not contained in R>0 g.
□
Note that a metric g ∈ M(X) is maximal if and only if{h | Isom(X, g) ⊂ Isom(X, h)} ⊂ [g],where [g] is the equivalent class with respect to ∼. Since R>0 g ⊂ [g], one has
Corollary 2.3. A complete isotropy irreducible metric is a maximal metric.
2.2. maximal metrics and self-similar solutions for metric evolution equations
Denote by S(X) the set of all symmetric (0, 2)-tensors on X. Let us consider amap R : M(X) → S(X). Then one can deﬁne a partial diﬀerential equation∂
gt = R(gt ).
∂t
For examples, the equation is called the Ricci ﬂow when the case R = −2Ric.
A solution {gt }t∈[0,T ) is called self-similar if [gt ] = [g0 ] for all t ∈ [0, T ). Also,if a metric g ∈ M(X) admits some self-similar solution {gt } with g = g0 , theng is called a soliton. For examples, a soliton for the Ricci ﬂow is usually calleda Ricci soliton. The study of self-similar solutions and solitons are important inorder to understand metric evolution equations.
By the property (1.2), one has

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Proposition 2.4. Let g ∈ M(X) be a maximal metric. If a solution {gt }t∈[0,T )∂for the equation ∂t
gt = R(gt ) with g0 = g preserves the isometry group in thesense that Isom(X, g0 ) ⊂ Isom(X, gt ) for all t ∈ [0, T ), then the solution {gt }t∈[0,T )is self-similar. That is, the initial metric g = g0 is soliton.
For examples, one knows that the solution {gt } ⊂ M(X) of the Ricci ﬂow(1.1) starting at g ∈ M(X) with bounded curvature exists ([23]), and preserves
the isometry group ([4]). This proves Proposition 1.2: a homogeneous maximal
metric is a Ricci soliton.
Let (X, g) be a homogeneous Riemannian manifold. Denote by G := Isom(X, g).
Denote by MG (X) and SG (X) the set of all G-invariant metrics and the set of
all G-invariant symmetric (0, 2)-tensors, respectively. Note that SG (X) is a ﬁnite dimensional vector space, and MG (X) is an open subset of SG (X). Hence,
MG (X) and SG (X) have a natural diﬀerentiable structure. Note that the groupof diﬀeomorphism Diﬀ(X) acts on S(X) and M(X) in the natural way. Assume that R : M(X) → S(X) is Diﬀ(X)-equivariant. Since G ⊂ Diﬀ(X), R
induces the map R : MG (X) → SG (X). The map R : MG (X) → SG (X) canbe regarded as a vector ﬁeld on MG (X). If the map R : MG (X) → SG (X) is
continuous, then there exists an integral curve {gt }t∈[0,T ) ⊂ MG (X) with initialmetric g. Then {gt }t∈[0,T ) ⊂ MG (X) is also a solution for the original diﬀerential∂
equation ∂t
gt = R(gt ). We call the solution {gt }t∈[0,T ) ⊂ MG (X) the homogeneous solution for the equation. Homogeneous solutions {gt }t∈[0,T ) preserve theisometry groups. Hence, by Proposition 2.4, one has
Proposition 2.5. Let g ∈ M(X) be a homogeneous maximal metric. Denoteby G := Isom(X, g). Consider a Diﬀ(X)-equivariant map R : M(X) → S(X).
Assume that the map R : MG (X) → SG (X) is continuous. Then a homogeneous∂
solution gt of the partial diﬀerential equation ∂t
gt = R(gt ) starting at g is selfsimilar. In other words, g is a soliton for the equation.
If R is deﬁned as the combination of some curvature tensors, then R is Diﬀ(X)equivariant, and the restriction map R : MG (X) → SG (X) is continuous. Hence
we can apply Proposition 2.5 for this case.
For a Riemannian metric g, denote by Rmg the Riemannian curvature tensor. Also, Ricg and scalg are the Ricci curvature and the scalar curvature of g,respectively. For a, b, c ∈ R, let us deﬁne the map R : M(X) → S(X) by
R(g) = −aRicg − bscalg · g − cRm2g(g ∈ M(X)).

Here, Rm2g is the (0, 2)-symmetric tensor given by
Rm2g (x, y) := −tr(Rmg (x, ⋆) ◦ Rmg (y, ⋆)),
where “⋆” means the metric contraction with respect to g. Then the equation∂g = R(gt ) is called∂t t• the Ricci ﬂow if a = 2, b = c = 0.

A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS

7• the Yamabe ﬂow if a = c = 0, b = 1.
• the Ricci-Bourguignon ﬂow if a = 2, b 6= 0, c = 0.
• the RG-2 ﬂow if a = 2, b = 0, c 6= 0.
In these cases, R is a combination of the Diﬀ(X)-equivariant curvature tensors.
Hence one has
Corollary 2.6. Let g ∈ M(X) be a homogeneous maximal metric. Then a∂homogeneous solution of the metric evolution equation ∂t
gt = R(gt ) with
R(g) = −aRicg − bscalg · g − cRm2g(g ∈ M(X))

starting at g is self-similar. In other words, a homogeneous maximal metric g isa soliton for this equation.
Also, one can apply Proposition 2.5 for the Bach ﬂow. For g ∈ M(X), deﬁneR(g) ∈ S(X) by
1
1
R(g)(x, y) =∇⋆ ∇• Wg (x, ⋆, y, •) +Ricg (⋆, •)Wg (x, ⋆, y, •),n−3n−2where Wg is the Weyl tensor with respect to g, and ⋆ and • mean the metriccontraction with respect to g. Then the (0, 2)-tensor R(g) is called the Bach∂tensor for g, and the equation ∂t
gt = −R(gt ) is called the Bach ﬂow.
The Bach tensor R is also deﬁned by the combination of the Diﬀ(X)-equivariant
curvature tensors. This yields that
Corollary 2.7. Let g ∈ M(X) be a homogeneous maximal metric. Then ahomogeneous solution of the Bach ﬂow starting at g is self-similar. In otherwords, a homogeneous maximal metric g is a soliton for the Bach ﬂow.
2.3. maximal metrics and maximal symmetry metrics
A left-invariant metric g on a Lie group G is called a maximal symmetry metric if
for all left-invariant metric h on G there exists φ ∈ Aut(G) such that Isom(G, h) ⊂Isom(G, φ.g), where φ.g := g(dφ−1 , dφ−1 ). Namely, a maximal symmetry metricg is the one whose isometry group is maximum among the set of left-invariant
metrics up to automorphism. The notion of maximal symmetry metric has been
introduced by Jablonski-Gordon ([6]).
Proposition 2.8 ([9]). A left-invariant Ricci soliton metric on a simply connected unimodular completely solvable Lie group is a maximal symmetry metric.
Hence, any maximal metric on a simply connected unimodular completely solvable Lie group has maximal symmetry. On the other hands, when G is not unimodular completely solvable, left-invariant maximal metrics on G are not necessarily maximally symmetric. For examples, let g := span{v1 , v2 , v3 } be a Lie
algebra whose nonzero bracket relation is given by[v1 , v2 ] = v2 .

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Note that g is a non-unimodular completely solvable Lie algebra. Let G be thesimply connected Lie group with Lie algebra g. Then Jablonski and Gordon haveshown that there are no maximal symmetry metrics on G ([6]). On the other
hand, the left-invariant metric with orthonormal frame {v1 , v2 , v3 } is isometric to
gw with w = (1, 0) in Theorem 1.6, and hence is a maximal metric.
2.4. maximal metrics have the maximal isometry groups
One of the interesting properties of maximal metrics g ∈ M(X) is the maximallityof the isometry groups in the sense that Isom(X, h, i) ⊂ Isom(X, h, i′ ) implies
Isom(X, h, i) = Isom(X, h, i′ ) for all Riemannian metric h, i′ on X. To see this,
we ﬁrstly give an easy lemma, which will be used repeatedly in this article.
Lemma 2.9. Let H be a Lie group with ﬁnitely many connected components, andassume that H and a Lie group H ′ are isomorphic with each other. If H ⊂ H ′then one has H = H ′ .
Note that the assumption of the ﬁniteness is essentially needed. For examples,2Z := {2z | z ∈ Z} ⊊ Z, and 2Z ∼= Z.
Proposition 2.10. Let h, i ∈ M(X) be a maximal metric whose isometry grouphas ﬁnitely many connected components. Then the isometry group Isom(X, h, i) is
maximal with respect to the inclusion “⊂”. In particular, a homogeneous maximal
metric has the maximal isometry group.
Proof. Take any Riemannian metric h, i′ with Isom(X, h, i) ⊂ Isom(X, h, i′ ). We
show that Isom(X, h, i) = Isom(X, h, i′ ). Since h, i is maximal, there exists λ > 0
and φ ∈ Diﬀ(X, h, i) such that φ is an isometry between (X, λh, i) and (X, h, i′ ).
Hence one hasIsom(X, h, i) = Isom(X, λh, i) = Isom(X, φ.h, i′ ) = φIsom(X, h, i′ )φ−1 .
Hence Isom(X, h, i) and Isom(X, h, i)′ are isomorphic with each other. By Lemma 2.9,
they coincide with each other.
□
Here, we mention the correspondence between maximal symmetry left-invariant
metrics and maximal isometry left-invariant metrics. Firstly, we show that
Proposition 2.11. Let h, i be a left-invariant metric on a connected Lie group G.
If h, i is a maximal symmetry metric, then h, i has the maximal isometry group.
Proof. Assume that h, i is maximally symmetric. Take any Riemannian metrich, i′ with Isom(G, h, i) ⊂ Isom(G, h, i′ ). Since h, i is left-invariant, one has h, i′
is also left-invariant. Then one has Isom(G, φ.h, i′ ) ⊂ Isom(G, h, i) for someφ ∈ Aut(G). This yields that
Isom(G, φ.h, i′ ) ⊂ Isom(G, h, i) ⊂ Isom(G, h, i′ ).
On the other hand, one has Isom(G, φ.h, i′ ) = Isom(G, h, i′ ) by Lemma 2.9. Thisconcludes that h, i has the maximal isometry group.
□A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS

9This concludes that the following diagram holds for arbitrary left-invariant
metrics on arbitrary Lie groups:
maximal⇒ maximal isometry⇓⇑
Ricci soliton
maximal symmetry
Figure 1. for general case

For some special cases, one can add several arrows to Figure 1. To see this,weshow the following:
Proposition 2.12. Assume that a connected Lie group G admits a maximal
symmetry left-invariant metric. Let h, i be a left-invariant metric on G. Thenh, i has a maximal isometry group if and only if h, i is a maximal symmetry leftinvariant metric.
Proof. We have shown “if part” in Proposition 2.11. We prove “only if part”.
Assume that h, i has the maximal isometry group. We show that h, i is a maximal
symmetry metric. Take any left-invariant metric g on G. We have only to showthat there exists some φ ∈ Aut(G) such that Isom(G, φ.g) ⊂ Isom(G, h, i). By
the assumption, there exists a maximal symmetry metric h, i′ on G. Then there
exists ϕ, ψ ∈ Aut(G) such that
Isom(ϕ.g) ⊂ Isom(G, h, i′ ),Isom(G, ψ.h, i) ⊂ Isom(G, h, i′ ).

This yields that Isom(G, h, i) ⊂ Isom(G, ψ −1 h, i′ ). Since h, i has the maximal
isometry group, one has Isom(G, ψ −1 .h, i′ ) = Isom(G, h, i). Now deﬁne φ ∈Aut(G) by φ := ψ −1 ϕ. One hasIsom(G, φ.g) = Isom(ψ −1 ϕ.g) ⊂ Isom(G, ψ −1 h, i′ ) = Isom(G, h, i),which completes the proof.

□

For examples, Jablonski and Gordon has shown that any unimodular completely solvable Lie groups always admit maximally symmetric left-invariant metrics. Hence, for the case of unimodular completely solvable Lie groups, one has
maximal⇒ maximal isometry⇓m
Ricci soliton ⇒ maximal symmetry
Figure 2. for unimodular completely solvable case

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3. Some general theory on isolated orbits
In this section, we construct some general theory on isolated orbits of proper
actions. Our goal is to show Proposition 3.4, which plays an important role toconstruct examples of maximal metrics.
Firstly, we introduce some notions of orbits of set-theoretical group actions.
Let G be a group, and assume that G acts on a set X. Denote by G\X theorbit space of the G-action. For orbits G.p, G.q ∈ G\X, denote by G.p ∼ G.q(resp. Gp < Gq ) if there exists q ′ ∈ G.q such that Gp = Gq (resp. G.p ⊂ G.q).
Then ∼ is an equivalent relation on G\X, and < is a preorder on G\X. An
orbit G.p is said to be of maximal orbit type if G.p is maximal with respect to<, namely, if G.p < G.q then G.p = G.q. Denote by [G.p] the equivalent class of
G.p with respect to ∼. An orbit G.p is called a solitary orbit if [G.p] = {G.p}.
By the deﬁnition, if G.p has a maximal orbit type then G.p is a solitary orbit.
The preorder < on G\X induces a preorder on the double coset space
G\X/∼ := {[G.p] | G.p ∈ G\X},which is also denoted by <. Namely, we denote by [G.p] < [G.q] if G.p < G.q ′for some q ′ ∈ G.q.
Now we consider group actions with topology. Let G be a topological group,
and assume that G acts on a topological space X continuously. We consider thenatural quotient topology on G\X. An orbit G.p is called an isolated orbit if
G.p ∈ G\X is an isolated point of the subset [G.p] ⊂ G\X, that is, there exists
some open subset U ⊂ G\X such that U ∩ [G.p] = {G.p}. In other words, anisolated orbit is a locally solitary orbit. Now one has(3.1)

maximal orbit type ⇒ solitary ⇒ isolated.

In general, these notions are not equivalent. However, these notions are equivalent
for some special case (cf. Proposition 3.4).
We review some fundamental facts on proper actions. A continuous action of
a topological group G on a topological space X is called proper if the map
G × X → X × X, (g, x) 7→ (g.x, x)
is a proper map (i.e. inverse images of compact subsets are also compact). By thedeﬁnition, one can see that isotropy subgroups of a proper action on a manifold
are compact, and hence have ﬁnitely many connected components. By using
Lemma 2.9, one can easily see that
Lemma 3.1. Let X be a manifold, and assume that a Lie group G acts on X
properly. Then the preordered set (G\X/∼ , <) is a partially ordered set. Namely,[G.p] < [G.q] and [G.q] < [G.p] imply [G.p] = [G.q].

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Now let (X, h, i) be a complete connected Riemannian manifold, and G ⊂Isom(X, h, i) be a closed subgroup. Then it is known that G acts on X isometrically and properly. Also, by the existence of slices for proper actions on manifolds(e.g. [2]), one has
Proposition 3.2. Let (X, h, i) be a complete connected Riemannian manifold,
and G ⊂ Isom(X, h, i) be a closed subgroup. For each orbit G.p ∈ G\X, there
exists an open neighborhood V ⊂ G\X of G.p such that G.x < G.p for all G.x ∈V.
For G.p ∈ G\X, denote by U(G.p) the set of all upper bounds of G.p, namely
U(G.p) := {G.q ∈ G\X | G.p < G.q}.
Then one has the following:
Lemma 3.3. Let (X, h, i) be a Riemannian manifold whose exponential map
expp : Tp X → X is a diﬀeomorphism for all p ∈ X. Let G ⊂ Isom(X, h, i) be aclosed subgroup. Then U(G.p) is a connected subset of G\X for all G.p ∈ G\X.
Proof. We prove that U(G.p) is path-connected. Take any G.q ∈ G\X with
G.p < G.q. We construct a path from G.p to G.q. That is, we show that there
exists a continuous curve γ : [0, T ] → G\X such that
(3.2)

γ(0) = G.p,γ(T ) = G.q,

G.p = γ(0) < γ(t) (t ∈ [0, T ]).

By the deﬁnition of <, there exists q ′ ∈ G.q such that Gp ⊂ Gq′ . Take any
geodesic c : [0, T ] → X with c(0) = p, c(T ) = q ′ . Since the exponential map
expp : Tp X → X is a diﬀeomorphism, such a geodesic c is unique. Note that
Gc(0) = Gp ⊂ Gq′ = Gc(T ) . That is, the action of Gp on X ﬁxes the starting
point c(0) and the end point c(T ) of c. We also note that the Gp -action sends
each geodesic to geodesic. Hence, by the uniqueness of the geodesic c, the Gp action must ﬁx the geodesic c pointwisely. Therefore, one has Gp ⊂ Gc(t) , namely
G.p < G.c(t) for all t ∈ [0, T ]. Let π : X → G\X be the projection, and deﬁne
γ := π ◦ c. Then one can see that γ satisﬁes (3.2).
□
By the above preparation, we prove the main subject of this section as follows:
Proposition 3.4. Let (X, h, i) be a Riemannian manifold whose exponential map
expp : Tp X → X is a diﬀeomorphism for all p ∈ X. Let G ⊂ Isom(X, h, i) be aclosed subgroup. If an orbit G.p is isolated, then G.p has a maximal orbit type.
In particular, three notions in (3.1) are equivalent with each other.
Proof. Assume that an orbit G.p is isolated. We have to show G.p has maximal
orbit type, namely, U(G.p) = {G.p}. By Lemma 3.3, one knows U(G.p) is connected. Therefore, in order to show U(G.p) = {G.p}, we have only to show that{G.p} is a clopen subset of U(G.p).

12

YUICHIRO TAKETOMIWe show that {G.p} is a closed subset of U(G.p). The orbit space G\X isHausdorﬀ since G-action is proper. Therefore, {G.p} ⊂ G\X is closed, andhence is closed in U(G.p).
We now verify that {G.p} is an open subset of U(G.p). To show this, weconstruct an open subset W of G\X such that W ∩ U(G.p) = {G.p}.
We ﬁrstly construct W . Since G.p is isolated, there exists an open subset U
such that U ∩ [G.p] = {G.p}. Also, by Proposition 3.2, there exists an open
neighborhood V of G.p such that G.x < G.p for all G.x ∈ V . Denote by W :=U ∩V.
We show that W ∩ U(G.p) = {G.p}. W ∩ U(G.p) ⊃ {G.p} is obvious. We
have only to verify that W ∩ U (G.p) ⊂ U ∩ [G.p] since U ∩ [G.p] = {G.p}. Take
any G.q ∈ W ∩ U(G.p). Since W ⊂ U , one has G.q ∈ U . We lastly show thatG.q ∈ [G.p]. Since G.q ∈ V , one has G.q < G.p. This yields that [G.q] < [G.q].
On the other hand, since G.q ∈ U(G.p), one has G.p < G.q. Hence one has[G.p] < [G.q]. Therefore, by Lemma 3.1, we have [G.p] = [G.q], and hence
G.q ∈ [G.p].
□
4. Some sufficient conditions for maximal left-invariant metrics
In this section, we prove Theorem 1.5 which gives a simple suﬃcient condition
for left-invariant metrics on a simply connected Lie group to be maximal.
4.1. An introduction to R>0 Aut(g)-actionsFirstly, we give a review on the study of moduli spaces of left-invariant metrics.
Let G be a simply connected Lie group with the Lie algebra g. Note that a
left-invariant metric on G is naturally identiﬁed with the inner product on g. Letm(g) be the set of all inner products on g. Then the general linear group GL(g)acts transitively on m(g) by
g.h·, ·i := hg −1 ·, g −1 ·i (g ∈ GL(g), h, i ∈ M(g)).
Denote by R>0 Aut(g) := {cφ ∈ GL(g) | c > 0, φ ∈ Aut(g)}. Note that the
subgroup R>0 Aut(g) is closed in GL(g), and acts on m(g).
Since G is simply connected, the diﬀerential map Aut(G) → Aut(g) is bijective.
This yields the following:
Lemma 4.1. Let G be a simply connected Lie group with the Lie algebra g. If
two inner products h, i, h, i′ ∈ m(g) belong to the same R>0 Aut(g)-orbit then two
left-invariant metrics h, i, h, i′ ∈ M(G) are isometric up to scaling.
Note that the converse is not true in general, but is true if g is unimodular
completely solvable case for instance. The orbit space R>0 Aut(g)\M(g) helps us
to classify left-invariant metrics on G, and some geometers have been studying
the R>0 Aut(g)-action (e.g. [11, 24]).

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Denote by O(h, i) ⊂ GL(g) the orthogonal group with respect to h, i ∈ m(g).
Now we investigate the stabilizer R>0 Aut(g) ∩ O(h, i). One can easily see thatAut(g) ∩ O(h, i) ⊂ R>0 Aut(g) ∩ O(h, i). In fact, the equality holds:
Lemma 4.2. Let g be a Lie algebra, and h, i be an inner product on g. Then one
has R>0 Aut(g) ∩ O(h, i) = Aut(g) ∩ O(h, i). In other words, the isotropy subgroupof the R>0 Aut(g)-action at h, i ∈ m(g) coincides with Aut(g) ∩ O(h, i).
Proof. We have to show R>0 Aut(g) ∩ O(h, i) ⊂ Aut(g) ∩ O(h, i) only. To provethis, let us prepare some notions.
We ﬁrstly deﬁne the action of GL(V ) on the set of Lie brackets on V byg.[, ] := g[g −1 , g −1 ] (g ∈ GL(V ), [, ] : Lie bracket).
Secondly, for a metric Lie algebra (V, [, ], h, i), denote by k[, ]k⟨,⟩ the norm of
the Lie bracket [, ], namely,X(4.1)
k[, ]k⟨,⟩ := ( h[vi , vj ], [vi , vj ]i)1/2 ,i,jwhere {v1 , . . . , vn } ⊂ V is an orthonormal basis with respect to h, i.
We are now in the position to show the assertion. For a Lie algebra g = (V, [, ])with an inner product h, i, one can see thatAut(g) = {g ∈ GL(V ) | g.[, ] = [, ]},(4.2)
R>0 Aut(g) = {g ∈ GL(V ) | k[, ]k⟨,⟩ · g.[, ] = kg.[, ]k⟨,⟩ · [, ]}.
On the other hand, one can see that kg.[, ]k⟨,⟩ = k[, ]k⟨,⟩ for all g ∈ O(h, i). These
conclude that R>0 Aut(g) ∩ O(h, i) ⊂ Aut(g) ∩ O(h, i).
□
Denote by sym(g) the vector space of all symmetric bilinear forms on g. Thenm(g) is an open subset of sym(g), and hence each tangent space T⟨,⟩ m(g) at
h, i ∈ m(g) is naturally identiﬁed with sym(g). A natural inner product (, )⟨,⟩ onT⟨,⟩ m(g) ∼= sym(g) is deﬁned as follows:Xθ(vi , vj )η(vi , vj ) (θ, η ∈ sym(g)),(θ, η)⟨,⟩ :=i

where {v1 , . . . , vn } is an orthonormal basis with respect to h, i. Then (m(g), (, )) isa GL(g)-homogeneous Riemannian manifold. Note that m(g) ∼= R>0 ×SL(n)/SO(n)by the de Rham decomposition, and hence m(g) is an Hadamard manifold.
Lemma 4.3. R>0 Aut(g) is a closed subgroup of the isometry group of (m(g), (, )).
In particular, the R>0 Aut(g)-action is a proper isometric action on the Hadamard
manifold m(g).
Proof. The de Rham decomposition of m(g) ∼= R>0 × SL(n)/SO(n) implies thatthe identity component of Isom(m(g), (, )) is GL+ (g) ∼= R>0 × SL(g). This yields
that GL(g) is a closed subgroup of Isom(m(g), (, )). On the other hand, by theexpression (4.2), R>0 Aut(g) is a closed subgroup of GL(g). Hence R>0 Aut(g) is
closed in Isom(m(g), (, )).
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YUICHIRO TAKETOMI

4.2. some sufficient conditions for maximal metrics
In this subsection, we show the ﬁrst assertion of Theorem 1.5 by proving Theorem 4.8, and show the second assertion by proving Theorem 4.9. Firstly, we givea following suﬃcient condition for left-invariant metrics on a simply connected
Lie group to be maximal.
Proposition 4.4. Let G be a simply connected Lie group, and h, i be a leftinvariant metric on G. Denote by g the Lie algebra of G. If the orbit R>0 Aut(g).h, i
has the maximal orbit type (see Section 3), then the equivalent class [h, i] ∈M(G)/∼ is a maximal element. In other words, h, i is a maximal metric.
Proof. Assume that R>0 Aut(g).h, i has the maximal orbit type. Take any Isom(G, h, i)invariant metric h, i′ ∈ M(G). Since Isom(G, h, i) ⊂ Isom(G, h, i′ ), the metric h, i′
is also a left-invariant metric on G. To show that h, i and h, i′ are isometric up toscaling, we have only to show that R>0 Aut(g).h, i = R>0 Aut(g).h, i′ . One hasIsom(G, h, i) ∩ Aut(G) ⊂ Isom(G, h, i′ ) ∩ Aut(G).
Also, since G is simply connected, one has Isom(G, g) ∩ Aut(G) ∼= O(g) ∩ Aut(g)for any left-invariant metric g via the diﬀerential at the unit element e ∈ G. Here,O(g) ⊂ GL(g) is the orthogonal group with respect to the inner product g. This
yields that
O(h, i) ∩ Aut(g) ⊂ O(h, i′ ) ∩ Aut(g).
Note that O(g) ∩ Aut(g) is the stabilizer of the R>0 Aut(g)-action at g ∈ m(g).
Hence one has R>0 Aut(g).h, i < R>0 Aut(g).h, i′ . By the maximallity, one has
R>0 Aut(g).h, i = R>0 Aut(g).h, i′ .
□If the orbit space R>0 Aut(g)\m(g) is small, then one can reasonably check the
maximality condition. For examples, Proposition 4.4 shows that if the R>0 Aut(g)action is transitive (i.e. the orbit space is a point), then any left-invariant metricon G is maximal. Note that Lie algebras g whose R>0 Aut(g)-actions are transitive
have been classiﬁed by Lauret as follows:
Theorem 4.5 ([13]). Let g be an n-dimensional Lie algebra. If the R>0 Aut(g)action is transitive, then g is isomorphic to one of the following three Lie algebras:(1) abelian Lie algebra Rn ,(2) almost abelian Lie algebra R ⋉φ Rn−1 with φ(t) := t · idRn−1 . In other words,
the Borel subalgebra of so(n − 1, 1).
(3) the product algebra Rn−3 ⊕ h3 of the abelian Lie algebra Rn−3 and the Heisenberg Lie algebra h3 .
Note that (1) and (2) in Theorem 4.5 are isotropy irreducible. On the other
hand, the case (3) is not isotropy irreducible. This gives ﬁrst nontrivial examplesof maximal metrics:

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Corollary 4.6. Let G := Rn−3 × H3 be a product Lie group of the abelian Liegroup Rn−3 and the Heisenberg Lie group H3 . Then any left-invariant metric on
G is a maximal metric which is not isotropy irreducible.
Remark 4.7. Note that a simply connected nilpotent Lie group G with leftinvariant metric h, i is isotropy irreducible if and only if G is abelian. By the
result of Wilson ([27]), the isotropy representation of (G, h, i) at the unit element
e is the action of Aut(g) ∩ O(h, i) on g, where g = Te G is the Lie algebra of G. If
g is non-abelian and nilpotent, there is a nontrivial center z(g) ⊂ g, which is aninvariant subspace of the isotropy representation.
Recall that R>0 Aut(g)-actions are proper isometric actions on the Hadamard
manifolds m(g). By applying Proposition 3.4 and Proposition 4.4, one has theﬁrst assertion of Theorem 1.5:
Theorem 4.8. Let G be a simply connected Lie group, and h, i be a left-invariant
metric on G. Denote by g the Lie algebra of G. If the orbit R>0 Aut(g).h, i is an
isolated orbit, then h, i is a maximal metric.
It has been known that singular orbits of cohomogeneity one actions are isolated
orbits. It has been shown that the cohomogeneity of R>0 Aut(g)-actions for 3dimensional solvable g are at most one, and some of them are cohomogeneity one
actions with singular orbits. For examples, the Lie algebra of the motion groupof R2 which is given by g := (span{v1 , v2 , v3 }, [, ]),[vi , vj ] = 0 (i, j ∈ {1, 2}),

[v1 , v3 ] = −v2 ,[v2 , v3 ] = v1

gives an example of R>0 Aut(g)-action with a singular orbit. For more details,see [8].
Note that an R>0 Aut(g)-orbit being an isolated orbit is not a necessary condition for left-invariant to be a maximal metric (see Remark 4.10). However, if
g is a unimodular completely solvable Lie algebra, then the isolation of the orbit
is also a necessary condition:
Theorem 4.9. Let G be a simply connected unimodular completely solvable Liegroup, and h, i be a left-invariant metric on G. Denote by g the Lie algebra of G.
Then h, i is a maximal metric if and only if the orbit R>0 Aut(g).h, i is an isolated
orbit.
Proof. Assume that h, i is maximal. By the diagram (3.1), we have only toshow that R>0 Aut(g).h, i has maximal orbit type. Namely O(h, i) ∩ Aut(g) ⊂O(h, i′ ) ∩ Aut(g) implies R>0 Aut(g).h, i = R>0 Aut(g).h, i′ .
Take any left-invariant metric h, i′ with O(h, i) ∩ Aut(g) ⊂ O(h, i′ ) ∩ Aut(g).
When G is simply connected unimodular completely solvable, it has been known
that
Isom(G, g) ∼= (O(g) ∩ Aut(g)) ⋉ LG

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for any left-invariant metric g on G, where LG is the group of left-translations of G([7]). This yields that O(h, i) ∩ Aut(g) ⊂ O(h, i′ ) ∩ Aut(g) implies Isom(G, h, i) ⊂Isom(G, h, i′ ). Since h, i is maximal, there exists λ > 0 such that (G, λh, i)and (G, h, i′ ) are isometric. It has been known that if two completely solvable Riemannian Lie groups (H, g) and (H ′ , g ′ ) are isometric then there exists
a group isomorphism φ : H → H ′ such that φ is also a Riemannian isometry ([1]). This yields that there exists an automorphism ψ : G → G suchthat ψ is an isometry between (G, λh, i) and (G, h, i′ ). This concludes that
R>0 Aut(g).h, i = R>0 Aut(g).h, i′ .
□Remark 4.10. The converse of Theorem 4.8 does not hold in general. For examples, let us consider the Lie algebra s := span{x1 , . . . , xn , y1 , . . . yn , z, v} by[xi , yi ] = z, [xi , v] = (1/2)xi , [yi , v] = (1/2)yi , [z, v] = z.
Denote by h, i an inner product on s with an orthonormal basis {x1 , . . . , yn , z, v}.
Then the simply connected Riemannian Lie group with respect to the metric Liealgebra (s, h, i) is isometric to complex hyperbolic space CH n up to scaling. Since
CH n is isotropy irreducible, then the left-invariant metric h, i is maximal. On
the other hand, one can see that the orbit R>0 Aut(s).h, i is not isolated.
5. examples
In this section, by applying Theorem 1.5, we prove Theorem 1.6 and Theorem 1.7 that give examples of left-invariant maximal metrics on some solvable
Lie groups.
5.1. general settings
According to Theorem 1.5, in order to construct examples of left-invariant maximal metrics, we have only to ﬁnd a metric Lie algebra (g, h, i) so that the orbit
R>0 Aut(g).h, i is an isolated orbit in M(g). The following characterization ofisolated orbits was obtained by the author:
Proposition 5.1 ([25]). Let X be a complete connected Riemannian manifold,
and G ⊂ Isom(X) be a closed subgroup. Then an orbit G.p is an isolated orbit if
and only if the slice representation at p has no nonzero ﬁxed normal vectors.
Here, the slice representation of an orbit G.p ⊂ X at a point p ∈ X is a linear
representation of the stabilizer Gp on the normal space (Tp G.p)⊥ by diﬀerential.
Therefore, we construct metric Lie algebras (g, h, i) such that the slice representation of the R>0 Aut(g)-action at h, i has no nonzero ﬁxed normal vectors. Recall
that the tangent space at h, i ∈ M(g) is naturally identiﬁed with the space of symmetric bilinear forms sym(g). By ﬁxing an orthonormal basis {v1 , . . . , vn } ⊂ g,
one can see that the normal space (T⟨,⟩ R>0 Aut(g).h, i)⊥ is given byX(5.1){θ ∈ sym(g) | ∀A ∈ R ⊕ Der(g),θ(vi , Avi ) = 0},i

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17

where R ⊕ Der(g) := {c · idg + D | c ∈ R, D ∈ Der(g)} is the Lie algebraof R>0 Aut(g). The slice representation at h, i is given by g.θ := θ(g −1 , g −1 ),where g ∈ Aut(g) ∩ O(h, i), and θ ∈ (T⟨,⟩ R>0 Aut(g).h, i)⊥ . In conclusion, we havethe following linear algebraic suﬃcient condition for left-invariant metrics to be
maximal:Proposition 5.2. Let g be an n-dimensional Lie algebra, B = {v1 , v2 , . . . , vn } bea basis of g, and G be the simply connected Lie group with the Lie algebra g. If
g and B satisfy(i) ∀g ∈ Aut(g) ∩ O(n),Pθ(g·, g·) = θ(·, ·),(5.2)
θ ∈ sym(g)= {0},(ii) ∀A ∈ R ⊕ Der(g),i θ(vi , Avi ) = 0
then the left-invariant metric h, i on G with orthonormal frame B is maximal.
To ﬁnd a Lie algebra with a basis which satisﬁes (5.2), let us introduce thefollowing notion:
Deﬁnition 5.3. Let g be an n-dimensional Lie algebra. A basis B ⊂ g is calleda 2-reversible basis if for any v, w ∈ B with v 6= w, there exists g ∈ Aut(g) ∩ O(n)and a ∈ {−1, 1} such that gv = av and gw = −aw.
For a 2-reversible basis B on g, and an Aut(g) ∩ O(n)-invariant bilinear form
θ : g × g → R, one hasθ(v, w) = θ(gv, gw) = θ(av, −aw) = −θ(v, w) (v, w ∈ B, v 6= w).
This yields that
Lemma 5.4. For a 2-reversible basis B ⊂ g, any Aut(g)∩O(n)-invariant bilinear
form θ : g × g → R is diagonal with respect to B.
For examples, the Ricci tensor Ric : g × g → R is Aut(g) ∩ O(n)-invariant
bilinear form. Hence a 2-reversible basis B is a special case of a Ricci diagonalbasis.
5.2. almost abelian Lie groups with diagonal extension
In this subsection, we prove Theorem 1.6. For (w2 , w3 , . . . , wn ) ∈ Rn−1 , we consider a Lie group structure ∗ on Rn by(x1 , x2 , . . . , xn ) ∗ (y1 , y2 , . . . , yn ) := (x1 + y1 , x2 + ew2 x1 y2 , . . . , xn + ewn x1 yn ).
Namely, (Rn , ∗) is a semi-direct product R ⋉ Rn−1 whose action of R on Rn−1 isgiven by
R → GL(n − 1, R),
t 7→ diag(ew2 t , ew3 t , . . . , ewn t ).
Lie groups with codimension one abelian normal subgroups such as (Rn , ∗) above
are called almost abelian Lie groups.
Deﬁne vector ﬁelds v1 , v2 , . . . , vn on Rn := {(x1 , x2 , . . . , xn ) | xi ∈ R} by
∂
∂
∂
(5.3)
v1 :=
, v2 := ew2 x1, . . . , vn := ewn x1
.
∂x1
∂x2
∂xn

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One can see that the vector ﬁelds v1 , v2 , . . . , vn are left-invariant with respect
to the Lie group structure ∗. Note that Bw := {v1 , v2 , . . . , vn } is a basis of
the Lie algebra of (Rn , ∗). The non-zero bracket relations of the Lie algebra
sw := span{v1 , v2 , . . . , vn } are given by(5.4)[v1 , v2 ] = w2 v2 ,[v1 , v3 ] = w3 v3 ,...,[v1 , vn ] = wn vn .

For w ∈ Rn−1 , denote by gw ∈ M(Rn ) the left-invariant metric on (Rn , ∗) withthe orthonormal frame Bw . The metric gw is explicitly given by(5.5)

gw := (dx1 )2 + e−2w2 x1 (dx2 )2 + · · · + e−2wn x1 (dxn )2 ,which is appeared in Theorem 1.6. By the result of Lauret in [14], one can easily
see that gw is a left-invariant Ricci soliton on the simply connected solvable Liegroup (Rn , ∗). Now we are in the position to prove Theorem 1.6
Proof of Theorem 1.6. Take any w := (w2 , w3 , . . . , wn ). We ﬁrstly show that thebasis Bw ⊂ sw is a 2-reversible basis. Take any vi , vj ∈ Bw with i < j. Letφ ∈ GL(n, R) be a diagonal matrix whose (j, j)-element is −1 and the others
are 1. Note that j > 1. By the bracket relations (5.4), one can see that φ ∈Aut(sw ) ∩ O(n), and one has φvi = vi and φvj = −vj .
Now we have only to show that the Lie algebra sw with the basis Bw satisﬁes thecondition (5.2) in Proposition 5.2. Take any θ ∈ sym(sw ) which satisﬁes (i) and(ii) in (5.2). By Lemma 5.4, θ is diagonal with respect to Bw . Hence we have only
to show that θ(vi , vi ) = 0 for each i ∈ {1, 2, . . . , n}. Take any i ∈ {1, 2, . . . , n}.
By the bracket relations (5.4), one can see that
E22 , E33 , . . . , Enn ∈ Der(sw ),where Epq ∈ gl(sw ) is a matrix whose (p, q)-entry is 1, and the others are 0.
Note that E11 = idsw − E22 − E33 − · · · − Enn ∈ R ⊕ Der(sw ). This yields thatEii ∈ R ⊕ Der(sw ). By the condition (ii), one hasX0=θ(vj , Eii vj ) = θ(vi , vi ).
jThis concludes that θ = 0.
We show the last assertion. Denote by αw := w2 + w3 + · · · + wn . Then one
can see that the Ricci tensor of gw is given by(5.6)

Ric(gw ) = diag(−|w|2 , −w2 αw , −w3 αw , . . . , −wn αw ).

This concludes that if wi 6= wj for some i, j then gw is not isotropy irreducible
since gw is not Einstein.
□
Remark 5.5. The equation (5.6) yields that, if n ≥ 3, gw gives continuous familyof maximal metrics such that gw ≁ gw′ for generic (w, w′ ) ∈ Rn−1 × Rn−1 .

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5.3. nilpotent Lie algebras attached with edge transitive graphs
In this subsection, we prove Theorem 1.7. Firstly, let us give a review on the
2-step nilpotent Lie algebras attached to graphs, introduced in [5]. Let V be aﬁnite set, and denote by PV2 := {e ⊂ V | ♯e = 2}. Then, for a subset E ⊂ PV2 , thepair G := (V, E) is called a (ﬁnite) simple graph with vertices V and edges E. Fora simple graph G = (V, E), a map d : E → V is called a direction of G if d(e) ∈ e
for all e ∈ E. Namely, a direction d determines a starting point for each edge.
Also, denote by d∗ : E → V the opposite direction of d (i.e. {d(e), d∗ (e)} = e foreach e ∈ E). A simple graph G with a direction Gd := (G, d) is called a directed
simple graph.
Deﬁnition 5.6. For a directed simple graph Gd = (G, d) with vertices V and edges
E, one can deﬁne a 2-step nilpotent Lie algebra structure [, ] on spanR (V ∪ E) as
follows:[d(e), d∗ (e)] := e, [e, ·] := 0 (e ∈ E),[v, w] := 0 ({v, w} ∈/ E).

We call the Lie algebra nGd := (spanR (V ∪ E), [, ]) the 2-step nilpotent Lie algebra
attached with G.
Note that nGd is abelian if and only if E = ∅. In the latter argument, we always
assume that E 6= ∅.
Proposition 5.7. For any directed graph Gd = (V, E, d), the basis V ∪ E ⊂ nGd
is a 2-reversible basis.
Proof. For v ∈ V, deﬁne rv ∈ Aut(nGd ) ∩ O(p + q) byrv (v) := −v,

′

′

′

rv (v ) := v (v ∈ V \ {v}),rv (e) :=−e (v ∈ e),
e
(v ∈
/ e).

We show that {rv | v ∈ V} reﬂects the basis V ∪ E.
Firstly, for v, v ′ ∈ V with v 6= v ′ , one has rv (v) = −v and rv (v ′ ) = v ′ .
Secondly, for v ∈ V, and e ∈ E, put v ′ ∈ e \ {v}. Then one has rv′ (v) = v andrv′ (e) = −e.
Lastly, for any e, e′ ∈ E with e 6= e′ , there exists a vertex v ∈ e \ e′ . Then one
has rv (e) = −e and rv (e′ ) = e′ .
□The following lemma asserts that invariant normal vectors θ ∈ (T⟨,⟩G R>0 Aut(nGd ).h, iG )⊥are completely determined by the diagonal parts θ(e, e) along the edges e ∈ E.
Lemma 5.8. Let Gd = (G, d) be a directed simple graph. Take any θ ∈ sym(nGd )which satisﬁes the condition (ii) in (5.2). Then one hasP(1) θ(v,
P v) = − e∋v θ(e, e) for all v ∈ V,(2) e∈E θ(e, e) = 0,

20

YUICHIRO TAKETOMIProof. We ﬁrstly prove (1). Take any v ∈ V. Deﬁne Dv ∈ Der(nGd ) by
(
(
′
v
(v
=
v)
e (v ∈ e)
Dv (v ′ ) :=(v ′ ∈ V), Dv (e) :=(e ∈ E).
′0 (v 6= v)
0 (v ∈/ e)P
Since θ is a normal vector at h, iG , one has x∈V∪E θ(x, Dv x) = 0 (see (5.2)).
This yields that
X
X
X
X
0=
θ(x, Dv x) =θ(v ′ , Dv v ′ ) +θ(e, Dv e) = θ(v, v) +θ(e, e).
v ′ ∈V

x∈V∪E

P

e∋v

e∈E

This concludes that θ(v, v) = − e∋v θ(e, e).
We secondly prove the assertion (2). Deﬁne D ∈ Der(nGd ) by
D(v) = v,

D(e) = 2e(v ∈ V, e ∈ E).

Then one has D − I ∈ R ⊕ Der(nGd ). Note that(D − I)(v) = 0,This yields that
0=

X
x∈V∪E

which completes the proof.

(D − I)(e) = eθ(x, (D − I)x) =(v ∈ V, e ∈ E).
X

θ(e, e),e∈E□

For two ﬁnite simple graphs G = (V, E) and G ′ = (V ′ , E ′ ), the bijective map
σ : V ∪ E → V ′ ∪ E ′ is called an isomorphism between G and G ′ if σ satisﬁes
σ(V) = V ′ and σ({v, w}) = {σ(v), σ(w)} for each {v, w} ∈ E. Denote by Aut(G)the group of self-isomorphisms of the graph G.
For each σ ∈ Aut(G), one can deﬁne σ̃ ∈ Aut(nGd ) as follows:σ(e)(if σ(d(e)) = d(σ(e))),(5.7)
σ̃(v) := σ(v), σ̃(e) :=−σ(e) (if σ(d(e)) 6= d(σ(e))),where v ∈ V, and e ∈ E. Then one has σ̃ ∈ Aut(nGd ) ∩ O(h, iG ). Namely, theautomorphism group Aut(G) of the “undirected” graph G acts on the normal
space (T⟨,⟩G R>0 Aut(nGd ).h, iG )⊥ by the slice representation.
A simple graph G is called an edge-transitive graph if the automorphism group
Aut(G) acts on E transitively. For more informations about edge-transitivegraphs, for examples, see [21]. It has been known that the left-invariant metrich, iG is a Ricci soliton if and only if the graph G is positive ([15]). One can easily
see that edge-transitive graphs are positive, and hence h, iG for edge-transitive G
are Ricci soliton.
For a directed simple graph Gd , denote by NGd the simply connected nilpotent
Lie group with Lie algebra nGd . Note that, if G = (V, E) has p vertices and q
edges, then NGd is diﬀeomorphic to Rp+q via the basis V ∪ E. Now we are in
the position to prove Theorem 1.7. We show the following precise version of the
Theorem 1.7.

A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS

21

Theorem 5.9. Let G = (V, E) be an edge-transitive graph. Then for any directiond : E → V, the left-invariant metric h, i ∈ M(NGd ) with the orthonormal frame
B := V ∪E is maximal. The metric h, i is isotropy irreducible if and only if E = ∅.
Proof. Take any θ ∈ sym(nGd ) which satisﬁes∀g ∈ Aut(g) ∩ O(n), θ(g·, g·) = θ(·, ·), ∀A ∈ R ⊕ Der(g),Xθ(vi , Avi ) = 0.

i

By Proposition 5.2, we have only to show that θ = 0. By Lemma 5.4, θ is diagonalwith respect to B. Also, by Lemma 5.8, the diagonal parts of θ are completely
controlled by the edge parts θ(e, e). Hence we have only to show that θ(e, e) = 0
for all e ∈ E.
Assume that G is edge-transitive. For any e, e′ ∈ E, there exists σ ∈ Aut(G)such that σ(e) = e′ , and henceθ(e′ , e′ ) = θ(σ(e), σ(e)) = θ(±σ̃(e), ±σ̃(e)) = θ(e, e).
This concludes that the function e 7→ θ(e, e) is constant. On the other hand,Pe∈E θ(e, e) = 0 by Lemma 5.8. Therefore, for each e ∈ E,X
θ(e, e) = ♯E · θ(e, e),0=e∈E

and hence θ(e, e) = 0.
We have shown the last assertion. See Remark 4.7.

□

For a directed graph Gd with G = (V, E), denote by h, iGd the left-invariant
metric on NGd with the orthonormal frame V ∪E. The above arguments yield that
if given an edge-transitive graph G with p vertices and q edges, and a direction d,
one can obtain maximal metrics h, iGd on Rp+q . On the other hand, the following
states that directions d do not aﬀect on the resulting left-invariant metric h, iGd .
Theorem 5.10 ([16, 27]). Let Gd and Gd′ ′ be two directed simple graphs. Thenthe corresponding two simply connected Riemannian Lie groups (NGd , h, iGd ) and(NGd′ ′ , h, iGd′ ′ ) are isometric if and only if G and G ′ are isomorphic as undirected
graphs.
Denote by Gp,q := {[G] | G = (V, E) : simple graph, ♯V = p, ♯E = q}, where[G] means the isomorphism class of G. By Theorem 5.10, the map
Gp,q → M(Rp+q )/∼ , [G] 7→ [h, iGd ]is well-deﬁned and injective. In conclusion, we obtain the following corollary for
Theorem 5.9 which states that one can construct as many maximal metrics on
the Euclidean spaces as many edge-transitive graphs, hence there are plenty of
maximal metrics on the Euclidean spaces.
Corollary 5.11. Denote by Gedgep,q := {[G] ∈ Gp,q | G : edge-transitive}. Then theedge
p+q
injective map Gp,q → M(R )/∼ maps each [G] to maximal elements [h, iGd ] inM(Rp+q )/∼ .

22

YUICHIRO TAKETOMIFor examples, the following three edge-transitive graphs give three diﬀerent
maximal metrics on R10 :(a) vertex: 6, edge: 4(b) vertex: 4, edge: 6(c) vertex: 5, edge: 5

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