A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS YUICHIRO TAKETOMI

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A MAXIMAL ELEMENT OF A MODULI SPACE OF

RIEMANNIAN METRICS

YUICHIRO TAKETOMI

Abstract. For a given smooth manifold, we consider the moduli space of

Riemannian 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 metric

that attains a maximal element with respect to the preorder a maximal metric.

Maximal metrics give nice examples of self-similar solutions for various metric

evolution equations such as the Ricci ﬂow. In this paper, we construct many

examples of maximal metrics on Euclidean spaces.

1. Introduction

Let Xbe 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 curvature

S:M(X)→R, g 7→ vol(X, g)2

n−1ZX

scalgdVg

has been studied actively for a compact manifold X. Here, vol(X, g) is the

Riemannian volume, scalgis the scalar curvature, and dVgis the Riemannian

volume element. For examples, Einstein metrics on a compact manifold Xare

characterized by critical points of ˜

S:M(X)→R. Since the normalized total

scalar curvature ˜

Sis invariant under the action of R>0×Diﬀ(X) on M(X), ˜

Scan

be regarded as the function on the moduli space M(X)/∼. Another important

example is the Ricci ﬂow

(1.1) ∂

∂tgt=−2Ricgt,

This work was supported by the Research Institute for Mathematical Sciences, an Inter-

national Joint Usage/Research Center located in Kyoto University. This work was partly

supported by Osaka Central Advanced Mathematical Institute: MEXT Joint Usage/Research

Center on Mathematics and Theoretical Physics JPMXP0619217849.

arXiv:2210.01483v1 [math.DG] 4 Oct 2022

2 YUICHIRO TAKETOMI

where Ricgis the Ricci tensor for a Riemannian metric g. Since the Ricci ten-

sor Ricgis invariant under the scaling (i.e. Riccg = Ricg) and diﬀeomorphisms

(i.e. Ricϕ∗g=ϕ∗Ricg, where ∗means the pullback), the Ricci ﬂow equation can

be regarded as a ﬂow on the moduli space M(X)/∼. Then, for examples, self-

similar solutions of the Ricci ﬂow are considered as stationary solutions of the

corresponding 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 size

of isometry groups. For [g],[h]∈M(X)/∼, denote by [g]≺[h] if Isom(X, g)⊂

Isom(X, h0) for some h0∈[h]. Then ≺deﬁnes an order on M(X)/∼. Note

that ≺is not a partial order but a preorder. That is, ≺satisﬁes reﬂexivity and

transitivity, 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 Rie-

mannian metrics via the size of isometry groups, and maximal metrics are “max-

imally nice” metrics with respect to this barometer.

Note that g∈M(X) is a maximal metric if and only if gsatisﬁes 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 ggive examples

of self-similar solutions for various metric evolution equations

∂

∂tgt=R(gt) (gt∈M(X)),

where Ris 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 gis a Ricci soliton.

For more details, see Subsection 2.2.

Remark 1.3.Homogeneous Ricci solitons have been studied actively. Recently, a

proof of the Alekseevskii conjecture has been announced by Lafuente and B¨ohm

([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

A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS 3

homogeneous Ricci soliton manifold is the Riemannian product of a compact Ein-

stein 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 that

a noncompact homogeneous Ricci soliton irreducible Riemannian manifold is dif-

feomorphic 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 mani-

folds 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,i0) implies Isom(X, h,i) =

Isom(X, h,i0) for all Riemannian metric h,i0on X. For a maximal metric g∈

M(X), if the number of connected components of Isom(X, g) is ﬁnite then ghas

a maximal isometry group. In particular,

Proposition 1.4. A homogeneous maximal metric ghas a maximal isometry

group.

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 construct

examples 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>0Aut(g) on the set of all inner

products 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 Gbe a simply connected Lie group, and h,ibe a left-invariant

metric on G. If the orbit R>0Aut(g).h,i ⊂ m(g)is an isolated orbit, then the left-

invariant metric h,iis maximal. The converse holds if Gis 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 gwon Rnwith

the Cartesian coordinate system (x1, x2, . . . , xn)by

gw:= (dx1)2+e−2w2x1(dx2)2+· · · +e−2wnx1(dxn)2.

Then gwis a maximal metric for all w∈Rn−1. If wi6=wjfor some i, j ∈

{2,3, . . . , n}, then gwis not isotropy irreducible.

4 YUICHIRO TAKETOMI

The Riemannian metric gwis isometric to a left-invariant metric on a certain

solvable Lie group. For more details, see Subsection 5.2. Note that the symmetric

group Sn−1acts on Rn−1naturally. For w, w0∈Rn−1,gwand gw0are isometric

with each other if and only if there exists some permutation σ∈Sn−1such that

σ.w =w0. Hence Theorem 1.6 gives continuously many examples of maximal

metrics.

The other examples are constructed by considering nilmanifolds attached with

graphs. By applying Theorem 1.5, we show that

Theorem 1.7. If given an edge-transitive graph Gwith pvertices and qedges,

one can construct maximal metrics on Rp+q. If q6= 0 then the metrics are not

isotropy irreducible.

A precise assertion of Theorem 1.7 will be given in Theorem 5.9. Note that,

for graphs Gand G0, the corresponding metrics are isometric with each other if

and only if Gand G0are 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 the

moduli space of Riemannian metrics on Xup to isometry and scaling, and ≺is

the preorder with respect to the size of the isometry groups. A maximal metric

g∈M(X) is the one whose equivalent class [g]∈M(X)/∼is a maximal element

with respect to ≺.

2.1. isotropy irreducible metrics and maximal metrics

Firstly, we explain that isotropy irreducible metrics are maximal metrics. Recall

that, for p∈Xin a Riemannian manifold (X, g), the action of the stabilizer

Isom(X, g)p:= {ϕ∈Isom(X, g)|ϕ(p) = p}on TpXby diﬀerential is called the

isotropy 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∈Xis an irreducible repre-

sentation. 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 homoge-

neous. For examples, see [26]. Strongly isotropy irreducible spaces which are some

special class of isotropy irreducible spaces have been classiﬁed independently by

Manturov ([17, 18, 19]), Wolf ([28, 29]) and Kr¨amer ([12]). Conclusively, isotropy

irreducible spaces have been classiﬁed by Wang-Ziller ([26]).

A MAXIMAL ELEMENT OF A MODULI SPACE OF RIEMANNIAN METRICS 5

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>0g.

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 Xis proper, there exists

a non-constant Isom(X, g)-invariant smooth function fon X. Then one has efg

is a smooth Isom(X, g)-invariant Riemannian metric which is not contained in

R>0g. 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 any

p∈X. Then the set of Isom(X, g)-invariant metrics on Xis naturally identiﬁed

with the set of inner products which are invariant under the isotropy represen-

tation on a tangent space TpX. Assume that (X, g) is not isotropy irreducible.

Then there exists a subspace V(TpXwhich is invariant under the isotropy

representation. Denote by g1and g2the restriction of gto Vand the normal

space V⊥, respectively. Then ag1+bg2is an Isom(X, g)-invariant metrics for all

a, b > 0. This implies that, for examples, 2g1+ 3g2is an Isom(X, g)-invariant

metrics which is not contained in R>0g.

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>0g⊂[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 evolu-

tion equations

Denote by S(X) the set of all symmetric (0,2)-tensors on X. Let us consider a

map R:M(X)→S(X). Then one can deﬁne a partial diﬀerential equation

∂

∂tgt=R(gt).

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, then

gis called a soliton. For examples, a soliton for the Ricci ﬂow is usually called

aRicci soliton. The study of self-similar solutions and solitons are important in

order to understand metric evolution equations.

By the property (1.2), one has

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AMAXIMALELEMENTOFAMODULISPACEOFRIEMANNIANMETRICSYUICHIROTAKETOMIAbstract.Foragivensmoothmanifold,weconsiderthemodulispaceofRiemannianmetricsuptoisometryandscaling.Onecandeneapreorderonthemodulispacebythesizeofisometrygroups.WecallaRiemannianmetricthatattainsamaximalelementwithrespecttothepreorderam...

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