Prescribed Riemannian Symmetries
Given a smooth free action of a compact connected Lie group 𝐺 on a smooth compact manifold 𝑀, we show that the space of 𝐺-invariant Riemannian metrics on 𝑀 whose automorphism group is precisely 𝐺 is open and dense in the space of all 𝐺-invariant metrics, provided the dimension of 𝑀 is ''su...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2021 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211319 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Prescribed Riemannian Symmetries. Alexandru Chirvasitu. SIGMA 17 (2021), 030, 17 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859657533638049792 |
|---|---|
| author | Chirvasitu, Alexandru |
| author_facet | Chirvasitu, Alexandru |
| citation_txt | Prescribed Riemannian Symmetries. Alexandru Chirvasitu. SIGMA 17 (2021), 030, 17 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Given a smooth free action of a compact connected Lie group 𝐺 on a smooth compact manifold 𝑀, we show that the space of 𝐺-invariant Riemannian metrics on 𝑀 whose automorphism group is precisely 𝐺 is open and dense in the space of all 𝐺-invariant metrics, provided the dimension of 𝑀 is ''sufficiently large'' compared to that of 𝐺. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way, we also show, under less restrictive conditions on both dimensions and actions, that the space of 𝐺-invariant metrics whose automorphism groups preserve the 𝐺-orbits is dense 𝐺δ in the space of all 𝐺-invariant metrics.
|
| first_indexed | 2026-03-14T17:00:27Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 030, 17 pages
Prescribed Riemannian Symmetries
Alexandru CHIRVASITU
Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
E-mail: achirvas@buffalo.edu
Received September 27, 2020, in final form March 10, 2021; Published online March 25, 2021
https://doi.org/10.3842/SIGMA.2021.030
Abstract. Given a smooth free action of a compact connected Lie group G on a smooth
compact manifoldM , we show that the space ofG-invariant Riemannian metrics onM whose
automorphism group is precisely G is open dense in the space of all G-invariant metrics,
provided the dimension of M is “sufficiently large” compared to that of G. As a consequence,
it follows that every compact connected Lie group can be realized as the automorphism group
of some compact connected Riemannian manifold; this recovers prior work by Bedford–
Dadok and Saerens–Zame under less stringent dimension conditions. Along the way we
also show, under less restrictive conditions on both dimensions and actions, that the space
of G-invariant metrics whose automorphism groups preserve the G-orbits is dense Gδ in the
space of all G-invariant metrics.
Key words: compact Lie group; Riemannian manifold; isometry group; isometric action;
principal action; principal orbit; scalar curvature; Ricci curvature
2020 Mathematics Subject Classification: 53B20; 58D17; 58D19; 57S15
1 Introduction
The present paper fits into the general theme of realizing a predetermined group as the symmetry
group of a structure (combinatorial, topological, geometric, etc.) of given type. Variations on the
theme abound in the literature, mostly (but by no means exclusively) in the context of finite
groups. To list a few instances:
(1) Every finite group is the automorphism group of
� a finite graph [14],
� even better, a finite 3-regular graph [15, Theorems 2.4 and 4.1],
� more generally, a finite graph with given connectivity or chromatic number, regular of
given degree, and a number of other such constraints [30, Theorem 1.2],
� some convex polytope, with the group acting either purely combinatorially [33, Theo-
rems 1 and 2] or isometrically [11, Theorem 1.1].
(2) More generally, arbitrary (possibly infinite) groups are graph isomorphism groups [31, Theo-
rem].
(3) Switching to the more topologically-flavored setup that informs this paper,
� Polish (i.e., separable completely metrizable) topological groups can be realized as
isometry groups of separable complete metric spaces ([16] and [22, Section 3]),
� in the same spirit, compact metrizable groups are isometry groups of compact metric
spaces [22, Theorem 1.2],
� the same goes for locally compact groups and spaces [21, Theorem 2.1],
� Lie groups are isometry groups of manifolds (equipped with metrics compatible with
the manifold structure) [27, Corollary 1.2 and discussion following Proposition 1.4].
mailto:achirvas@buffalo.edu
https://doi.org/10.3842/SIGMA.2021.030
2 A. Chirvasitu
This paper was originally motivated by the following problem, posed as open in [22, Section 4]
(and appearing also as [27, Question Q4]):
Question 1.1. Is it the case that every compact Lie group is the isometry group of some compact
Riemannian manifold?
The answer turns out to be affirmative, as confirmed in both [3, Theorem 3] and [32, Theo-
rem 1] (I am grateful to one of the referees for bringing this work to my attention; I was not
aware of it while writing an early draft).
With this question as the initial motivating driver, one can then pose the natural follow-up
problem (to be made precise) of “how large” the set of desirable Riemannian metrics is, given
a manifold equipped with an action by a compact Lie group. The results below all revolve
around the general principle that given an (always smooth, for us) action of G on M , “most”
G-invariant Riemannian metrics on M are maximally rigid. In the case G = {1} this same
generic rigidity principle informs [12, Corollary 8.2 and Proposition 8.3], which we paraphrase
slightly as
Theorem 1.2. Let M be a compact smooth manifold. The set of Riemannian metrics on M
with trivial isometry group is open in the space of all Riemannian metrics, and it is dense if all
connected components of M have dimension ≥ 2.
The version of this result proven below (Theorems 3.2 and 4.3), involving a G-action, reads
as follows (withMG(M) denoting the space of G-invariant metrics on M , equipped with its C∞
topology):
Theorem 1.3. Let G be a compact Lie group acting smoothly on the compact manifold M .
(a) If
� the action is principal (Definition 3.1), and
� all connected components of M have dimension ≥ 3 + dimG
then the space of G-invariant Riemannian metrics whose isometry groups leave all G-orbits
invariant is a dense Gδ subset of MG(M).
(b) If furthermore
� G is connected,
� the action is free, and
� all connected components of M also have dimension ≥ 2 dimG+ 1
then the space of G-invariant metrics on M whose isometry group is precisely G is open
dense in MG(M).
In Section 2 we gather a number of preliminary remarks of use in the sequel (on topology,
Riemannian geometry, etc.).
Section 3 revolves around vertical Riemannian metrics: given an action of G on M , these are
the G-invariant metrics whose isometry groups preserve the G-orbits (as sets, not necessarily
pointwise). The term is inspired by the theory of fibrations / submersions (e.g., [5, Chapter 9]):
the G-orbits are the fibers of the fibration M → M/G, so the vectors tangent to the orbits
are vertical in fibration-specific terminology. The main result in that section is Theorem 3.2,
matching (a) of Theorem 1.3 above.
Naturally, if a G-invariant metric on M is maximally rigid in the sense that its isometry
group is precisely G (and not larger), it will also be vertical. For that reason, Section 3 serves as
Prescribed Riemannian Symmetries 3
preparation for Section 4. In the latter we focus on maximally rigid G-invariant metrics on M .
Here the main result is Theorem 4.3, corresponding to part (b) of Theorem 1.3 above.
As for Question 1.1, Corollary 4.4 below recovers that affirmative answer for connected com-
pact Lie groups. There are trade-offs as compared to [3, Theorem 3] and [32, Theorem 1]: while
the latter make no connectedness assumptions, the Riemannian manifolds obtained here (with
prescribed symmetry group) tend to have smaller dimension.
2 Preliminaries
We will need some background on Riemannian geometry, as covered well in numerous sources: [4,
5, 10, 19, 28] will do for instance (as does [9, Appendix A] for a quick reference), and we cite
some of these more precisely in the discussion below. We use some of the standard conventions.
Having fixed a coordinate patch of the Riemannian manifold M with coordinates
xi, 1 ≤ i ≤ n, n := dimM
we denote
� by δij the Kronecker delta, 1 for i = j and 0 otherwise,
� by gij the Riemannian metric tensor,
� by gk` the inverse of gij ,
� by R`ijk the curvature (3, 1)-tensor,
� by
Rik = Rjijk (2.1)
the Ricci (2, 0)-tensor, with Einstein summation convention (i.e., summing over the repe-
ated j in (2.1)),
� by R (unadorned) the scalar curvature, trRij , a function on M ,
� on one occasion, by Zij the traceless Ricci (2, 0)-tensor
Zij = Rij −
1
n
Rgij . (2.2)
See for instance [5, Chapter 1] or [28, Chapter 3] for a recollection of the various notions.
We also adopt the usual convention on raising and lowering indices via gk` and gij respectively,
for instance as in [5, Section 1.42]: for a tensor Aj−··· we set
A−i··· := gijA
j−
···
(summation over the repeated j, as always). Analogous formulas hold for raising rather than
lowering an index, with gk` in place of gij . We will refer again, for instance, to the Ricci
(1, 1)-tensor
Rij = gikRkj
(see [5, Remark 1.91]). It induces an operator on each tangent space TpM , p ∈ M of a Rie-
mannian manifold, and that operator is self-adjoint (i.e., symmetric with respect to the Hilbert
space structure on TpM imposed by the Riemannian metric). The symmetry, concretely, simply
means that
Rij = Rji .
4 A. Chirvasitu
“Smooth” always means C∞. The main manifold under consideration (typically denoted
by M) can be assumed boundary-less, but various auxiliary submanifolds thereof will, in general,
have boundaries or be non-compact: in those cases the arguments will be local in nature, so the
non-compactness and/or presence of a boundary will not make a difference.
For a smooth manifold M , we follow [12] in denoting by M :=M(M) the space of smooth
Riemannian structures on M . In the presence of a smooth action of a (typically compact) Lie
group G on M we amplify this notation by writing
MG :=MG(M)
for the space of (always smooth) G-invariant Riemannian metrics on M .
The following piece of terminology is justified by the example of a fibration M → M/G
induced by a free G-action on M , with fibers ∼= G regarded as “vertical” in a pictorial rendition
of that fibration.
Definition 2.1. Let G be a Lie group acting smoothly on the smooth manifold M . Vectors
in TM tangent to G-orbits are vertical.
A G-invariant Riemannian structure on M is vertical if its automorphism group leaves every
G-orbit invariant. We denote by
Mv
G(M) ⊂MG(M)
the space of G-invariant vertical Riemannian metrics.
Note thatM(M) is open in the Polish (i.e., separable completely metrizable) space Γ(T⊗2M)
of smooth sections of the tensor square bundle T⊗2M , and hence is itself Polish [34, Appen-
dix, Proposition A.1]. MG(M) is also Polish, for instance because it is closed in the Polish
space M(M) [6, Chapter IX, Section 6.1, Proposition 1a)].
Recall:
Definition 2.2. A subset of a topological space is
� meager or of first category if it is a countable union of nowhere dense sets,
� non-meager or of second category if it is not meager,
� residual if its complement is meager.
A topological space is Baire (or a Baire space) if meager sets have empty interior.
Cf. [26, Definitions 11.6.1 and 11.6.5].
According to the Baire category theorem [26, Theorem 11.7.2] complete metric spaces are
Baire. SinceMG(M) is Polish, that result allows us to regard residual subsets thereof as “large”:
they are certainly dense, but being residual says more than that.
As usual (e.g., [13, Section 1.3]) Fσ-subsets of a topological space are countable union of closed
subsets, while Gδ-subsets are countable intersections of open subsets. Their relevance here stems
from the fact that in a Baire space a countable intersection of open dense subsets is residual and
hence dense.
3 Vertical metrics
The reader might find [1, 7, 18] particularly useful in parsing the material below, on Lie-group
actions on manifolds.
Theorem 3.2 below is an equivariant version of the result that “generically”, Riemannian
manifolds are rigid (i.e., have trivial isometry groups); this is [12, Proposition 8.3], which can
Prescribed Riemannian Symmetries 5
be recovered from Theorem 3.2 by setting G = {1}. Recall Definition 2.1 above for notation.
We also need the following notion (see, e.g., [2, Proposition I.2.5 and the discussion following it,
and Remark I.2.7]).
Definition 3.1. Let G be a compact Lie group acting smoothly on a smooth manifold M so
that M/G is connected.
(a) An orbit Gp is principal if either of the two following equivalent conditions holds:
� the points q ∈M whose isotropy groups Gq are in the same conjugacy class as Gp form
a dense open subset of M ,
� the action of Gp on the quotient TpM/Tp(Gp) of tangent spaces is trivial.
(b) The action is principal if all of its orbits are.
(c) In general (i.e., for possibly-disconnected M/G), the components of the action are the actions
of G on the preimages of the connected components of M/G. Every orbit is an orbit of some
component, and hence the notion of principality makes sense for orbits in full generality.
(d) Similarly, a general action is principal if its components are.
Theorem 3.2. Let M be a compact smooth manifold equipped with a principal G-action by
a compact Lie group G. If
dimMi ≥ 3 + dimG (3.1)
for every connected component Mi ⊂ M then the space Mv
G(M) of G-invariant vertical Rie-
mannian metrics on M in the sense of Definition 2.1 is a dense Gδ-subset of MG(M).
Remark 3.3. As observed in the statement of [12, Proposition 8.3], some requirement (3.1) on
dimensions is necessary: when G is trivial, the circle M ∼= S1 has isometry group O(2) for any
Riemannian metric. Since in this case ‘vertical’ simply means ‘with trivial automorphism group’,
there are no vertical metrics at all.
This example rules out a 1+ correction term in (3.1), but not a 2+ correction term; indeed,
I do not know whether the result is sharp in this sense.
Proof of Theorem 3.2: the Gδ claim. Let orbi, i ∈ Z≥0 be a countable set of orbits that
is dense in M/G, and Un, n ∈ Z≥0 a countable set of G-invariant open subsets of M which
constitute fundamental systems of neighborhoods of the orbi. Set
Fi,n := {g ∈MG(M) | the automorphism group aut(g) moves some point of orbi
out of Un}.
We now have
(i) Each Fi,n is closed. To sketch this briefly, let
Fi,n 3 gα −→
α
g ∈MG(M)
be a convergent net. By assumption, each gα admits an automorphism γα that moves some
point pα ∈ orbi outside Un. Because for large enough α the metrics gα are uniformly close
to g, so are the global geodesic metrics they induce on M , and hence the union of all
isometry groups of gα (again, assuming large α) will be equicontinuous [24, Section 45,
Definition].
It now follows from the Arzela–Ascoli theorem (e.g., in the variant appearing as [24,
Theorem 47.1]) that {γα} is relatively compact, as is {γ−1α } in the uniform topology on
6 A. Chirvasitu
self-maps of M . This implies that we can find a subnet γβ convergent to an isometry γ
of g. Further passing to a subnet thereof if necessary, we can furthermore assume that pβ
converges to some point p which of course still belongs to the closed set orbi. Finally,
M \ Un 3 γβpβ → γp
and hence this latter point will again belong to the closed set M \ Un.
(ii) The complement MG(M) \Mv
G(M) is the union of the Fi,n.
Jointly, (i) and (ii) imply the desired Gδ-ness conclusion. �
The proof of (the rest of) Theorem 3.2 will require some preparation, in part to recall,
somewhat informally, the proof strategy for [12, Proposition 8.3]. That proof proceeds as follows.
(a) An arbitrary Riemannian metric gij on M is first perturbed slightly so that the maximum
over p ∈M of the largest eigenvalue
max spec(ric(p))
of the symmetric operator
ric(p) := Rij(p) : TpM → TpM
is achieved at a unique point p, and the perturbation is confined to an arbitrarily small
neighborhood U of p.
(b) With this in hand, every isometry of M with respect to the new metric will fix that unique
point p.
(c) The procedure is repeated on small spheres around p avoiding U , ensuring that the maximal
eigenvalue of Rij on such a sphere is achieved at a unique point, which will then again be
fixed by every Riemannian isometry.
(d) Repeating the procedure a large (but finite) number of times, one obtains a metric whose
isometry group fixes at least dimM + 1 “independent” points of M . It then follows that
the isometry group must be trivial (e.g., [25, Theorem 3]).
The proof of Theorem 3.2 appearing below follows essentially the same plan, with some
modifications. For one thing, in place of the maximal eigenvalue we consider other numerical
invariants of a self-adjoint operator on a (real) Hilbert space:
Notation 3.4. Let T : Rn → Rn be a symmetric operator. We write
� ‖T‖ for its norm with respect to any real Hilbert space structure on Rn; it is the largest
|λ| for λ ranging over the spectrum spec(T ).
� spr(T ) for the spread of T , i.e., the length of the smallest interval containing spec(T ).
We will be interested in maximizing the norm or spread of the operators ric(p) instead. Note
that in general, for a Riemannian manifold M ,
max
p∈M
spr(ric(p)) = 0
precisely when each operator ric(p) is a scalar multiple of the identity or, equivalently, the Ricci
(2, 0)-tensor Rij is a “conformal multiple” of the metric gij :
∀ p ∈M, Rij(p) = f(p)gij(p) (3.2)
for some function f : M → R. Assuming M is connected, this is
Prescribed Riemannian Symmetries 7
� no restriction at all when dimM = 2 (i.e., it is automatic) [5, Remark 1.96(a)],
� equivalent to M being an Einstein manifold when dimM ≥ 3, i.e., the function f in (3.2)
is in fact constant [5, Theorem 1.97].
Notation 3.5. For a smooth manifold M equipped with a smooth action by a Lie group G we
introduce the following notation.
� NSRG(M) ⊂MG(M) is the set of G-invariant Riemannian structures satisfying
spr(ric(p)) > 0 over a dense set of p ∈M.
The symbol stands for “non-scalar Ricci”, based on the fact that spr(ric(p)) vanishes
precisely when the operator ric(p) : TpM → TpM is a scalar multiple of the identity.
� Similarly, NZRG(M) (for “non-zero”) is the set of G-invariant Riemannian structures
such that
ric(p) 6= 0 over a dense set of p ∈M.
� NZSG(M) (for “non-zero scalar”) is the set of G-invariant Riemannian structures such
that
R(p) = tr(ric(p)) 6= 0 over a dense set of p ∈M.
� For a subset U ⊆M , we write SRUG(M) for the set of G-invariant structures for which
spr(ric(p)) = 0, ∀ p ∈ U
(i.e., the Ricci tensor is scalar along U).
� Finally, we set
SRG(M) := SRMG (M).
Proposition 3.6. Let M be a smooth compact manifold with connected components Mi, and
equipped with a smooth action by a compact Lie group G.
(a) If
dimMi ≥ 2 + dimG, ∀ i
the set NZRG(M) is residual in MG(M) in the sense of Definition 2.2.
(b) The same goes for NZSG(M).
(c) If furthermore we have dimMi ≥ 3 for all i then the space NSRG(M) is residual inMG(M)
in the sense of Definition 2.2.
Proof. We prove (c), while only very briefly sketching how the (simpler) proofs for parts (a)
and (b) can be adapted from this.
(c) The complement
MG(M) \ NSRG(M)
is the union, over all open U ⊆ M , of the sets SRUG(M) introduced in Notation 3.5. Since we
can furthermore range U over some countable base for the topology of M , it will be enough to
prove that for every non-empty open U the set SRUG(M) is nowhere dense in MG(M). Since
that set is closed, what we want to argue is that it has empty interior. In other words:
8 A. Chirvasitu
Claim 3.7. A metric g ∈ SRUG(M) has arbitrarily small deformations outside that set.
We can see this by effecting a conformal deformation
g 7→ g′ := ϕ−2g,
where ϕ is a strictly positive, G-invariant function on M that is C∞-close to the constant
function 1.
We can assume that U is G-invariant. According to the slice theorem for G-actions (e.g., [20,
Théorème, p. 139], [23, Theorem 1] or [2, Theorem I.2.1] among others) every point p ∈ U has
a G-invariant “tubular” neighborhood contained in U , G-equivariantly diffeomorphic to G×GpV ,
where
� Gp ⊆ G is the isotropy group at p,
� V is the quotient space TpM/Tp(Gp) (Gp being the orbit through p),
� the Gp-action on V is the differential of the Gp-action on M obtained by restricting that
of G.
Furthermore, it follows from [2, Proposition I.2.5] that there is a dense set of points p for
which the linear action of Gp on V is trivial (i.e., those lying on principal orbits in the sense
of Definition 3.1). For such a p ∈ U (which we henceforth fix), the tubular neighborhood G×GpV
is in fact diffeomorphic to the product manifold Gp × V ; we frequently identify the two in the
discussion below. We can then select our scaling function ϕ so that
� it is identically 1 outside some G-invariant neighborhood of Gp whose closure is contained
in Gp× V ,
� on Gp× V it depends only on local coordinates on V , and is thus G-invariant.
Additionally, we have to choose ϕ so as to achieve the desired outcome that g′ have non-scalar
Ricci (1, 1)-tensor in U . By [5, equation (1.161b)] the conformal transformation rules for the
traceless Ricci tensor (2.2) are of the form
Z ′ = Z + (some multiple of g) +
dimM − 2
ϕ
Hess(ϕ), (3.3)
where Hess denotes the Hessian defined [5, Section 1.54] as a (2, 0)-tensor by
Hess(ϕ)(X,Y ) = X(Y ϕ)− (∇XY )(ϕ),
where ∇ denotes the Levi-Civita connection (denoted by the same symbol in [19, Section IV.2]
and by DXY in [5, Section 1.41]).
Since dimM ≥ 3, it will be enough to choose ϕ so that Hessian fails to be a scalar multiple
of the metric g at some point in U . In normal [5, Section 1.44] local coordinates Hess(ϕ) is
expressible as the familiar Hessian matrix with entries
Hess(ϕ)i,j =
∂2ϕ
∂xi∂xj
. (3.4)
Since (with Mi ⊂M being the component that contains p) we have
dimV = dimMi − dimGp ≥ dimMi − dimG ≥ 2
by assumption, we can certainly arrange for second partial derivatives with respect to the coor-
dinates xi on V so that the bilinear form with matrix (3.4) is not a scalar multiple of (gij)i,j .
This proves the claim and hence the result.
Prescribed Riemannian Symmetries 9
(b) We can follow the same strategy as above, this time replacing (3.3) with its scalar-
curvature version [5, Theorem 1.159(f)]: if we conformally scale the metric g to g′ = e2fg then
the relation between the two scalar curvatures R′ (new) and R (old) is
R′ = e−2f
(
R+ 2(n− 1)∆f − (n− 2)(n− 1)|df |2
)
,
where
� n is the dimension of the underlying manifold,
� ∆f is the Laplacian of f [5, Section 1.54c],
� |df | denotes the length of the gradient of f [5, Section 1.54a] in the metric g.
(a) This follows from parts (b) and (c): the former trivially covers components of dimen-
sion ≥ 3, whereas the latter ensures non-vanishing on dimension-2 components, where Rij =
1
2Rgij . �
Lemma 3.8 implements (a) (and (b)) in the above discussion, following the statement of
Theorem 3.2; its proof is very much in the spirit of that of [12, Proposition 8.3].
Lemma 3.8. Let G be a compact Lie group acting smoothly and isometrically on a Riemannian
manifold (M, g) with components of dimension ≥ 2 + dimG. Then, there is a point p ∈M such
that
� one can find G-invariant metrics g′ on M arbitrarily close to g,
� achieving the maximal absolute value of its scalar curvature on a unique G-orbit in an arbi-
trarily small G-invariant neighborhood U of Gp, and hence,
� so that the isometry group aut(g′) leaves that orbit invariant.
Moreover, if g 6∈ SRG(M) then we can ensure g′ = g outside the arbitrarily-small neighborhood U
of Gp.
Proof. By part (b) of Proposition 3.6 we can perturb g (arbitrarily) slightly so as to ensure the
scalar curvature
R(p) = trric(p)
is non-zero for most p. We retain this assumption on g throughout the rest of the proof.
Now let p ∈ M be a point where the maximal absolute value |R(q)|, q ∈ M is achieved
(it will be the point p in the statement), and fix a G-invariant neighborhood U of Gp. Consider
a smooth function
ψ : R≥0 → R≥1
that is
� C∞-close to the constant function 1,
� equal to some constant slightly larger than 1 on a small interval [0, r],
� equal to 1 on [r + ε,∞).
One then obtains a smooth G-invariant function ϕ on M , C∞-close to 1, by
ϕ(x) := ψ(distance from x to the orbit Gp), ∀x ∈M.
We assume r in the above discussion is small enough that ϕ is identically 1 off U .
10 A. Chirvasitu
Finally, consider the G-invariant conformal rescaling g1 := ϕ−2g. Because it scales g by the
constant ψ(0)−2 < 1 in a neighborhood of Gp, it scales the operator ric(p) (and hence its trace)
by the inverse scalar ψ(0)2 > 1. Since g1 ∼= g off U , the new metric achieves its maximal
|R(q)|, q ∈M (3.5)
somewhere in U .
Now repeat the procedure, as in the proof of [12, Proposition 8.3]: pick q ∈ U maximizing (3.5)
for g1, choose a neighborhood U1 of q less than half the size of U with respect to some fixed
metric inducing the topology of M , and perturb g1 to g2 so that
� the perturbation g2 − g1 is less than half the size of g1 − g in some metric inducing the
C∞ topology on the space of Riemannian structures,
� g2 = g1 off U1, and
� for g2 the maximal value of (3.5) is achieved in U2.
Continuing in this fashion, the limit
g′ := lim
n→∞
gn
will be a G-invariant metric close to g whose maximal (3.5) is achieved on a unique orbit
contained in the original (arbitrarily small) neighborhood U of p. It follows that orbit must be
preserved by the isometry group of g′, as desired.
As for the last statement (on g 6∈ SRG(M)), it is clear from the proof: the argument produces
metrics identical to g off U after the initial step of perturbing g away from SRG(M). �
Proof of Theorem 3.2. By passing to the components of the action in the sense of Defi-
nition 3.1, we may as well assume that the orbit space M/G is connected. Furthermore,
by Lemma 3.8 we can assume that our metric g achieves its maximal scalar curvature along
a single orbit Gp (for some p ∈M).
Now consider the geodesics emanating from p, orthogonal to Gp (we refer to such geodesics
as horizontal, in keeping with the spirit of Definition 2.1). Denoting by dg the distance induced
by the metric g, for sufficiently small r > 0 the tubular neighborhood
Gp≤r := {q ∈M | dg(q,Gp) ≤ r}
is (by the principality of the action) diffeomorphic to Gp×H≤r, where
� H is the union of the horizontal geodesics emanating from p, and hence a manifold close
enough to Gp,
� H≤r is, as the notation suggests, the subset of H at distance dg ≤ r from the orbit Gp
(or equivalently, from p).
Horizontal geodesics are orthogonal to all G-orbits they encounter (e.g., [5, Lemma 9.44] or [17,
Section 1.1]), and we can obtain G-invariant Riemannian structures by deforming the metric g
along the manifold H comprising the horizontal geodesics (sufficiently close to Gp so as not
to run into injectivity-radius issues) and keeping it invariant along the G-orbits. Explicitly, at
a point q ∈ Hr we can split the tangent space TqM as
TqM = Tq(Gq)⊕ TqH,
decompose the matrix of the Riemannian metric g correspondingly as a block matrix(
Av B
Bt Ah
)
Prescribed Riemannian Symmetries 11
(with the top left and bottom right corners representing, respectively, the restrictions of g to Gq
and H), and deforming only the lower right-hand corner Ah sufficiently slightly so as to ensure
the resulting matrix still represents a positive symmetric bilinear form.
The isometry group aut(g) leaves Gp invariant, and hence the isotropy subgroup aut(g)p
preserves every p-centered ball H≤r in H. Now choose small r, ε > 0 and deform the metric
slightly in H≤2r so that
� the perturbed metric coincides with the old metric g outside H≤r+ε and inside H≤r−ε,
� inside the annulus H[r−ε,r+ε] the perturbation is spherical, in the sense that we choose
geodesic spherical coordinates [8, Section III.1] in H≤r centered at p, with a radial coor-
dinate and (dimH−1) “angular” coordinates, and deform the metric only along the latter,
� the perturbed metric on the sphere Hr has trivial isometry group (this is possible because
that sphere is at least 2-dimensional by (3.1), and hence [12, Proposition 8.3] applies).
For the resulting G-invariant metric g′ the manifold H consisting of horizontal geodesics ema-
nating from p still bears that description because of the spherical character of the deformation.
By construction, the isotropy group aut(g′)p will then fix Hr identically (i.e., pointwise). But in
that case
� aut(g′)p leaves invariant the G-orbit of every point in the tubular neighborhood GHr
of Gp,
� and hence so does
aut(g′) = G · aut(g′)p.
Note that the latter product is not direct, and ‘G’ is a stand-in for its image in the automorphism
group of g′ (the action of G is not assumed faithful here).
Since we are assuming the orbit space M/G is connected, all orbits are reachable from Gp
by horizontal geodesics emanating from it. Since aut(g′)p acts trivially on the initial segments
of those geodesics it acts trivially on horizontal geodesics period, meaning that all orbits are left
invariant by aut(g′). �
3.1 Some remarks on the literature
The discussion above gives a brief review of the proof of [12, Proposition 8.3]. For this reader,
at least, that proof presented a difficulty that appeared not to be immediately addressed by the
text in loc.cit. Specifically, the proof proceeds, as indicated above, by
(1) first deforming a metric g so as to produce a globally-invariant point p (i.e., one fixed by all
isometries), and then
(2) deforming the metric again around a radius-r sphere Sp,r centered at p so as to produce
a point q where the Ricci (1, 1)-tensor ric achieves its unique maximal spectral value
along Sr(p).
The isometry group of the metric obtained after step (1) will leave p invariant, and hence also
S := Sr(p) (which in [12] would be denoted by Arp). If ric were to achieve its maximal spectral
value at a unique point q ∈ S at this stage, then q would be invariant under the isometry group.
The problem, though, is that q is produced after further deformation, whereupon S need not
remain a p-centered sphere.
In other words, I see no reason (without further elaboration) why the metric produced after (2)
should leave S invariant (and hence q on it). There are ways to handle this:
12 A. Chirvasitu
Deforming outside a ball. The alteration of the metric “around S” (as it is phrased
on [12, p. 36], with Aρq in place of S) might be interpreted as an alteration only outside the
ball Br(p) bounded by S. This is possible, since the alteration in question consists of adding
to the (2, 0)-tensor g another tensor whose 2nd derivatives with respect to a system of normal
coordinates satisfy certain inequalities (see [12, equation (8.4)]).
This would ensure that after the deformation in (2) the radius r-sphere centered at p retains
its identity.
An inductive approach. Alternatively, one could proceed inductively on dimension, by
� first proving the claim separately for surfaces, and then
� finding p as above, and then modifying the metric only on geodesic spheres around p as in
the proof of Theorem 3.2, making use of spherical coordinates.
4 Maximal rigidity
As indicated in the Introduction, the initial motivation for the results above was to produce G-
invariant metrics whose isometry group is precisely G; they should, in other words, be maximally
rigid subject to the requirement that they be G-invariant (hence the title of the present section).
This also justifies
Notation 4.1. Given a faithful isometric action of a Lie group G on a Riemannian manifold M ,
the spaceMmax
G (M) of maximally rigid G-invariant metrics consists of those g ∈MG(M) whose
isometry group is precisely G.
The same notation (and terminology) applies to arbitrary (non-faithful) actions: if H E G is
the kernel of the action, then by definition
Mmax
G (M) =Mmax
G/H(M).
Since we can harmlessly pass to faithful actions by passing to the quotient by the kernel of
the action, we typically assume faithfulness throughout.
One cannot hope for metrics produced as in Theorem 3.2 to be maximally rigid in full
generality, for arbitrary compact Lie groups. Indeed, most finite groups G will fail in that
respect:
Example 4.2. Let G be a compact Lie group with ≥ 3 connected components Gi, acting in the
obvious fashion on M := G ×N for some manifold N . Then, for any G-invariant Riemannian
structure g on M , the automorphism group aut(g) can permute the manifolds Gi×N for γ ∈ N
arbitrarily.
Now, if G0 ⊂ G is the identity component, then the action of G on the set of manifolds
Gi ×N is isomorphic (as a permutation action) to the regular action of G/G0. Since the latter
is strictly smaller than the symmetric group S(G/G0) of the set G/G0, we have
S(G/G0) ⊂ aut(g) but S(G/G0) 6⊆ G ⊂ aut(g).
In particular, for suchG (and actions) we can never obtainG = aut(g) for a suitable Riemannian
metric g.
It turns out, though, that the disconnectedness of G in Example 4.2 is the only issue:
Theorem 4.3. Let G be a compact connected Lie group acting freely and smoothly on a compact
smooth manifold M . Then, the following statements hold.
Prescribed Riemannian Symmetries 13
� The subset
Mmax
G (M) ⊆MG(M) (4.1)
is open.
� If furthermore the components Mi of M satisfy the dimension inequality
dimMi ≥ max(3 + dimG, 2 dimG+ 1) (4.2)
then (4.1) is dense.
As an immediate consequence we have
Corollary 4.4. Every compact connected Lie group arises as the isometry group of some compact
Riemannian manifold.
Proof. In Theorem 4.3, simply take M = G × N equipped with the obvious action on the
left-hand factor for some connected manifold N of sufficiently large dimension. �
This answers the question in [22, Section 4] (and [27, Question Q4]) affirmatively.
Remark 4.5. Since G is connected, it operates on each connected component of M . Restricting
our attention to an individual component, we can assume that M is connected; we do this
throughout the present section. With this connectedness assumption in place, an isometry of M
is trivial if and only if
� it fixes some point p (arbitrary, chosen beforehand), and
� it induces the trivial linear action on TpM .
Proof of Theorem 4.3: openness. This follows from the upper semicontinuity of the auto-
morphism group of Riemannian structures. Let g ∈Mmax
G (M). According to the aforementioned
semicontinuity result [12, Theorem 8.1], for g′ ∈MG(M) sufficiently close to g we have
σaut(g′)σ−1 ⊆ aut(g) = G (4.3)
for some diffeomorphism σ of M . The left hand side is a subgroup of diff(M) (group of dif-
feomorphisms) containing the Lie group
σGσ−1 ⊂ diff(M)
because, g′ being G-invariant, aut(g′) contains G. Since Lie groups cannot contain proper
isomorphic copies of themselves (4.3) must be an equality. It follows that so too is
G ⊆ aut(g′),
again for reasons of size: G and aut(g′) are Lie groups with the same dimension and the same
number of components, one containing the other. �
We have the following characterization of maximally rigid actions.
Lemma 4.6. A vertical free action of a compact Lie group G on a connected manifold M is
maximally rigid if and only if either of the following equivalent statements holds:
(a) the action of the isometry group is free, i.e., the isotropy group of every point is trivial,
(b) the isotropy group of a single arbitrary point p ∈M is trivial.
14 A. Chirvasitu
Proof. We only prove equivalence to (b), leaving the other point to the reader.
For a vertical metric g ∈Mv
G(M) an arbitrary point p ∈M will be moved by every isometry σ
to a point q on the same orbit Gp. We can then translate q back to p via the G-action, i.e., by
some element γ ∈ G. Then, σ belongs to G if and only if γσ does. Since the action is free, the
isotropy group Gp is trivial. We already know that γσ is in the isotropy group aut(g)p, so
σ ∈ G ⇐⇒ σγ ∈ G ⇐⇒ σγ ∈ Gp ⇐⇒ σγ = 1.
Since, as σ ranges over aut(g), elements of the form σγ range over aut(g)p, this proves the
equivalence between maximal rigidity and (b). �
We will often keep this characterization in mind in the arguments below, sometimes implicitly.
Note that even though Theorem 3.2 only says that the vertical metrics form a Gδ (rather
than open) set, in the context of that proof we have quite a bit of freedom in varying g so as
to keep it vertical. Specifically, if, as in that proof, we assume the maximal scalar curvature is
achieved along a unique orbit Gp (as we will), then all metrics g′
� sufficiently C∞-close to g,
� coinciding with g close to Gp
will be vertical. This is because, again as in the aforementioned proof, the corresponding “hori-
zontal” manifold H through p (i.e., the union of the geodesics emanating from p and orthogonal
to Gp) will have trivial isometry group by [12, Proposition 8.3]. For these reasons, we need not
worry below, in the proof of Theorem 4.3, about breaking the verticality of our slightly-deformed
Riemannian structures.
Proof of Theorem 4.3: density. According to Theorem 3.2 we can deform an arbitrary met-
ric arbitrarily slightly so as to render it vertical, so we work with vertical metrics g to begin
with. In fact, we will assume (via Lemma 3.8) that the maximal scalar curvature of g is achieved
along a unique orbit Gp, and hence that orbit is left invariant.
We also reprise some of the notation (and setup) from the proof of Theorem 3.2: H will be
a manifold consisting of sufficiently short geodesic arcs based at p and orthogonal to the orbit Gp,
we work inside small balls H≤r therein, etc. When we want to indicate the dependence of H
on g and/or p we decorate H with those subscripts, as in Hg, Hp or, maximally, Hg,p.
From Remark 4.5 and Lemma 4.6 we know that it suffices to find metrics g′, close to g,
for which the isotropy group of some (or any) q ∈ M acts trivially on the tangent space TqM .
The isotropy group aut(g)p of p acts trivially on
� the horizontal manifold Hg,p at p and hence on every Tq(Hg,p) for q thereon,
� on the horizontal manifold Hg,q at q ∈ Hg,p, if q is sufficiently close to p, because in that
case the restricted metric onHg,q will be close to that on its diffeomorphic counterpartHg,p,
and hence will be rigid by [12, Corollary 8.2 and Proposition 8.3].1
Claim 4.7. g can always be deformed slightly so as to ensure that for q ∈ Hg,p close to p the
subspaces
Tq(Gq)
⊥ and Tq(Hg,p) ⊂ TqM (4.4)
are in general position, i.e., intersect minimally.
1The proof of [12, Proposition 8.3], asserting density, does not require that the manifold be boundary-less.
On the other hand, while the openness result [12, Corollary 8.2] is nominally proved for boundary-less manifolds
(though see [12, p. 11, footnote 3]), one can simply regard Hg,p as a subset of such a manifold: Riemannian
structures always extend from compact manifolds with boundary to compact manifolds without boundary, e.g.,
by [29, Theorem A].
Prescribed Riemannian Symmetries 15
Wrapping up assuming the claim. Since
� they always intersect at least along the line in Tq(Hg,p) tangent to the geodesic connecting p
and q, and
� we have
dimTq(Gq)
⊥ + dimTq(Hg,p)− 1 = 2(dimM − dimG)− 1 ≥ dimM
(by (4.2)), general position means that
Tq(Gq)
⊥ + Tq(Hg,p) = TqM.
In conclusion, upon performing a small deformation of g the group aut(g)p fixes q and acts
trivially on TqM , and is thus trivial. The conclusion follows, finishing the proof of the theorem.
Proof of the claim. This asserted our ability to deform g so as to have (4.4) placed in general
position. To see this, note first that for any g′ ∈MG(M) the map π : M →M/G is a Riemannian
submersion in the sense of [5, Definition 9.8] and conversely (e.g., by [5, Section 9.15]), in order
to specify a G-invariant metric on M we need to fix
� a Riemannian structure on M/G,
� smoothly-varying G-invariant Riemannian structures on the fibers (isomorphic to G) of
M →M/G,
� a G-invariant distribution H ⊂ TM (i.e., a smoothly-varying choice of subspaces Hx ⊂
TxM for x ∈M) complementary to the vertical distribution V consisting of vectors tangent
to fibers.
(Hg will then consist of the tangent vectors orthogonal to the fibers.) Correspondingly, our
desired modification of g will
� leave the already-existing Riemannian structure on M/G unaffected,
� leave the already-existing metrics on the fibers unaffected,
� alter only the horizontal distribution Hg attached to g slightly, to Hg′ .
Recall that H consists of geodesics emitted from p and orthogonal to Gp, and we chose q ∈ H
some small distance r away from p. The tangent space Tq(Hg,p) is spanned by the line tangent
to the geodesic pq and the tangent space Tq(Hg,p,r) where, consistently with the notation H≤r
above,
Hg,p,r := {x ∈ Hg,p | dg(p, x) = r}
is the radius-r sphere centered at p along H. The line tangent to the geodesic pq will always
be orthogonal to Tq(Gq) (a geodesic horizontal at one point is horizontal everywhere: [5, Lem-
ma 9.44]), but the crucial observation is that by deforming g slightly, we can
(a) keep Tq(Gq)
⊥ invariant,
(b) make Tq(Hg,p,r) sweep out an open subset of the relevant Grassmannian, hence the desired
generic-position conclusion.
To achieve these last two goals ((a) and (b)) note first that denoting as above by
π : M →M/G
the canonical projection, the geodesics p → x for p to points x ∈ Hg,p,r are the horizontal
lifts of the geodesics in M/G connecting π(p) to the points π(x) on the radius-r sphere Sπ(p),r
around it. Now choose any submanifold S of M that
16 A. Chirvasitu
� is C∞-close to Hg,p,≤2r (in particular, it is transverse to the G-orbits ≤ 2r away from Gp
and has the same dimension as Hg,p,≤2r),
� is horizontal (i.e., orthogonal to the G-orbits) along the geodesic line connecting p and q,
and
� coincides with Hg,p,≤2r off Hg,p,≤r.
We can now declare the tangent spaces to S to be horizontal (for a new metric g′ on M),
obtaining a G-invariant distribution on the tubular neighborhood
{x ∈M | dg(x,Gp) ≤ 2r}
by operating with G. Because we imposed the condition that S = Hg,p,≤2r off Hg,p,≤r, this glues
with g to obtain a globally-defined G-invariant metric g′ on M that perturbs g slightly.
With respect to g′ the new horizontal lifts of the geodesics
π(p)→ π(x) ∈ Sπ(x),r
are their lifts to S = Hg′,p,≤r (rather than the old Hg,p,≤r). Clearly, this gives us sufficient
freedom to move the tangent space Tq(Hg′,p,r) within a small neighborhood of the old Tq(Hg,p,r),
as desired. �
Acknowledgements
This work is partially supported by NSF grants DMS-1801011 and DMS-2001128. I am indebted
to the anonymous referees for numerous suggestions contributing to the improved quality of the
initial draft. In particular, I would have remained unacquainted with [3, 32] were it not for one
of the referee reports.
References
[1] Alexandrino M.M., Bettiol R.G., Lie groups and geometric aspects of isometric actions, Springer, Cham,
2015.
[2] Audin M., The topology of torus actions on symplectic manifolds, Progress in Mathematics, Vol. 93,
Birkhäuser Verlag, Basel, 1991.
[3] Bedford E., Dadok J., Bounded domains with prescribed group of automorphisms, Comment. Math. Helv.
62 (1987), 561–572.
[4] Berger M., A panoramic view of Riemannian geometry, Springer-Verlag, Berlin, 2003.
[5] Besse A.L., Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 10, Springer-
Verlag, Berlin, 1987.
[6] Bourbaki N., Elements of mathematics. General topology. Part 2, Hermann, Paris, Addison-Wesley Pub-
lishing Co., Reading, Mass. – London – Don Mills, Ont., 1966.
[7] Bredon G.E., Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46,
Academic Press, New York – London, 1972.
[8] Chavel I., Riemannian geometry – a modern introduction, Cambridge Tracts in Mathematics, Vol. 108,
Cambridge University Press, Cambridge, 1993.
[9] Chow B., Knopf D., The Ricci flow: an introduction, Mathematical Surveys and Monographs, Vol. 110,
Amer. Math. Soc., Providence, RI, 2004.
[10] do Carmo M.P., Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc.,
Boston, MA, 1992.
[11] Doignon J.P., Any finite group is the group of some binary, convex polytope, Discrete Comput. Geom. 59
(2018), 451–460, arXiv:1602.02987.
https://doi.org/10.1007/978-3-319-16613-1
https://doi.org/10.1007/978-3-0348-7221-8
https://doi.org/10.1007/BF02564462
https://doi.org/10.1007/978-3-642-18245-7
https://doi.org/10.1007/978-3-540-74311-8
https://doi.org/10.1007/978-3-540-74311-8
https://doi.org/10.1090/surv/110
https://doi.org/10.1007/s00454-017-9945-0
https://arxiv.org/abs/1602.02987
Prescribed Riemannian Symmetries 17
[12] Ebin D.G., The manifold of Riemannian metrics, in Global Analysis (Proc. Sympos. Pure Math., Vol. XV,
Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, 11–40.
[13] Engelking R., General topology, 2nd ed., Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag,
Berlin, 1989.
[14] Frucht R., Herstellung von Graphen mit vorgegebener abstrakter Gruppe, Compositio Math. 6 (1939), 239–
250.
[15] Frucht R., Graphs of degree three with a given abstract group, Canad. J. Math. 1 (1949), 365–378.
[16] Gao S., Kechris A.S., On the classification of Polish metric spaces up to isometry, Mem. Amer. Math. Soc.
161 (2003), viii+78 pages.
[17] Haefliger A., Feuilletages riemanniens, Astérisque 177–178 (1989), Exp. No. 707, 183–197.
[18] Kobayashi S., Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag,
Berlin, 1995.
[19] Kobayashi S., Nomizu K., Foundations of differential geometry, Vol. I, Wiley Classics Library, John Wiley
& Sons, Inc., New York, 1996.
[20] Koszul J.L., Sur certains groupes de transformations de Lie, in Géométrie différentielle. Colloques Interna-
tionaux du Centre National de la Recherche Scientifique, Strasbourg, 1953, Centre National de la Recherche
Scientifique, Paris, 1953, 137–141.
[21] Malicki M., Solecki S., Isometry groups of separable metric spaces, Math. Proc. Cambridge Philos. Soc. 146
(2009), 67–81.
[22] Melleray J., Compact metrizable groups are isometry groups of compact metric spaces, Proc. Amer. Math.
Soc. 136 (2008), 1451–1455, arXiv:math.GR/0505509.
[23] Montgomery D., Yang C.T., The existence of a slice, Ann. of Math. 65 (1957), 108–116.
[24] Munkres J.R., Topology, Prentice Hall, Inc., Upper Saddle River, NJ, 2000.
[25] Myers S.B., Steenrod N.E., The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939),
400–416.
[26] Narici L., Beckenstein E., Topological vector spaces, 2nd ed., Pure and Applied Mathematics (Boca Raton),
Vol. 296, CRC Press, Boca Raton, FL, 2011.
[27] Niemiec P., Isometry groups of proper metric spaces, Trans. Amer. Math. Soc. 366 (2014), 2597–2623,
arXiv:1201.5675.
[28] Petersen P., Riemannian geometry, 3rd ed., Graduate Texts in Mathematics, Vol. 171, Springer, Cham, 2016.
[29] Pigola S., Veronelli G., The smooth Riemannian extension problem, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
20 (2020), 1507–1551, arXiv:1606.08320.
[30] Sabidussi G., Graphs with given group and given graph-theoretical properties, Canad. J. Math. 9 (1957),
515–525.
[31] Sabidussi G., Graphs with given infinite group, Monatsh. Math. 64 (1960), 64–67.
[32] Saerens R., Zame W.R., The isometry groups of manifolds and the automorphism groups of domains, Trans.
Amer. Math. Soc. 301 (1987), 413–429.
[33] Schulte E., Williams G.I., Polytopes with preassigned automorphism groups, Discrete Comput. Geom. 54
(2015), 444–458, arXiv:1505.06253.
[34] Trèves F., Topological vector spaces, distributions and kernels, Dover Publications, Inc., Mineola, NY, 2006.
https://doi.org/10.4153/cjm-1949-033-6
https://doi.org/10.1090/memo/0766
https://doi.org/10.1007/978-3-642-61981-6
https://doi.org/10.1017/S0305004108001631
https://doi.org/10.1090/S0002-9939-07-08727-8
https://doi.org/10.1090/S0002-9939-07-08727-8
https://arxiv.org/abs/math.GR/0505509
https://doi.org/10.2307/1969667
https://doi.org/10.2307/1968928
https://doi.org/10.1090/S0002-9947-2013-05941-7
https://arxiv.org/abs/1201.5675
https://doi.org/10.1007/978-3-319-26654-1
https://arxiv.org/abs/1606.08320
https://doi.org/10.4153/CJM-1957-060-7
https://doi.org/10.1007/BF01319053
https://doi.org/10.2307/2000347
https://doi.org/10.2307/2000347
https://doi.org/10.1007/s00454-015-9710-1
https://arxiv.org/abs/1505.06253
1 Introduction
2 Preliminaries
3 Vertical metrics
3.1 Some remarks on the literature
4 Maximal rigidity
References
|
| id | nasplib_isofts_kiev_ua-123456789-211319 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T17:00:27Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Chirvasitu, Alexandru 2025-12-29T11:10:23Z 2021 Prescribed Riemannian Symmetries. Alexandru Chirvasitu. SIGMA 17 (2021), 030, 17 pages 1815-0659 2020 Mathematics Subject Classification: 53B20; 58D17; 58D19; 57S15 arXiv:2008.10072 https://nasplib.isofts.kiev.ua/handle/123456789/211319 https://doi.org/10.3842/SIGMA.2021.030 Given a smooth free action of a compact connected Lie group 𝐺 on a smooth compact manifold 𝑀, we show that the space of 𝐺-invariant Riemannian metrics on 𝑀 whose automorphism group is precisely 𝐺 is open and dense in the space of all 𝐺-invariant metrics, provided the dimension of 𝑀 is ''sufficiently large'' compared to that of 𝐺. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way, we also show, under less restrictive conditions on both dimensions and actions, that the space of 𝐺-invariant metrics whose automorphism groups preserve the 𝐺-orbits is dense 𝐺δ in the space of all 𝐺-invariant metrics. This work is partially supported by NSF grants DMS-1801011 and DMS-2001128. I am indebted to the anonymous referees for numerous suggestions contributing to the improved quality of the initial draft. In particular, I would have remained unacquainted with [3, 32] were it not for one of the referee reports. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Prescribed Riemannian Symmetries Article published earlier |
| spellingShingle | Prescribed Riemannian Symmetries Chirvasitu, Alexandru |
| title | Prescribed Riemannian Symmetries |
| title_full | Prescribed Riemannian Symmetries |
| title_fullStr | Prescribed Riemannian Symmetries |
| title_full_unstemmed | Prescribed Riemannian Symmetries |
| title_short | Prescribed Riemannian Symmetries |
| title_sort | prescribed riemannian symmetries |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211319 |
| work_keys_str_mv | AT chirvasitualexandru prescribedriemanniansymmetries |