Prescribed Riemannian symmetries

Given a smooth free action of a compact connected Lie group $G$ on a smooth manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ 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. 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 $G_{\delta}$ in the space of all $G$-invariant metrics.


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 [10]; • even better, a finite 3-regular graph [11,Theorems 2.4
(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 ( [12] and [16, §3]); • in the same spirit, compact metrizable groups are isometry groups of compact metric spaces [ • all connected components of M have dimension ≥ 3 + dim G then the space of G-invariant Riemannian metrics whose isometry groups leave all G-orbits invariant is a dense G δ subset of M G (M ).
• the action is free and • all connected components of M also have dimension ≥ 2 dim G + 1 then the space of G-invariant metrics on M whose isometry group is precisely G is open dense in M G (M ).
In Section 1 we gather a number of preliminary remarks of use in the sequel (on topology, Riemannian geometry, etc.). Section 2 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. [3,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 2.2, matching (a) of Theorem 0.3 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 2 serves as preparation for Section 3. In the latter we focus on maximally rigid G-invariant metrics on M . Here the main result is Theorem 3.3, corresponding to part (b) of Theorem 0.3 above.

Preliminaries
We will need some background on Riemannian geometry, as covered well in numerous sources: [14,3,6,2,20] will do for instance (as does [5, 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 • by δ ij the Kronecker delta, 1 for i = j and 0 otherwise; • by g ij the Riemannian metric tensor; • by g kℓ the inverse of g ij ; • by R ℓ ijk the curvature (3, 1)-tensor; the Ricci (2, 0)-tensor, with Einstein summation convention (i.e. summing over the repeated j in (1-1)).
• by R (unadorned) the scalar curvature, tr R ij , a function on M ; • on one occasion, by Z ij the traceless Ricci (2, 0)-tensor See for instance [3,Chapter 1] or [20,Chapter 3] for a recollection of the various notions. We also adopt the usual convention on raising and lowering indices via g kℓ and g ij respectively, for instance as in [3, §1.42]: for a tensor A j− ··· we set

···
(summation over the repeated j, as always). Analogous formulas hold for raising rather than lowering an index, with g kℓ in place of g ij . We will refer again, for instance, to the Ricci (1, 1)tensor R i j = g ik R kj (see [3,Remark 1.91]). It induces an operator on each tangent space T p M , p ∈ M of a Riemannian manifold, and that operator is self-adjoint (i.e. symmetric0 with respect to the Hilbert space structure on T p M imposed by the Riemannian metric. The symmetry, concretely, simply means that For a smooth manifold M , we follow [8] 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 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. Recall:

Definition 1.2 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.
According to the Baire category theorem ([18, Theorem 11.7.2]) complete metric spaces are Baire. Since M G (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. [9, §1.3]) F σ -subsets of a topological space are countable union of closed subsets, while G δ -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.

Vertical metrics
The following statement is an equivariant version of the result that "generically", Riemannian manifolds are rigid (i.e. have trivial isometry groups); this is [8,Proposition 8.3], which can be recovered from Theorem 2.2 by setting G = {1}. Recall Definition 1.1 above for notation. We also need the following notion (see e.g. [1, Proposition I.2.5 and the discussion following it, and Remark I.2.7]). Definition 2.1 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 G q are in the same conjugacy class as G p form a dense open subset of M ; • the action of G p on the quotient T p M/T p (Gp) of tangent spaces is trivial.

Remark 2.3
As observed in the statement of [8,Proposition 8.3], the requirement (2-1) on dimensions is necessary: when G is trivial, the circle M ∼ = S 1 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.
Proof of Theorem 2.2: the G δ claim Let orb i , i ∈ Z ≥0 be a countable set of orbits that is dense in M/G, and U n , n ∈ Z ≥0 a countable set of G-invariant open subsets of M which constitute fundamental systems of neighborhoods of the orb i . Setting is the union of the F i,n . This concludes the proof of G δ -ness.
The proof of (the rest of) Theorem 2.2 will require some preparation, in part to recall, somewhat informally, the proof strategy for [8,Proposition 8.3]. That proof proceeds as follows.
(a) An arbitrary Riemannian metric g ij on M is first perturbed slightly so that the maximum over p ∈ M of the largest eigenvalue max spec(R i j (p)) of the symmetric operator 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 R i j 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 dim M + 1 "independent" points of M . It then follows that the isometry group must be trivial (e.g. [17,Theorem 3]).
The proof of Theorem 2.2 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 2.4 Let T : R n → R n be a symmetric operator. We write • T for its norm with respect to any real Hilbert space structure on R n ; 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 constant or, equivalently, the Ricci (2, 0)-tensor R ij is a "conformal multiple" of the metric g ij : for some function f : M → R. Assuming M is connected, this is Notation 2.5 For a smooth manifold M equipped with a smooth action by a Lie group G we introduce the following notation. The symbol stands for 'non-constant Ricci', based on the fact that spr(ric(p)) precisely when the operator ric(p) : • Similarly, N ZR G (M ) (for 'non-zero') is the set of G-invariant Riemannian structures such that ric(p) = 0 over a dense set of p ∈ M.
• N ZS G (M ) (for 'non-zero scalar') is the set of G-invariant Riemannian structures such that R(p) = tr(ric(p)) = 0 over a dense set of p ∈ M.
• For a subset U ⊆ M , we write CR U G (M ) for the set of G-invariant structures for which ric(p) is constant for p ∈ U .  Since that set is closed, what we want to argue is that it has empty interior. In other words: Claim: A metric g ∈ CR U G (M ) has arbitrarily small deformations outside that set. We can see this by effecting a conformal deformation 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 ([1, Theorem I.2.1]) every point p ∈ U has a G-invariant "tubular" neighborhood contained in U , G-equivariantly diffeomorphic to G × Gp V , where • G p ⊆ G is the isotropy group at p; • V is the quotient space T p M/T p (Gp) (Gp being the orbit through p); • the G p -action on V is the differential of the G p -action on M obtained by restricting that of G.
Furthermore, it follows from [1, Proposition I.2.5] that there is a dense set of points p for which the linear action of G p on V is trivial (i.e. those lying on principal orbits in the sense of Definition 2.1). For such a p ∈ U , the tubular neighborhood G × Gp V is in fact diffeomorphic to the product manifold Gp × V . 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 nonconstant Ricci (1, 1)-tensor in U . By [3, equation (1.161b)] the conformal transformation rules for the traceless Ricci tensor (1-2) are of the form where Hess denotes the Hessian defined ([3, §1.54]) as a (2, 0)-tensor by Since dim M ≥ 3, it will be enough to choose ϕ so that that Hessian fails to be a scalar multiple of the metric g at some point in U . In normal ([3, §1.44]) local coordinates Hess(ϕ) is expressible as the familiar Hessian matrix with entries (2-5) Since (with M i ⊂ M being the component that contains p) we have by assumption, we can certainly arrange for second partial derivatives with respect to the coordinates x i on V so that the bilinear form with matrix (2-5) is not a scalar multiple of (g ij ) i,j . This proves the claim and hence the result.
Our implementation of (a) (and (b)) in the above discussion following the statement of Theorem 2.2 is Lemma 2.7 below; its proof is very much in the spirit of that of [8,Proposition 8.3].
Lemma 2.7 Let G be a compact Lie group acting smoothly and isometrically on a Riemannian manifold (M, g) with components of dimension ≥ 2 + dim G. 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 arbitrarily small G-invariant neighborhood U of Gp, and hence • so that the isometry group aut(g ′ ) leaves that orbit invariant.
Moreover, if g ∈ CR G (M ) then we can ensure g ′ ∼ = g outside the arbitrarily-small neighborhood U of Gp.
Proof By part (b) of Proposition 2.6 we can perturb g (arbitrarily) slightly so as to ensure the scalar curvature R(p) = tr ric(p) 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 required in the statement), and fix a G-invariant neighborhood U of Gp. Consider a smooth function ψ : 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 We assume r in the above discussion is small enough that ϕ is identically 1 off U . Finally, consider the G-invariant conformal rescaling g 1 := ϕ −2 g. 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 g 1 ∼ = g off U , the new metric achieves its maximal |R(q)|, q ∈ M (2-6) somewhere in U . Now repeat the procedure, as in the proof of [8, Proposition 8.3]: pick q ∈ U maximizing (2-6) for g 1 , choose a neighborhood U 1 of q less than half the size of U with respect to some fixed metric inducing the topology of M , and perturb g 1 to g 2 so that • the perturbation g 2 − g 1 is less than half the size of g 1 − g in some metric inducing the C ∞ topology on the space of Riemannian structures, • g 2 ∼ = g 1 off U 1 , and • for g 2 the maximal value of (2-6) is achieved in U 2 .
Continuing in this fashion, the limit g ′ := lim n→∞ g n will be a G-invariant metric close to g whose maximal (2-6) is achieved on a unique orbit contained in the original (arbitrarily small) neighborhood U of p. It follows that that orbit must be preserved by the isometry group of g ′ , as desired.
As for the last statement (on g ∈ CR G (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 CR G (M ).
Proof of Theorem 2.2 By passing to the components of the action in the sense of Definition 2.1, we may as well assume that the orbit space M/G is connected. Furthermore, by Lemma 2.7 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 1.1). Denoting by d g the distance induced by the metric g, for sufficiently small r > 0 the tubular neighborhood • 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 d g ≤ r from the orbit Gp (or equivalently, from p).
Horizontal geodesics are orthogonal to all G-orbits they encounter (e.g. [3,Lemma 9.44] or [13, §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 G p so as not to run into injectivity-radius issues) and keeping it invariant along the G-orbits. Explicitly, at a point q ∈ H r we can split the tangent space T q M as decompose the matrix of the Riemannian metric g correspondingly as a block matrix (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 A h 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 ([4, §III.1]) in H ≤r centered at p, with a radial coordinate and (dim H −1) "angular" coordinates, and deform the metric only along the latter.
• the perturbed metric on the sphere H r has trivial isometry group (this is possible because that sphere is at least 2-dimensional by , and hence [8,Proposition 8.3]

applies).
For the resulting G-invariant metric g ′ the manifold H consisting of horizontal geodesics emanating 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 H r identically (i.e. pointwise). But in that case • aut(g ′ ) p leaves invariant the G-orbit of every point in the tubular neighborhood GH r of Gp, • and hence so does aut(g ′ ) = G · aut(g ′ ) p .
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 ′ ).

Some remarks on the literature
The discussion above gives a brief review of the proof of [8,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 S p,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 S r (p).
The isometry group of the metric obtained after step (1) will leave p invariant, and hence also S := S r (p) (which in [8] would be denoted by A r p ). 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 now 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:

Deforming outside a ball.
The alteration of the metric "around S" (as it is phrased on [8, p.36], with A ρ q in place of S) might be interpreted as an alteration only outside the ball B r (p) bounded by S. This is possible, since the alteration in question consists of adding to the (2, 0)-tensor g another tensor whose 2 nd derivatives with respect to a system of normal coordinates satisfy certain inequalities (see [8, 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 2.2, making use of spherical coordinates.

Maximal rigidity
As indicated in the Introduction, the initial motivation for the results above was to produce Ginvariant 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 3.1 Given a faithful isometric action of a Lie group G on a Riemannian manifold M , the space M max G (M ) of maximally rigid G-invariant metrics consists of those g ∈ M G (M ) whose isometry group is precisely G.
The same notation (and terminology) applies to arbitrary (non-faithful) actions: if H G is the kernel of the action, then by definition 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 2.2 to be maximally rigid in full generality, for arbitrary compact Lie groups. Indeed, most finite groups G will fail in that respect: Example 3.2 Let G be a compact Lie group with ≥ 3 connected components G i , 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 G i × N for γ ∈ N arbitrarily. Now, if G 0 ⊂ G is the identity component, then the action of G on the set of manifolds G i × N is isomorphic (as a permutation action) to the regular action of G/G 0 . Since the latter is strictly smaller than the symmetric group S(G/G 0 ) of the set G/G 0 , we have In particular, for such G (and actions) we can never obtain G = aut(g) for a suitable Riemannian metric g.
It turns out, though, that the disconnectedness of G in Example 3.2 is the only issue: Theorem 3.3 Let G be a compact connected Lie group acting freely and smoothly on a compact smooth manifold M . Then, the following statements hold.
is open.
• If furthermore the components M i of M satisfy the dimension inequality then (3-1) is dense.
As an immediate consequence we have for some diffeomorphism σ of M . The left hand side is a subgroup of diff(M ) (group of diffeomorphisms) containing the Lie group because, g ′ being G-invariant, aut(g ′ ) contains G. Since Lie groups cannot contain proper isomorphic copies of themselves (3-3) must be an equality. It follows that so too is 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. (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.
Proof We only prove equivalence to (b), leaving the other point to the reader. For a vertical metric g ∈ M v 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 G p is trivial. We already know that γσ is in the isotropy group aut(g) p , so σ ∈ G ⇐⇒ σγ ∈ G ⇐⇒ σγ ∈ G p ⇐⇒ 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 2.2 only says that the vertical metrics form a G δ (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 ′ (H g 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 H g attached to g slightly, to H g ′ .
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 T q (H g,p ) is spanned by the line tangent to the geodesic pq and the tangent space T q (H g,p,r ) where, consistently with the notation H ≤r above, H g,p,r := {x ∈ H p,g | d g (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 T q (Gq) (a geodesic horizontal at one point is horizontal everywhere: [3, Lemma 9.44]), but the crucial observation is that by deforming g slightly, we can (a) keep T q (Gq) ⊥ invariant; (b) make T q (H g,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 ∈ H g,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 S π(p),r around it. Now choose any manifold S ⊂ H g,p,≤2r that • is C ∞ -close to H g,p,≤2r (in particular, it is transverse to the G-orbits ≤ 2r away from Gp); • is horizontal (i.e. orthogonal to the G-orbits) along the geodesic line connecting p and q, and • coincides with H g,p,≤2r off H g,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 by operating with G. Because we imposed the condition that S = H g,p,≤2r off H g,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) ∈ S π(x),r are their lifts to S = H g ′ ,p,≤r (rather than the old H g,p,≤r ). Clearly, this gives us sufficient freedom to move the tangent space T q (H g ′ ,p,r ) within a small neighborhood of the old T q (H g,p,r ), as desired.