Prescribed Riemannian Symmetries

Prescribed Riemannian Symmetries

Alexandru CHIRVASITU

A. Chirvasitu

Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA
<EMAIL_ADDRESS>
Received September 27, 2020, in final form March 10, 2021; Published online
March 25, 2021

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 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 $G_{\delta}$ in the space
of all $G$-invariant metrics.

compact Lie group; Riemannian manifold; isometry group; isometric action;
principal action; principal orbit; scalar curvature; Ricci curvature

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. (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,
Theorems 1 and 2] or isometrically [11, Theorem 1.1].

  2. (2)

More generally, arbitrary (possibly infinite) groups are graph isomorphism
groups [31, Theorem].

  3. (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].

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, Theorem 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 $\geq 2$.

The version of this result proven below (Theorems 3.2 and 4.3), involving a
$G$-action, reads as follows (with $\mathcal{M}_{G}(M)$ denoting the space of
$G$-invariant metrics on $M$, equipped with its $C^{\infty}$ topology):

######  Theorem 1.3.

Let $G$ be a compact Lie group acting smoothly on the compact manifold $M$.

  1. $(a)$

If

     * •

the action is principal $($Definition $\ref{def:princ})$, and

     * •

all connected components of $M$ have dimension $\geq 3+\dim G$

then the space of $G$-invariant Riemannian metrics whose isometry groups leave
all $G$-orbits invariant is a dense $G_{\delta}$ subset of
$\mathcal{M}_{G}(M)$.

  2. $(b)$

If furthermore

     * •

$G$ is connected,

     * •

the action is free, and

     * •

all connected components of $M$ also have dimension $\geq 2\dim G+1$

then the space of $G$-invariant metrics on $M$ whose isometry group is
precisely $G$ is open dense in $\mathcal{M}_{G}(M)$.

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 $M\to M/G$, 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 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 compact 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

$\displaystyle x^{i},\qquad 1\leq i\leq n,\qquad n:=\dim M$

we denote

  * •

by $\delta_{ij}$ the Kronecker delta, $1$ for $i=j$ and $0$ otherwise,

  * •

by $g_{ij}$ the Riemannian metric tensor,

  * •

by $g^{k\ell}$ the inverse of $g_{ij}$,

  * •

by $R_{ijk}^{\ell}$ the curvature $(3,1)$-tensor,

  * •

by

$\displaystyle R_{ik}=R_{ijk}^{j}$ (2.1)

the Ricci $(2,0)$-tensor, with Einstein summation convention (i.e., summing
over the repeated $j$ in (2.1)),

  * •

by $R$ (unadorned) the scalar curvature, $\operatorname{tr}R_{ij}$, a function
on $M$,

  * •

on one occasion, by $Z_{ij}$ the traceless Ricci $(2,0)$-tensor

$\displaystyle Z_{ij}=R_{ij}-\frac{1}{n}Rg_{ij}.$ (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
$g^{k\ell}$ and $g_{ij}$ respectively, for instance as in [5, Section 1.42]:
for a tensor $A_{\cdots}^{j-}$ we set

$\displaystyle A_{i\cdots}^{-}:=g_{ij}A_{\cdots}^{j-}$

(summation over the repeated $j$, as always). Analogous formulas hold for
raising rather than lowering an index, with $g^{k\ell}$ in place of $g_{ij}$.
We will refer again, for instance, to the Ricci $(1,1)$-tensor

$\displaystyle R^{i}_{j}=g^{ik}R_{kj}$

(see [5, Remark 1.91]). It induces an operator on each tangent space $T_{p}M$,
$p\in M$ of a Riemannian manifold, and that operator is self-adjoint (i.e.,
symmetric with respect to the Hilbert space structure on $T_{p}M$ imposed by
the Riemannian metric). The symmetry, concretely, simply means that

$\displaystyle R^{i}_{j}=R_{i}^{j}.$

“Smooth” always means $C^{\infty}$. 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
$\mathcal{M}:=\mathcal{M}(M)$ 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

$\displaystyle\mathcal{M}_{G}:=\mathcal{M}_{G}(M)$

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
$M\to M/G$ induced by a free $G$-action on $M$, with fibers $\cong G$ 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

$\displaystyle\mathcal{M}_{G}^{v}(M)\subset\mathcal{M}_{G}(M)$

the space of $G$-invariant vertical Riemannian metrics.

Note that $\mathcal{M}(M)$ is open in the Polish (i.e., separable completely
metrizable) space $\Gamma(T^{\otimes 2}M)$ of smooth sections of the tensor
square bundle $T^{\otimes 2}M$, and hence is itself Polish [34, Appendix,
Proposition A.1]. $\mathcal{M}_{G}(M)$ is also Polish, for instance because it
is closed in the Polish space $\mathcal{M}(M)$ [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. Since $\mathcal{M}_{G}(M)$ 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., [13, Section 1.3]) $F_{\sigma}$-subsets of a topological space
are countable union of closed subsets, while $G_{\delta}$-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 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.

  1. ($a$)

An orbit $Gp$ is principal if either of the two following equivalent
conditions holds:

     * •

the points $q\in M$ 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.

  2. ($b$)

The action is principal if all of its orbits are.

  3. ($c$)

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.

  4. ($d$)

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

$\displaystyle\dim M_{i}\geq 3+\dim G$ (3.1)

for every connected component $M_{i}\subset M$ then the space
$\mathcal{M}^{v}_{G}(M)$ of $G$-invariant vertical Riemannian metrics on $M$
in the sense of Definition 2.1 is a dense $G_{\delta}$-subset of
$\mathcal{M}_{G}(M)$.

######  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
$M\cong\mathbb{S}^{1}$ 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.

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 $\boldsymbol{G_{\delta}}$ claim.

Let $\textsc{orb}_{i}$, $i\in\mathbb{Z}_{\geq 0}$ be a countable set of orbits
that is dense in $M/G$, and $U_{n}$, $n\in\mathbb{Z}_{\geq 0}$ a countable set
of $G$-invariant open subsets of $M$ which constitute fundamental systems of
neighborhoods of the $\textsc{orb}_{i}$. Set

$\displaystyle\mathcal{F}_{i,n}:=\\{g\in\mathcal{M}_{G}(M)\,|\,\text{the
automorphism group }\textsc{aut}(g)\text{ moves some point of
}\textsc{orb}_{i}$ $\displaystyle\hphantom{\mathcal{F}_{i,n}:=\\{}\text{ out
of }U_{n}\\}.$

We now have

  1. $(i)$

Each $\mathcal{F}_{i,n}$ is closed. To sketch this briefly, let

$\displaystyle\mathcal{F}_{i,n}\ni
g_{\alpha}\underset{\alpha}{\longrightarrow}g\in\mathcal{M}_{G}(M)$

be a convergent net. By assumption, each $g_{\alpha}$ admits an automorphism
$\gamma_{\alpha}$ that moves some point $p_{\alpha}\in\textsc{orb}_{i}$
outside $U_{n}$. Because for large enough $\alpha$ the metrics $g_{\alpha}$
are uniformly close to $g$, so are the global geodesic metrics they induce on
$M$, and hence the union of all isometry groups of $g_{\alpha}$ (again,
assuming large $\alpha$) 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 $\\{\gamma_{\alpha}\\}$ is relatively compact, as
is $\\{\gamma_{\alpha}^{-1}\\}$ in the uniform topology on self-maps of $M$.
This implies that we can find a subnet $\gamma_{\beta}$ convergent to an
isometry $\gamma$ of $g$. Further passing to a subnet thereof if necessary, we
can furthermore assume that $p_{\beta}$ converges to some point $p$ which of
course still belongs to the closed set $\textsc{orb}_{i}$. Finally,

$\displaystyle M\setminus U_{n}\ni\gamma_{\beta}p_{\beta}\to\gamma p$

and hence this latter point will again belong to the closed set $M\setminus
U_{n}$.

  2. $(ii)$

The complement $\mathcal{M}_{G}(M)\setminus\mathcal{M}^{v}_{G}(M)$ is the
union of the $\mathcal{F}_{i,n}$.

Jointly, (i) and (ii) imply the desired $G_{\delta}$-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.

  1. $(a)$

An arbitrary Riemannian metric $g_{ij}$ on $M$ is first perturbed slightly so
that the maximum over $p\in M$ of the largest eigenvalue

$\displaystyle\max~{}\textsc{spec}(\textsc{ric}(p))$

of the symmetric operator

$\displaystyle\textsc{ric}(p):=R^{i}_{j}(p)\colon T_{p}M\to T_{p}M$

is achieved at a unique point $p$, and the perturbation is confined to an
arbitrarily small neighborhood $U$ of $p$.

  2. $(b)$

With this in hand, every isometry of $M$ with respect to the new metric will
fix that unique point $p$.

  3. $(c)$

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.

  4. $(d)$

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., [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 $T\colon\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a symmetric operator. We write

  * •

$\|T\|$ for its norm with respect to any real Hilbert space structure on
$\mathbb{R}^{n}$; it is the largest $|\lambda|$ for $\lambda$ ranging over the
spectrum $\textsc{spec}(T)$.

  * •

$\textsc{spr}(T)$ for the spread of $T$, i.e., the length of the smallest
interval containing $\textsc{spec}(T)$.

We will be interested in maximizing the norm or spread of the operators
$\textsc{ric}(p)$ instead. Note that in general, for a Riemannian manifold
$M$,

$\displaystyle\max_{p\in M}~{}\textsc{spr}(\textsc{ric}(p))=0$

precisely when each operator $\textsc{ric}(p)$ is a scalar multiple of the
identity or, equivalently, the Ricci $(2,0)$-tensor $R_{ij}$ is a “conformal
multiple” of the metric $g_{ij}$:

$\displaystyle\forall\,p\in M,\qquad R_{ij}(p)=f(p)g_{ij}(p)$ (3.2)

for some function $f\colon M\to\mathbb{R}$. Assuming $M$ is connected, this is

  * •

no restriction at all when $\dim M=2$ (i.e., it is automatic) [5, Remark
1.96(a)],

  * •

equivalent to $M$ being an Einstein manifold when $\dim M\geq 3$, 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.

  * •

$\mathcal{NSR}_{G}(M)\subset\mathcal{M}_{G}(M)$ is the set of $G$-invariant
Riemannian structures satisfying

$\displaystyle\textsc{spr}(\textsc{ric}(p))>0\qquad\text{over a dense set
of}\quad p\in M.$

The symbol stands for “non-scalar Ricci”, based on the fact that
$\textsc{spr}(\textsc{ric}(p))$ vanishes precisely when the operator
$\textsc{ric}(p)\colon T_{p}M\to T_{p}M$ is a scalar multiple of the identity.

  * •

Similarly, $\mathcal{NZR}_{G}(M)$ (for “non-zero”) is the set of $G$-invariant
Riemannian structures such that

$\displaystyle\textsc{ric}(p)\neq 0\qquad\text{over a dense set of}\quad p\in
M.$

  * •

$\mathcal{NZS}_{G}(M)$ (for “non-zero scalar”) is the set of $G$-invariant
Riemannian structures such that

$\displaystyle R(p)=\mathrm{tr}(\textsc{ric}(p))\neq 0\qquad\text{over a dense
set of}\quad p\in M.$

  * •

For a subset $U\subseteq M$, we write $\mathcal{SR}^{U}_{G}(M)$ for the set of
$G$-invariant structures for which

$\displaystyle\textsc{spr}(\textsc{ric}(p))=0,\qquad\forall\,p\in U$

(i.e., the Ricci tensor is scalar along $U$).

  * •

Finally, we set

$\displaystyle\mathcal{SR}_{G}(M):=\mathcal{SR}_{G}^{M}(M).$

######  Proposition 3.6.

Let $M$ be a smooth compact manifold with connected components $M_{i}$, and
equipped with a smooth action by a compact Lie group $G$.

  1. $(a)$

If

$\displaystyle\dim M_{i}\geq 2+\dim G,\qquad\forall\,i$

the set $\mathcal{NZR}_{G}(M)$ is residual in $\mathcal{M}_{G}(M)$ in the
sense of Definition 2.2.

  2. $(b)$

The same goes for $\mathcal{NZS}_{G}(M)$.

  3. $(c)$

If furthermore we have $\dim M_{i}\geq 3$ for all $i$ then the space
$\mathcal{NSR}_{G}(M)$ is residual in $\mathcal{M}_{G}(M)$ 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

$\displaystyle\mathcal{M}_{G}(M)\setminus\mathcal{NSR}_{G}(M)$

is the union, over all open $U\subseteq M$, of the sets
$\mathcal{SR}^{U}_{G}(M)$ 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 $\mathcal{SR}^{U}_{G}(M)$
is nowhere dense in $\mathcal{M}_{G}(M)$. Since that set is closed, what we
want to argue is that it has empty interior. In other words:

######  Claim 3.7.

A metric $g\in\mathcal{SR}^{U}_{G}(M)$ has arbitrarily small deformations
outside that set.

We can see this by effecting a conformal deformation

$\displaystyle g\mapsto g^{\prime}:=\varphi^{-2}g,$

where $\varphi$ is a strictly positive, $G$-invariant function on $M$ that is
$C^{\infty}$-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 $p\in U$ has a $G$-invariant “tubular”
neighborhood contained in $U$, $G$-equivariantly diffeomorphic to
$G\times_{G_{p}}V$, where

  * •

$G_{p}\subseteq G$ 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 [2, 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 3.1). For such a
$p\in U$ (which we henceforth fix), the tubular neighborhood
$G\times_{G_{p}}V$ is in fact diffeomorphic to the product manifold $Gp\times
V$; we frequently identify the two in the discussion below. We can then select
our scaling function $\varphi$ so that

  * •

it is identically $1$ outside some $G$-invariant neighborhood of $Gp$ whose
closure is contained in $Gp\times V$,

  * •

on $Gp\times V$ it depends only on local coordinates on $V$, and is thus
$G$-invariant.

Additionally, we have to choose $\varphi$ so as to achieve the desired outcome
that $g^{\prime}$ 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

$\displaystyle Z^{\prime}=Z+(\text{some multiple of }g)+\frac{\dim
M-2}{\varphi}\operatorname{Hess}(\varphi),$ (3.3)

where $\operatorname{Hess}$ denotes the Hessian defined [5, Section 1.54] as a
$(2,0)$-tensor by

$\displaystyle\operatorname{Hess}(\varphi)(X,Y)=X(Y\varphi)-(\nabla_{X}Y)(\varphi),$

where $\nabla$ denotes the Levi-Civita connection (denoted by the same symbol
in [19, Section IV.2] and by $D_{X}Y$ in [5, Section 1.41]).

Since $\dim M\geq 3$, it will be enough to choose $\varphi$ 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 $\operatorname{Hess}(\varphi)$ is
expressible as the familiar Hessian matrix with entries

$\displaystyle\operatorname{Hess}(\varphi)_{i,j}=\frac{\partial^{2}\varphi}{\partial
x^{i}\partial x^{j}}.$ (3.4)

Since (with $M_{i}\subset M$ being the component that contains $p$) we have

$\displaystyle\dim V=\dim M_{i}-\dim Gp\geq\dim M_{i}-\dim G\geq 2$

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 (3.4) is not a scalar multiple of $(g_{ij})_{i,j}$. This proves the
claim and hence the result.

(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 $g^{\prime}=e^{2f}g$ then the relation between the two
scalar curvatures $R^{\prime}$ (new) and $R$ (old) is

$\displaystyle R^{\prime}=e^{-2f}\left(R+2(n-1)\Delta f-(n-2)(n-1)|{\rm
d}f|^{2}\right),$

where

  * •

$n$ is the dimension of the underlying manifold,

  * •

$\Delta f$ is the Laplacian of $f$ [5, Section 1.54c],

  * •

$|{\rm d}f|$ 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 dimension $\geq 3$, whereas the latter ensures non-vanishing on
dimension-2 components, where $R_{ij}=\frac{1}{2}Rg_{ij}$. ∎

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 $\geq 2+\dim G$.
Then, there is a point $p\in M$ such that

  * •

one can find $G$-invariant metrics $g^{\prime}$ 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 $\textsc{aut}(g^{\prime})$ leaves that orbit
invariant.

Moreover, if $g\not\in\mathcal{SR}_{G}(M)$ then we can ensure $g^{\prime}=g$
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

$\displaystyle R(p)=\operatorname{tr}\textsc{ric}(p)$

is non-zero for most $p$. We retain this assumption on $g$ throughout the rest
of the proof.

Now let $p\in M$ be a point where the maximal absolute value $|R(q)|$, $q\in
M$ is achieved (it will be the point $p$ in the statement), and fix a
$G$-invariant neighborhood $U$ of $Gp$. Consider a smooth function

$\displaystyle\psi\colon\ \mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 1}$

that is

  * •

$C^{\infty}$-close to the constant function $1$,

  * •

equal to some constant slightly larger than $1$ on a small interval $[0,r]$,

  * •

equal to $1$ on $[r+\varepsilon,\infty)$.

One then obtains a smooth $G$-invariant function $\varphi$ on $M$,
$C^{\infty}$-close to $1$, by

$\displaystyle\varphi(x):=\psi(\text{distance from $x$ to the orbit
$Gp$}),\qquad\forall\,x\in M.$

We assume $r$ in the above discussion is small enough that $\varphi$ is
identically $1$ off $U$.

Finally, consider the $G$-invariant conformal rescaling
$g_{1}:=\varphi^{-2}g$. Because it scales $g$ by the constant $\psi(0)^{-2}<1$
in a neighborhood of $Gp$, it scales the operator $\textsc{ric}(p)$ (and hence
its trace) by the inverse scalar $\psi(0)^{2}>1$. Since $g_{1}\cong g$ off
$U$, the new metric achieves its maximal

$\displaystyle|R(q)|,\qquad q\in M$ (3.5)

somewhere in $U$.

Now repeat the procedure, as in the proof of [12, Proposition 8.3]: pick $q\in
U$ maximizing (3.5) 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 $C^{\infty}$ topology on the space of Riemannian
structures,

  * •

$g_{2}=g_{1}$ off $U_{1}$, and

  * •

for $g_{2}$ the maximal value of (3.5) is achieved in $U_{2}$.

Continuing in this fashion, the limit

$\displaystyle g^{\prime}:=\lim_{n\to\infty}g_{n}$

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
$g^{\prime}$, as desired.

As for the last statement (on $g\not\in\mathcal{SR}_{G}(M)$), it is clear from
the proof: the argument produces metrics identical to $g$ off $U$ after the
initial step of perturbing $g$ away from $\mathcal{SR}_{G}(M)$. ∎

###### Proof of Theorem 3.2.

By passing to the components of the action in the sense of Definition 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 $p\in M$).

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 $d_{g}$ the distance induced by the metric $g$, for sufficiently
small $r>0$ the tubular neighborhood

$\displaystyle Gp_{\leq r}:=\\{q\in M\,|\,d_{g}(q,Gp)\leq r\\}$

is (by the principality of the action) diffeomorphic to $Gp\times H_{\leq r}$,
where

  * •

$H$ is the union of the horizontal geodesics emanating from $p$, and hence a
manifold close enough to $Gp$,

  * •

$H_{\leq r}$ is, as the notation suggests, the subset of $H$ at distance
$d_{g}\leq r$ 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 $G_{p}$ so as not
to run into injectivity-radius issues) and keeping it invariant along the
$G$-orbits. Explicitly, at a point $q\in H_{r}$ we can split the tangent space
$T_{q}M$ as

$\displaystyle T_{q}M=T_{q}(Gq)\oplus T_{q}H,$

decompose the matrix of the Riemannian metric $g$ correspondingly as a block
matrix

$\displaystyle\begin{pmatrix}A_{v}&B\\\ B^{t}&A_{h}\end{pmatrix}$

(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 $\textsc{aut}(g)$ leaves $Gp$ invariant, and hence the
isotropy subgroup $\textsc{aut}(g)_{p}$ preserves every $p$-centered ball
$H_{\leq r}$ in $H$. Now choose small $r,\varepsilon>0$ and deform the metric
slightly in $H_{\leq 2r}$ so that

  * •

the perturbed metric coincides with the old metric $g$ outside $H_{\leq
r+\varepsilon}$ and inside $H_{\leq r-\varepsilon}$,

  * •

inside the annulus $H_{[r-\varepsilon,r+\varepsilon]}$ the perturbation is
spherical, in the sense that we choose geodesic spherical coordinates [8,
Section III.1] in $H_{\leq r}$ 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 (3.1), and hence
[12, Proposition 8.3] applies).

For the resulting $G$-invariant metric $g^{\prime}$ 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 $\textsc{aut}(g^{\prime})_{p}$ will then fix
$H_{r}$ identically (i.e., pointwise). But in that case

  * •

$\textsc{aut}(g^{\prime})_{p}$ leaves invariant the $G$-orbit of every point
in the tubular neighborhood $GH_{r}$ of $Gp$,

  * •

and hence so does

$\displaystyle\textsc{aut}(g^{\prime})=G\cdot\textsc{aut}(g^{\prime})_{p}.$

Note that the latter product is not direct, and ‘$G$’ is a stand-in for its
image in the automorphism group of $g^{\prime}$ (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
$\textsc{aut}(g^{\prime})_{p}$ acts trivially on the initial segments of those
geodesics it acts trivially on horizontal geodesics period, meaning that all
orbits are left invariant by $\textsc{aut}(g^{\prime})$. ∎

###  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. (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. (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 [12] would be denoted by
$A_{p}^{r}$). If ric were to achieve its maximal spectral value at a unique
point $q\in S$ 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:

Deforming outside a ball. The alteration of the metric “around $S$” (as it is
phrased on [12, p. 36], with $A_{q}^{\rho}$ 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 [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 space $\mathcal{M}_{G}^{\max}(M)$ of maximally rigid $G$-invariant
metrics consists of those $g\in\mathcal{M}_{G}(M)$ whose isometry group is
precisely $G$.

The same notation (and terminology) applies to arbitrary (non-faithful)
actions: if $H\trianglelefteq G$ is the kernel of the action, then by
definition

$\displaystyle\mathcal{M}_{G}^{\max}(M)=\mathcal{M}_{G/H}^{\max}(M).$

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 $\geq 3$ connected components $G_{i}$,
acting in the obvious fashion on $M:=G\times N$ for some manifold $N$. Then,
for any $G$-invariant Riemannian structure $g$ on $M$, the automorphism group
$\textsc{aut}(g)$ can permute the manifolds $G_{i}\times N$ for $\gamma\in N$
arbitrarily.

Now, if $G_{0}\subset G$ is the identity component, then the action of $G$ on
the set of manifolds $G_{i}\times N$ 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

$\displaystyle S(G/G_{0})\subset\textsc{aut}(g)\quad\text{but}\quad
S(G/G_{0})\not\subseteq G\subset\textsc{aut}(g).$

In particular, for such $G$ (and actions) we can never obtain
$G=\textsc{aut}(g)$ 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.

  * •

The subset

$\displaystyle\mathcal{M}^{\max}_{G}(M)\subseteq\mathcal{M}_{G}(M)$ (4.1)

is open.

  * •

If furthermore the components $M_{i}$ of $M$ satisfy the dimension inequality

$\displaystyle\dim M_{i}\geq\max(3+\dim G,2\dim G+1)$ (4.2)

then $(\ref{eq:maxrigmetrics})$ 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 $M=G\times N$ 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 $T_{p}M$.

###### Proof of Theorem 4.3: openness.

This follows from the upper semicontinuity of the automorphism group of
Riemannian structures. Let $g\in\mathcal{M}_{G}^{\max}(M)$. According to the
aforementioned semicontinuity result [12, Theorem 8.1], for
$g^{\prime}\in\mathcal{M}_{G}(M)$ sufficiently close to $g$ we have

$\displaystyle\sigma\textsc{aut}(g^{\prime})\sigma^{-1}\subseteq\textsc{aut}(g)=G$
(4.3)

for some diffeomorphism $\sigma$ of $M$. The left hand side is a subgroup of
$\textsc{diff}(M)$ (group of diffeomorphisms) containing the Lie group

$\displaystyle\sigma G\sigma^{-1}\subset\textsc{diff}(M)$

because, $g^{\prime}$ being $G$-invariant, $\textsc{aut}(g^{\prime})$ contains
$G$. Since Lie groups cannot contain proper isomorphic copies of themselves
(4.3) must be an equality. It follows that so too is

$\displaystyle G\subseteq\textsc{aut}(g^{\prime}),$

again for reasons of size: $G$ and $\textsc{aut}(g^{\prime})$ 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:

  1. $(a)$

the action of the isometry group is free, i.e., the isotropy group of every
point is trivial,

  2. $(b)$

the isotropy group of a single arbitrary point $p\in M$ is trivial.

###### Proof.

We only prove equivalence to (b), leaving the other point to the reader.

For a vertical metric $g\in\mathcal{M}_{G}^{v}(M)$ an arbitrary point $p\in M$
will be moved by every isometry $\sigma$ 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 $\gamma\in G$. Then, $\sigma$ belongs to $G$ if and only if
$\gamma\sigma$ does. Since the action is free, the isotropy group $G_{p}$ is
trivial. We already know that $\gamma\sigma$ is in the isotropy group
$\textsc{aut}(g)_{p}$, so

$\displaystyle\sigma\in G\iff\sigma\gamma\in G\iff\sigma\gamma\in
G_{p}\iff\sigma\gamma=1.$

Since, as $\sigma$ ranges over $\textsc{aut}(g)$, elements of the form
$\sigma\gamma$ range over $\textsc{aut}(g)_{p}$, 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
$G_{\delta}$ (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 $g^{\prime}$

  * •

sufficiently $C^{\infty}$-close to $g$,

  * •

coinciding with $g$ close to $Gp$

will be vertical. This is because, again as in the aforementioned proof, the
corresponding “horizontal” 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 metric 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
$H_{\leq r}$ therein, etc. When we want to indicate the dependence of $H$ on
$g$ and/or $p$ we decorate $H$ with those subscripts, as in $H_{g}$, $H_{p}$
or, maximally, $H_{g,p}$.

From Remark 4.5 and Lemma 4.6 we know that it suffices to find metrics
$g^{\prime}$, close to $g$, for which the isotropy group of some (or any)
$q\in M$ acts trivially on the tangent space $T_{q}M$. The isotropy group
$\textsc{aut}(g)_{p}$ of $p$ acts trivially on

  * •

the horizontal manifold $H_{g,p}$ at $p$ and hence on every $T_{q}(H_{g,p})$
for $q$ thereon,

  * •

on the horizontal manifold $H_{g,q}$ at $q\in H_{g,p}$, if $q$ is sufficiently
close to $p$, because in that case the restricted metric on $H_{g,q}$ will be
close to that on its diffeomorphic counterpart $H_{g,p}$, and hence will be
rigid by [12, Corollary 8.2 and Proposition 8.3].111The 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 $H_{g,p}$ 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].

######  Claim 4.7.

$g$ can always be deformed slightly so as to ensure that for $q\in H_{g,p}$
close to $p$ the subspaces

$\displaystyle T_{q}(Gq)^{\perp}\qquad\text{and}\qquad T_{q}(H_{g,p})\subset
T_{q}M$ (4.4)

are in general position, i.e., intersect minimally.

Wrapping up assuming the claim. Since

  * •

they always intersect at least along the line in $T_{q}(H_{g,p})$ tangent to
the geodesic connecting $p$ and $q$, and

  * •

we have

$\displaystyle\dim T_{q}(Gq)^{\perp}+\dim T_{q}(H_{g,p})-1=2(\dim M-\dim
G)-1\geq\dim M$

(by (4.2)), general position means that

$\displaystyle T_{q}(Gq)^{\perp}+T_{q}(H_{g,p})=T_{q}M.$

In conclusion, upon performing a small deformation of $g$ the group
$\textsc{aut}(g)_{p}$ fixes $q$ and acts trivially on $T_{q}M$, 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
$g^{\prime}\in\mathcal{M}_{G}(M)$ the map $\pi\colon M\to M/G$ 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 $M\to M/G$,

  * •

a $G$-invariant distribution $\mathcal{H}\subset TM$ (i.e., a smoothly-varying
choice of subspaces $\mathcal{H}_{x}\subset T_{x}M$ for $x\in M$)
complementary to the vertical distribution $\mathcal{V}$ consisting of vectors
tangent to fibers.

($\mathcal{H}_{g}$ 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 $\mathcal{H}_{g}$ attached to $g$
slightly, to $\mathcal{H}_{g^{\prime}}$.

Recall that $H$ consists of geodesics emitted from $p$ and orthogonal to $Gp$,
and we chose $q\in H$ 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
$H_{\leq r}$ above,

$\displaystyle H_{g,p,r}:=\\{x\in H_{g,p}\,|\,d_{g}(p,x)=r\\}$

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: [5, Lemma 9.44]), but the crucial
observation is that by deforming $g$ slightly, we can

  1. $(a)$

keep $T_{q}(Gq)^{\perp}$ invariant,

  2. $(b)$

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

$\displaystyle\pi\colon\ M\to M/G$

the canonical projection, the geodesics $p\to x$ for $p$ to points $x\in
H_{g,p,r}$ are the horizontal lifts of the geodesics in $M/G$ connecting
$\pi(p)$ to the points $\pi(x)$ on the radius-$r$ sphere $S_{\pi(p),r}$ around
it. Now choose any submanifold $S$ of $M$ that

  * •

is $C^{\infty}$-close to $H_{g,p,\leq 2r}$ (in particular, it is transverse to
the $G$-orbits $\leq 2r$ away from $Gp$ and has the same dimension as
$H_{g,p,\leq 2r}$),

  * •

is horizontal (i.e., orthogonal to the $G$-orbits) along the geodesic line
connecting $p$ and $q$, and

  * •

coincides with $H_{g,p,\leq 2r}$ off $H_{g,p,\leq r}$.

We can now declare the tangent spaces to $S$ to be horizontal (for a new
metric $g^{\prime}$ on $M$), obtaining a $G$-invariant distribution on the
tubular neighborhood

$\displaystyle\\{x\in M\,|\,d_{g}(x,Gp)\leq 2r\\}$

by operating with $G$. Because we imposed the condition that $S=H_{g,p,\leq
2r}$ off $H_{g,p,\leq r}$, this glues with $g$ to obtain a globally-defined
$G$-invariant metric $g^{\prime}$ on $M$ that perturbs $g$ slightly.

With respect to $g^{\prime}$ the new horizontal lifts of the geodesics

$\displaystyle\pi(p)\to\pi(x)\in S_{\pi(x),r}$

are their lifts to $S=H_{g^{\prime},p,\leq r}$ (rather than the old
$H_{g,p,\leq r}$). Clearly, this gives us sufficient freedom to move the
tangent space $T_{q}(H_{g^{\prime},p,r})$ within a small neighborhood of the
old $T_{q}(H_{g,p,r})$, 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.

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