We show that a negative Einstein manifold admitting a proper isometric action
of a connected unimodular Lie group with compact, possibly singular, orbit
space splits isometrically as a product of a symmetric space and a compact
negative Einstein manifold. The proof involves the theory of polar actions,
Lie-theoretic arguments and maximum principles.

###### Contents

  1. 1 Introduction
  2. 2 The Einstein equation with symmetries
  3. 3 Volume densities on Riemannian manifolds with symmetries
  4. 4 Negative Ricci curvature and unimodular symmetry
  5. 5 Einstein manifolds with semisimple symmetry
  6. 6 The Lie-theoretic structure of the orbits
  7. 7 The orbits are symmetric spaces
  8. 8 Isometric splitting
  9. 9 Ricci flat manifolds with cocompact symmetry
  10. A Proper isometric actions and polar actions
  11. B The frame bundle
  12. C Some results on semisimple Lie groups and algebras

##  1\. Introduction

Let $(M^{n},g)$ be a complete, non-compact Einstein manifold with cocompact
symmetries, meaning that a connected Lie group $\mathsf{L}$ acts properly and
isometrically on $(M^{n},g)$ with a compact orbit space. Note that $M$, and
hence also $\mathsf{L}$, must be non-compact by Bochner’s theorem [Boc48]. Our
main result is the following ‘splitting theorem’ when the isometry group is
unimodular:

######  Theorem A.

Let $(M^{n},g)$ be a complete Einstein manifold with negative scalar
curvature, and let $\mathsf{L}$ be a connected unimodular Lie group acting
cocompactly, properly and isometrically. Then, $\mathsf{L}$ is semisimple, and
there is an isometric splitting of $M$ as $B\times\mathsf{L}/\mathsf{K}$,
where $B$ is a compact Einstein manifold and $\mathsf{L}/\mathsf{K}$ is a
symmetric space of non-compact type.

A connected Lie group $\mathsf{L}$ is unimodular if and only if its left Haar
measure is also right invariant. Nilpotent, semisimple and reductive Lie
groups are unimodular. Concerning examples of compact Einstein manifold with
negative scalar curvature we refer to [Aub78, Yau78, And06, Bam12, FP20].
Moreover, let us mention that in the particular case of homogeneous manifolds
(i.e. when $\mathsf{L}$ acts transitively), A follows from the recent proof of
the Alekseevskii conjecture in [BL23].

The first conclusion in Theorem A, that $\mathsf{L}$ must be semisimple, was
known for homogeneous spaces [DM88], and more generally under the assumption
that all orbits are principal [BL23], i.e. when the orbit space
$\mathsf{L}\backslash M$ is smooth. We would like to emphasize that, in
general, the orbit space can be a non-smooth metric space, due to the presence
of singular and exceptional orbits in $M$. The second conclusion —the
isometric splitting— is new even if the orbit space is smooth.

Assuming compactness of the orbit space in Theorem A is necessary: the
horospheres in real hyperbolic space ${\mathbb{H}}^{n}$ are the orbits of a
proper, isometric, ${\mathbb{R}}^{n-1}$-action, with orbit space
${\mathbb{R}}$, but clearly they do not split isometrically. More generally,
any negative Einstein homogeneous space can be seen as the orbit of an
isometric action in an irreducible symmetric space [Jab18]. Further
inhomogeneous examples may be found in [Böh99, CP02, DW11, BDW15, CDJL21,
Win21] and [CS21, MR22, Tho23], with compact and non-compact symmetry group,
respectively.

Regarding the unimodularity assumption, we propose the following

######  Conjecture 1.1.

Let $(M^{n},g)$ be a complete negative Einstein manifold admitting a proper,
isometric, cocompact action by a connected Lie group. Then, $M^{n}$ splits
isometrically as a product of a compact Einstein manifold and a non-compact
homogeneous Einstein space.

In the Ricci flat case, from the Cheeger-Gromoll splitting theorem one can
deduce the following:

######  Theorem B.

Let $(M^{n},g)$ be a complete, Ricci flat Riemannian manifold, and let
$\mathsf{G}$ be a Lie group acting cocompactly, properly and isometrically.
Then, the universal cover splits isometrically as a Riemannian product
$\tilde{M}=\bar{M}\times{\mathbb{R}}^{k}$, with $\bar{M}$ compact.

For an effective action it also follows that
$\mathsf{G}_{0}\subset\operatorname{Isom}({\mathbb{R}}^{k})$. Concerning
examples of compact Ricci flat manifolds we refer to [Joy07]. Moreover, let us
recall that homogeneous Ricci flat manifolds are flat [AK75].

Assuming compactness of the orbit space is again necessary: already in
dimension four there exist toric gravitational instantons (i.e. Ricci flat and
with $T^{2}$-symmetry), the Riemannian Kerr family (including Schwarzschild’s
metric) and the Chen-Teo family: see [CT11, BG23] and references therein.
Concerning cohomogeneity one examples, see [DW98, Böh99, BDW15, Chi22].

Regarding the sign of the Einstein constant, in the positive case numerous
examples of non-product Einstein manifolds with symmetry are known, e.g.
[Pag78, Böh98, WW98, FH17, Wan12].

We now briefly outline the proof of A. By the Slice Theorem, there is an open
and dense subset $M_{\sf reg}\subset M$ on which the quotient map
$\pi^{\mathsf{L}}:M\to\mathsf{L}\backslash M$ is a Riemannian submersion.
Using O’Neill’s curvature formulas, the Einstein equation becomes a coupled
system of elliptic PDE’s, involving the Lie-theoretic data of the group
$\mathsf{L}$. We analise this system by applying the maximum principle to
various functions which satisfy interesting differential inequalities. Note
that due to the possible non-smoothness of the orbit space a maximum principle
can only be applied to functions which are known to admit a local maximum in
the interior (smooth) part $\pi^{\mathsf{L}}(M_{\sf reg})$ of
$\mathsf{L}\backslash M$ (and not at boundary points). We would like to stress
that this is not just a technicality but a serious issue, which we were only
able to overcome in case that $\mathsf{L}$ is unimodular: see Lemma 4.2.

As a first step, we prove in Theorem 4.1 that $\mathsf{L}$ must be semisimple,
assuming only that the Ricci curvature is negative. To overcome the non-
smoothness of the orbit space, we argue on the frame bundle $\pi:FM\to M$,
which is a principal $\mathsf{O}(n)$-bundle. The natural lifted action of
$\mathsf{L}$ on $FM$ is proper, isometric and free, thus the quotient
$\mathsf{L}\backslash FM$ is a smooth Riemannian manifold with a right
isometric and proper $\mathsf{O}(n)$-action for which $\mathsf{L}\backslash
FM/\mathsf{O}(n)\simeq\mathsf{L}\backslash M$. Arguing as in the proof of
[BL23, Thm. 7.1], we consider the so-called $\beta^{+}$-volume density, a
certain weighted determinant of the orbit metrics; if $\mathsf{L}$ is abelian,
this is precisely the relative volume density of the orbits, cf. [Ron98,
Lot20]. The crucial difference here is that the volume density must be
computed with respect to the pull-back metric $\pi^{*}g$, which is only
positive semi-definite. At present we are only able to extend the
$\beta^{+}$-volume density functional to possibly degenerate metrics, assuming
unimodularity of $\mathsf{L}$.

In a second step, we consider all possible Iwasawa decompositions
$\mathsf{L}=\mathsf{K}\mathsf{A}\mathsf{N}$ for the semisimple Lie group
$\mathsf{L}$. By applying the results of [BL23] to the action of each of the
_Borel subgroups_ $\mathsf{G}:=\mathsf{A}\mathsf{N}$, we are able to deduce
that the action of $\mathsf{L}$ is polar (i.e. the distribution
$(\ker\pi^{\mathsf{L}})^{\perp}$ is integrable). Next, we apply a
generalisation of the algebraic curvature estimates from [BL23, $\S$12] that
holds without homogeneity, and show that the (identity components of the)
minimal parabolic subgroups $\mathsf{Q}\leq\mathsf{L}$ act transitively on
$\mathsf{L}$-orbits. If $\mathsf{L}$ is split-semisimple (e.g.
$\mathsf{L}=\mathsf{SL}(n,{\mathbb{R}})$) then $\mathsf{Q}=\mathsf{G}$, and it
follows that the $\mathsf{L}$-orbits are symmetric spaces. The same is true in
general, but to prove this we perform a delicate analysis of the Lie-algebraic
structure of the orbits, and use a Bochner-type argument inspired by the work
of Jablonski and Petersen [JP17] in the homogeneous case. In the final step,
following ideas in [NT18], we apply the Bochner technique for Killing fields
restricted to a section of the $\mathsf{L}$-action, and we conclude that the
orbits are totally geodesic.

The short proof of Theorem B is just a standard application of the Cheeger-
Gromoll splitting theorem [CG72] and the Bochner technique for Killing fields:
see $\S$9.

The article is organised as follows. In $\S$2 we review some known results on
the Einstein equation in the presence of symmetries (including the main
technical results from [BL23]). In $\S$3, we introduce weighted determinants
and their corresponding volume densities on the orbits of an isometric action.
The proof of A is done step by step in $\S\S$4–8. B is proved in $\S$9.
Finally, we also include some appendices in which we review some basic
terminology about proper isometric actions and polar actions (Appendix A), the
frame bundle geometry (Appendix B) and some important properties of semisimple
Lie groups (Appendix C) .

For most of the article, the notation will be consistent with [BL23], except
for the mean curvature vector of the orbits, which we will denote by
${\operatorname{H}}$ (and not $N$). Recall that we denote shape operators by
$L_{X}$, and Lie derivatives by $\mathcal{L}_{X}$.

##  2\. The Einstein equation with symmetries

Let $\mathsf{G}$ be a connected Lie group acting properly, effectively and
isometrically on a complete Riemannian manifold $(M^{n},g)$. We refer the
reader to Appendix A for basic properties about these type of actions. The
Slice Theorem implies that the orbit space $B=\mathsf{G}\backslash M_{\sf
reg}$ corresponding to the regular part $M_{\sf reg}$ is a smooth manifold.
O’Neill’s curvature formulas yield an expression for the Ricci curvature in
vertical direcions,

(1) $\operatorname{ric}_{g}(U,U)=\operatorname{ric}^{\mathsf{G}}(U,U)+\langle
L_{{\operatorname{H}}^{\mathsf{G}}}U,U\rangle+\|AU\|^{2}-\sum_{i}\left\langle\left(\nabla_{X_{i}}L\right)_{X_{i}}U,U\right\rangle,$

in horizontal directions,

(2)
$\operatorname{ric}_{g}(X,X)=\operatorname{ric}^{B}(X,X)-2\|A_{X}\|^{2}-\|L_{X}\|^{2}+\langle\nabla_{X}{\operatorname{H}}^{\mathsf{G}},X\rangle,$

and also an off-diagonal equation which we will not use in this paper. See
[BL23, Thm. 2.2]. Regarding notation, $\\{X_{i}\\}$ is any horizontal local
orthonormal frame; $U\in\Gamma({\mathcal{V}}^{\mathsf{G}})$ (resp.
$X\in\Gamma(\mathcal{H}^{\mathsf{G}})$) denotes a vertical (resp. horizontal)
vector field; $A=A^{\mathsf{G}}$ is O’Neill’s integrability tensor; $L_{X}$
the shape operator of the orbits in the direction of $X$ and
${\operatorname{H}}^{\mathsf{G}}$ their mean curvature vector;
$\operatorname{ric}^{\mathsf{G}}$ denotes the Ricci curvature of the orbits,
$\operatorname{ric}^{B}$ that of the quotient metric on $B$; and finally, the
pointwise norms are computed via

$\|AU\|:=\sum_{i}\|A_{X_{i}}U\|^{2},\qquad\|A_{X}\|^{2}:=\sum_{i}\|A_{X}X_{i}\|^{2}=\sum_{r}\|A_{X}U_{r}\|^{2},\qquad\|L_{X}\|^{2}:=\sum_{r}\|L_{X}U_{r}\|^{2},$

where $\\{U_{r}\\}$ is any vertical orthonormal frame.

We denote by $\mathsf{N}\lhd\mathsf{G}$ the nilradical of $\mathsf{G}$, and we
use the superscript $\mathsf{N}$ to denote objects corresponding to the
Riemannian submersion associated with the induced action of $\mathsf{N}$ on
$M$:

$\pi^{\mathsf{N}}:M\to\mathsf{N}\backslash M=:P.$

Assuming that $\mathsf{N}$ acts freely and polarly, $\pi^{\mathsf{N}}$ becomes
a principal $\mathsf{N}$-bundle, and there exists a section
$\sigma:\Sigma_{\mathsf{N}}\to M$ for the $\mathsf{N}$-action with polar group
$\Pi:=\mathsf{N}_{\Sigma}$. In this context, the $\mathsf{N}$-invariant metric
$g$ on $M$ can be characterised by the following data:

  * •

a Riemannian metric $g^{P}$ on $P$;

  * •

a flat principal connection $\mathcal{H}^{\mathsf{N}}$ on
$M\to\mathsf{N}\backslash M$;

  * •

a $\Pi$-equivariant map
$h:\Sigma_{\mathsf{N}}\to\operatorname{Sym}_{+}^{2}(\mathfrak{n}^{*})$,
defined by

$s\mapsto h(s):\mathfrak{n}\times\mathfrak{n}\to{\mathbb{R}},\qquad
h(s)(U,V):=g(U^{*},V^{*})_{\sigma(s)}.$

where $U^{*},V^{*}$ denote the Kiling fields on $M$ associated to
$U,V\in\mathfrak{n}$.

In the last bullet point, the action of $\Pi\leq\mathsf{N}$ on inner products
is given by

$\rho\cdot{\langle\,\cdot\,,\cdot\,\rangle}=\left\langle\operatorname{Ad}(\rho^{-1})\,\cdot\,,\operatorname{Ad}(\rho^{-1})\,\cdot\,\right\rangle,\qquad\rho\in\Pi,\quad{\langle\,\cdot\,,\cdot\,\rangle}\in\operatorname{Sym}_{+}^{2}(\mathfrak{n}^{*}).$

By abuse of notation, we also denote by $g_{p}^{\mathsf{N}}$ the metric
induced by $g$ on the orbit $\mathsf{N}\cdot p$. For the most general version
of the above characterisation of invariant metrics (without assuming that the
action is free and polar) we refer the reader to [BH87].

######  Remark 2.1.

The _quotient metric_ $g^{P}$ is characterised by the fact that
$\pi^{\mathsf{N}}$ is a Riemannian submersion. Also, recall that after fixing
a background inner product ${\langle\,\cdot\,,\cdot\,\rangle}$ on
$\mathfrak{n}$, we have an identification
$\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})\simeq\mathsf{GL}^{+}(\mathfrak{n})/\mathsf{SO}(\mathfrak{n},{\langle\,\cdot\,,\cdot\,\rangle})$,
a symmetric space of non-compact type.

The main technical results of [BL23] essentially state the negative Einstein
equation with cocompact $\mathsf{G}$-symmetry implies that $\mathsf{N}$ acts
polarly, with Ricci soliton orbits, $h$ is a harmonic map between Riemannian
manifolds [ES64] (considering on
$\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$ the symmetric metric), and the
quotient metric satisfies an Einstein-like equation. More precisely:

######  Theorem 2.2.

[BL23] Let $(M^{n},g)$, $\mathsf{G}$ be as above, assume that $(M^{n},g)$ is
Einstein with $\operatorname{ric}_{g}=-g$, $M_{\sf reg}=M$, and that
$B=\mathsf{G}\backslash M$ is compact. Then, the induced isometric action of
the nilradical $\mathsf{N}\leq\mathsf{G}$ on $(M^{n},g)$ satisfies the
following:

  1. (i)

The action of $\mathsf{N}$ is polar (equivalently, $\mathcal{H}^{\mathsf{N}}$
is flat: $A^{\mathsf{N}}=0$);

  2. (ii)

The $\mathsf{N}$-orbits are locally isometric to nilsolitons, and we have

(3)
$\operatorname{ric}^{\mathsf{N}}=-g^{\mathsf{N}}-\tfrac{1}{2}\,\mathcal{L}_{{\operatorname{H}}^{\mathsf{N}}}\left(g^{\mathsf{N}}\right),\qquad\operatorname{scal}^{\mathsf{N}}=-\dim\mathfrak{n}+\operatorname{tr}\beta^{+},\qquad
L_{{\operatorname{H}}^{\mathsf{N}}}=-\beta^{+}.$

  3. (iii)

For any $\mathsf{N}$-section $\Sigma_{\mathsf{N}}$, the ‘orbit metrics map’
$h:\Sigma_{\mathsf{N}}\to\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})\simeq\mathsf{GL}^{+}(\mathfrak{n})/\mathsf{SO}(\mathfrak{n})$
is a harmonic map;

  4. (iv)

The Ricci curvature of the base satisfies

$\operatorname{ric}^{P}=-g^{P}+L^{*}L,$

where $L^{*}L\in\operatorname{End}(TP)$ is given by
$\langle(L^{*}L)X,Y\rangle=\operatorname{tr}(L_{X}\,L_{Y})$,
$X,Y\in\Gamma(TP)$.

If in addition $\mathsf{G}$ is a completely solvable Lie group admitting a
non-flat left-invariant Einstein metric, and acts almost-freely on $M$, then:

  1. (v)

${\operatorname{H}}^{\mathsf{N}}$ is tangent to the $\mathsf{G}$-orbits.

###### Proof.

Item i is [BL23, Thm. 8.1(i)], and Item ii follows from [BL23, Thm. 8.1(iii),
Prop. 9.7 $\&$ Cor. 9.8]. For Item iii, using the information from the
previous two items in (1) (for the $\mathsf{N}$-submersion) yields

$\sum_{i}\left(\nabla_{X_{i}}L\right)_{X_{i}}=0.$

for a local, orthonormal, $\mathsf{N}$-horizontal frame $\\{X_{i}\\}$.
Rewriting this tensorial expression in local coordinates, the claim follows
from [Lot07, Prop. 4.17]. Item iv follows from (2) using that
$A^{\mathsf{N}}=0$ and
$\nabla{\operatorname{H}}^{\mathsf{N}}|_{\mathcal{H}^{\mathsf{N}}}=0$ ([BL23,
Thm. 8.1., (ii)], cf. also [BL23, Cor. 9.9]). Finally, Item v is [BL23, Thm.
11.3]. ∎

######  Remark 2.3.

When $\mathsf{N}$ is simply connected, there exists a vertical vector field
$X_{\beta^{+}}\in\Gamma({\mathcal{V}}^{\mathsf{N}})$ which on each orbit
satisfies
$\mathcal{L}_{{\operatorname{H}}^{\mathsf{N}}}(g^{\mathsf{N}})=\mathcal{L}_{X_{\beta^{+}}}(g^{\mathsf{N}})$.
Thus, in this case, the orbits are indeed Ricci solitons (cf. [Lau01]). In
general they are only locally isometric to one, since the soliton vector field
does not commute with deck transformations [Lau11].

##  3\. Volume densities on Riemannian manifolds with symmetries

The goal of this section is to introduce a set of weighted volume densities
for the orbits of an isometric group action in a complete Riemannian manifold.
These will play a crucial role in the proof of our main results. They were
first considered in [BL18] (in the homogeneous case), and later refined in
[BL23]. Essentially, they generalise the usual Riemannian volume density
function $\sqrt{\det g_{ij}}$ by introducing weights. However, the need for
‘fixing the right gauge’ makes them very hard to define explicitly in terms of
the metric coefficients $g_{ij}$. This is illustrated by the simplest non-
trivial case in Example 3.10. Further complications arise when studying their
behavior near singular orbits, a case which was not treated in [BL23]. To
overcome these issues, we use a parametrisation of the space of inner products
(5) by lower-triangular matrices, that allows us to introduce the ‘weighted
determinant’ function.

###  3.1. The volume density of the orbits

Consider a proper isometric action of a unimodular Lie group $\mathsf{N}$ on a
Riemannian manifold $(M^{n},g)$, with discrete principal isotropy groups. A
choice of basis $\\{U_{i}\\}_{i=1}^{\dim\mathfrak{n}}$ for $\mathfrak{n}$
yields, on the principal part, a global vertical frame
$\\{U_{i}^{*}\\}_{i=1}^{\dim\mathfrak{n}}$ consisting of Killing fields. We
define the volume density of the orbits via
$v_{\mathsf{N}}:M\to{\mathbb{R}}_{>0}$,

$v_{\mathsf{N}}(p):=\sqrt{\det g_{ij}(p)},\qquad
g_{ij}(p):=g(U_{i}^{*},U_{j}^{*})_{p}.$

Notice that $v^{2}_{\mathsf{N}}$ is well-defined and smooth on all of $M$,
whereas $v_{\mathsf{N}}$ is only smooth on the regular part $M_{\sf reg}$,
i.e. where the Killing fields do not vanish. Of course $v_{\mathsf{N}}$ is
still a continuous function on all of $M$, and it vanishes precisely on
singular orbits.

######  Lemma 3.1.

If $\mathsf{N}$ is unimodular then the volume density $v_{\mathsf{N}}$ is
$\mathsf{N}$-invariant.

###### Proof.

For $q=x\cdot p$, $x\in\mathsf{N}$, we have

$(dL_{x})_{p}(U^{*})_{p}=(\operatorname{Ad}(x)U)^{*}_{q},$

see e.g. [BL22, Appendix A], thus using that $L_{x}$ is an isometry we obtain

$g_{ij}(p)=g((dL_{x})_{p}(U_{i}^{*})_{p},(dL_{x})_{p}(U_{j}^{*})_{p})=g((\operatorname{Ad}(x)U_{i})^{*},(\operatorname{Ad}(x)U_{j})^{*})_{q}.$

Unimodularity now implies that

$v_{\mathsf{N}}(p)=(\det\operatorname{Ad}(x))v_{\mathsf{N}}(q)=v_{\mathsf{N}}(q).$

∎

Notice that Lemma 3.1 does not hold if $\mathsf{N}$ is not unimodular.

We say an isometry $f\in\operatorname{Isom}(M,g)$ normalises $\mathsf{N}$, if
the image of $\mathsf{N}$ in $\operatorname{Isom}(M,g)$ under the action map
is normalised by $f$. In this case, $f_{*}$ maps $\mathsf{N}$-Killing fields
to $\mathsf{N}$-Killing fields, and hence it defines a linear map

$\operatorname{Ad}(f)|_{\mathfrak{n}}:\mathfrak{n}\to\mathfrak{n},$

where $\operatorname{Ad}$ is the adjoint representation of
$\operatorname{Isom}(M,g)$. The same argument as in Lemma 3.1 shows:

######  Lemma 3.2.

If $f\in\operatorname{Isom}(M,g)$ normalises $\mathsf{N}$ then

$v_{\mathsf{N}}(f(p))=\det\left(\operatorname{Ad}(f)|_{\mathfrak{n}}\right)^{-1}v_{\mathsf{N}}(p).$

It is well-known (see e.g. [BL23, Lemma 3.2]) that the mean curvature vector
${\operatorname{H}}^{\mathsf{N}}$ of the principal $\mathsf{N}$-orbits is
given by

(4) ${\operatorname{H}}^{\mathsf{N}}=-\nabla\log v_{\mathsf{N}}.$

It also follows from a standard computation that the first-variation of $\log
v_{\mathsf{N}}$ is

$X(\log v_{\mathsf{N}})=\operatorname{tr}L_{X},\qquad
X\in\mathcal{H}^{\mathsf{N}}.$

###  3.2. The weigthed determinant

Our goal in this section is to introduce an extension of the determinant
function on $n\times n$ real matrices. Recall that the absolute value of the
usual determinant function,
$|{\det}|:\mathsf{GL}_{n}({\mathbb{R}})\to{\mathbb{R}}^{+}$, could be
characterised as the unique Lie group homomorphism whose differential at the
identity is the trace
$A\in\mathfrak{gl}_{n}({\mathbb{R}})\mapsto\operatorname{tr}A$ (the latter is
a Lie algebra homomorphism since commutators are traceless).

Let now $W\in\mathfrak{gl}_{n}({\mathbb{R}})$ be a fixed diagonal matrix, the
‘weights’. We have the corresponding subgroups of
$\mathsf{GL}_{n}({\mathbb{R}})$:
$\mathsf{G}_{W},\mathsf{U}_{W},\mathsf{Q}_{W},\mathsf{K}_{W}$ defined in
[BL18, $\S$4] (cf. also [BL20, $\S$7], [HSS08, $\S$4]), and their Lie algebras
denoted by the corresponding gothic letters. Roughly speaking, if
$V_{1}\oplus\cdots\oplus V_{l}$ is the eigenspace decomposition of $W$,
corresponding respectively to the different eigenvalues $w_{1}<\cdots<w_{l}$,
then $\mathfrak{u}_{W}$ consists of all block-lower-triangular matrices,
$\mathsf{U}_{W}:=\exp(\mathfrak{u}_{W})$, and

$\mathsf{G}_{W}:=Z(W)=\\{h\in\mathsf{GL}_{n}({\mathbb{R}}):hW=Wh\\},\qquad\mathsf{Q}_{W}:=\mathsf{G}_{W}\mathsf{U}_{W},\qquad\mathsf{K}_{W}:=\mathsf{Q}_{W}\cap\mathsf{O}(n).$

######  Lemma 3.3.

The linear functional

$\operatorname{tr}_{W}:\mathfrak{q}_{W}\to{\mathbb{R}},\qquad
A\mapsto\operatorname{tr}_{W}(A):=\operatorname{tr}(AW),$

is a Lie algebra homomorphism.

###### Proof.

Recall that $\mathfrak{q}_{W}=\mathfrak{g}_{W}\oplus\mathfrak{u}_{W}$,
$[\mathfrak{q}_{W},\mathfrak{q}_{W}]=[\mathfrak{g}_{W},\mathfrak{g}_{W}]\oplus\mathfrak{u}_{W}$.
Since the elements of $\mathfrak{u}_{W}$ are strictly lower triangular
matrices, $\operatorname{tr}\mathfrak{u}_{W}W=0$. On the other hand, on
$\mathfrak{g}_{W}$ the result follows by applying, on each of the eigenspaces
of $W$, the fact that the commutators are traceless. ∎

######  Proposition 3.4.

There exists a unique Lie group homomorphism
$\det_{W}:\mathsf{Q}_{W}\to{\mathbb{R}}^{+}$ whose differential at the
identity is $\operatorname{tr}_{W}:\mathfrak{q}_{W}\to{\mathbb{R}}$.

###### Proof.

Let $V_{1}\oplus\cdots\oplus V_{l}$ be the eigenspace decomposition of $W$,
corresponding respectively to the different eigenvalues $w_{1}<\cdots<w_{l}$.
For $q=hu\in\mathsf{Q}_{W}$, $h\in\mathsf{G}_{W}$, $u\in\mathsf{U}_{W}$, we
define

${\det}_{W}(q):=\prod_{i=1}^{l}|{\det h_{i}}|^{w_{i}},$

where $h_{i}:=h|_{V_{i}}$. A straightforward computation shows that its
differential at the identity is indeed $\operatorname{tr}_{W}$. To see that it
is a group morphism, notice that the decomposition
$\mathsf{Q}_{W}=\mathsf{G}_{W}\mathsf{U}_{W}$ induces a Lie group isomorphism

$\mathsf{Q}_{W}/\mathsf{U}_{W}\simeq\mathsf{G}_{W}\simeq\mathsf{GL}(V_{1})\times\cdots\times\mathsf{GL}(V_{l}).$

Then $\det_{W}$ is nothing but the composition of these isomorphisms with
taking the product of the absolute value of the usual determinant function
(raised to the corresponding power) on each factor $\mathsf{GL}(V_{i})$. ∎

######  Definition 3.5.

We call $\det_{W}$ the _weighted determinant_ function with weights
$W\in\mathfrak{gl}_{n}({\mathbb{R}})$.

Notice that for any $k\in\mathsf{K}_{W}\leq\mathsf{G}_{W}$ one has
$\det_{W}(k)=1$.

######  Remark 3.6.

If we set $W={\operatorname{Id}}$ then we recover the usual determinant
function. On the other hand, for a generic $W$ with $n$ different eigenvalues
$w_{1}<\cdots<w_{n}$, we have

${\det}_{W}:\mathsf{Q}_{W}=\\{q\in\mathsf{GL}_{n}({\mathbb{R}}):q_{ij}=0\,\hbox{if
}j>i\\}\to{\mathbb{R}}^{+},\qquad{\det}_{W}(q)=\prod_{i=1}^{n}|q_{ii}|^{w_{i}}.$

###  3.3. The weighted volume density

We now extend the weighted determinant function to bilinear forms. Let
$\mathfrak{n}$ be a real vector space and consider the set
$\operatorname{Sym}^{2}_{\geq 0}(\mathfrak{n}^{*})$ of positive semi-definite
scalar products on $\mathfrak{n}$. Let us fix a background, positive-definite
scalar product $\bar{h}$ on $\mathfrak{n}$, a $\bar{h}$-orthonormal basis
$\\{\bar{e}_{i}\\}_{i=1}^{d}$, and a set of weights $W=(w_{1},\ldots,w_{d})$
with $w_{1}\leq\cdots\leq w_{d}$. Any positive definite
$h\in\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$ can be written as

(5) $h=q\cdot\bar{h}:=\bar{h}(q^{-1}\cdot,q^{-1}\cdot),\qquad
q\in\mathsf{Q}_{W}.$

It is not hard to see that the association $h\mapsto q$ can be chosen so that
it defines a smooth map

(6) $\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})\to\mathsf{Q}_{W}.$

Indeed, with respect to the fixed basis $\\{\bar{e}_{i}\\}$, we may take $q$
to be lower triangular and with positive diagonal entries (see [BL23, $\S$5]).

######  Definition 3.7.

The weighted volume density of
$h\in\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$ is defined as

$v_{W}(h):=\begin{cases}{\det}_{-W}(q),\qquad&\hbox{if }h=q\cdot\bar{h}\hbox{
is positive definite;}\\\ 0,\qquad&\hbox{otherwise}.\end{cases}$

The minus sign in $-W$ and the absence of a square root can be explained by
the fact that $q$ scales inverse quadratically with respect to $h$. We notice
that $v_{W}$ is well-defined: indeed, if
$h=q_{1}\cdot\bar{h}=q_{2}\cdot\bar{h}$ then
$q_{2}^{-1}q_{1}\in\mathsf{O}(\mathfrak{n},\bar{h})\cap\mathsf{Q}_{W}=\mathsf{K}_{W}$,
thus $\det_{-W}(q_{1})=\det_{-W}(q_{2})$.

######  Proposition 3.8.

The weighted volume density is well-defined and smooth on positive-definite
inner products. Moreover, if $W>0$ then $v_{W}$ can be continuously extended
to $\operatorname{Sym}^{2}_{\geq 0}(\mathfrak{n}^{*})$.

###### Proof.

If $h=q\cdot\bar{h}=q^{\prime}\cdot\bar{h}$ then $q^{\prime}=qk$,
$k\in\mathsf{Q}_{W}\cap\mathsf{O}(\mathfrak{n},\bar{h})=\mathsf{K}_{W}$. Since
$\det_{-W}$ is multiplicative, we have
$\det_{-W}(q^{\prime})=\det_{-W}(q)\det_{-W}(k)$, with $\det_{-W}(k)=1$
because the restriction of $k$ to each eigenspace of $W$ is orthogonal. Hence,
$v_{W}$ is well defined.

Finally, thanks to smoothness of (6) and of $\det_{-W}$, $v_{W}$ is smooth on
the open and dense subset of positive definite metrics. Thus, to extend
continuously to all of $\operatorname{Sym}^{2}_{\geq 0}(\mathfrak{n}^{*})$ we
must show that $v_{W}(h_{k})\to 0$ for any sequence
$(h_{k})_{k\in{\mathbb{N}}}$ in $\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$
with $h_{k}\to h_{\infty}\in\operatorname{Sym}^{2}_{\geq
0}(\mathfrak{n}^{*})\setminus\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$,
provided $W>0$.

To that end, write $h_{k}=q_{k}\cdot\bar{h}$, with $q_{k}=\exp(E^{(k)})$,
$E^{(k)}\in\mathfrak{q}_{W}$, lower-triangular with respect to
$\\{\bar{e}_{i}\\}$, and with diagonal entries $E^{(k)}_{ii}\in{\mathbb{R}}$,
$i=1,\ldots,\dim\mathfrak{n}$. Since

$h_{k}(\bar{e}_{i},\bar{e}_{i})=\|\exp(-E^{(k)})\bar{e}_{i}\|_{\bar{h}}^{2}=e^{-2E^{(k)}_{ii}}+\hbox{
non-negative terms}$

converges to $h_{\infty}(\bar{e}_{i},\bar{e}_{i})$ as $k\to\infty$, the
sequences of eigenvalues $\\{E^{(k)}_{ii}\\}_{k\in{\mathbb{N}}}$ are uniformly
bounded below. On the other hand, if along a subsequence all the eigenvalues
are bounded above, then the limit metric $h_{\infty}$ would be positive
definite. Hence

$\max\left\\{E^{(k)}_{ii}:1\leq
i\leq\dim\mathfrak{n}\right\\}\overset{k\to\infty}{\longrightarrow}+\infty.$

Since $w_{i}>0$ for all $i$, the above discussion yields that
$\operatorname{tr}_{W}E_{k}\to\infty$ as $k\to\infty$. Therefore,

$v_{W}(h_{k})={\det}_{-W}(q_{k})=\exp(-\operatorname{tr}E^{(k)}W)\to 0.$

∎

######  Example 3.9.

When $\mathfrak{n}=\operatorname{Lie}(\mathsf{N})$ we have that
$v_{{\operatorname{Id}}}=v_{\mathsf{N}}$. Indeed, if $\\{U_{i}^{*}\\}$ is
$\bar{h}$-orthonormal and $h=q\cdot\bar{h}$ denotes the inner product induced
by $g$ on $\mathfrak{n}$ via evaluation of Killing fields at $p$, then

$g_{ij}(p)=\bar{h}(q^{-1}U_{i},q^{-1}U_{j})=(qq^{T})^{-1}_{ij},$

thus $(\det g_{ij}(p))^{1/2}=(\det q)^{-1}$.

###  3.4. The $\beta^{+}$-weighted volume density

Let $\mathsf{N}$ be a connected nilpotent Lie group with Lie algebra
$\mathfrak{n}$. After fixing a background left-invariant metric on
$\mathsf{N}$, defined by an inner product
$\bar{h}\in\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$, consider the
associated $\bar{h}$-self-adjoint _stratum label_
$\beta\in\mathfrak{gl}(\mathfrak{n})$, chosen so that the Lie bracket
$[\cdot\,,\cdot]_{\mathfrak{n}}$ of $\mathfrak{n}$ is gauged correctly, see
[BL23, Appendix D]. We also set

$\beta^{+}:=(\operatorname{tr}\beta^{2})^{-1}\,\beta+{\operatorname{Id}}_{\mathfrak{n}},$

which is known to be positive definite since $\mathfrak{n}$ is nilpotent
[Lau10]. Applying Definition 3.7 to the set of weights $W$ given by the
eigenvalues of $\beta^{+}$ in non-decreasing order we obtain the _$\beta^{+}$
-weighted volume density_

$v_{\beta^{+}}:\operatorname{Sym}^{2}_{\geq
0}(\mathfrak{n}^{*})\to{\mathbb{R}}.$

######  Example 3.10.

Let $\mathfrak{n}$ be the $3$-dimensional Heisenberg Lie algebra, spanned by a
basis $\\{e_{1},e_{2},e_{3}\\}$ where the only non-zero bracket is
$[e_{1},e_{2}]=e_{3}$. Let $\bar{h}$ make that basis orthonormal. The stratum
label is $\beta=\operatorname{diag}(-1,-1,1)$ [Lau03] gives the correct gauge
for the Lie bracket (this follows from the fact that
$\beta^{+}\in\operatorname{Der}(\mathfrak{n})$), and we have

$\beta^{+}=\operatorname{diag}\left(\tfrac{2}{3},\tfrac{2}{3},\tfrac{4}{3}\right).$

For any other inner product $h$ on $\mathfrak{n}$ we may perform a Gram-
Schmidt orthogonalization process on the basis $\\{e_{i}\\}$, starting from
$e_{3}$ and going backwards. In this way it is not hard to see that the
entries of the matrix $q$ relating $h$ with $\bar{h}$ via
$h=\bar{h}(q^{-1}\cdot,q^{-1}\cdot)$ satisfies

$\displaystyle q_{33}$ $\displaystyle=h(e_{3},e_{3})^{-1/2},$ $\displaystyle
q_{22}$ $\displaystyle=h(u_{2},u_{2})^{-1/2},\qquad
u_{2}=e_{2}-\frac{h_{23}}{h_{33}}e_{3},$ $\displaystyle q_{11}$
$\displaystyle=h(u_{1},u_{1})^{-1/2},\qquad
u_{1}=e_{1}-\frac{h_{12}}{h_{22}}e_{2}-\frac{h_{13}}{h_{33}}e_{3}.$

Here $h_{ij}:=h(e_{i},e_{j})$. Finally, the $\beta^{+}$-weighted volume
density is given by

$v_{\beta^{+}}(h)=\left(h(u_{1},u_{1})\,h(u_{2},u_{2})\,h(e_{3},e_{3})^{2}\right)^{1/3}.$

In the particular case of a _diagonal_ metric $h$, i.e. one for which
$\\{e_{i}\\}$ is orthogonal, we have

$v_{\beta^{+}}(h)=\left(h_{11}\,h_{22}\,h_{33}^{2}\right)^{1/3}.$

######  Lemma 3.11.

If for some positive-definite inner products $h_{1},h_{2}$ there exists
$\varphi\in\operatorname{Aut}(\mathfrak{n})$ defining a linear isometry
$\varphi:(\mathfrak{n},h_{1})\to(\mathfrak{n},h_{2})$, then

$v_{\beta^{+}}(h_{2})=\det(\varphi)^{-1}\,v_{\beta^{+}}(h_{1}).$

In particular, $v_{\beta^{+}}$ is
$(\operatorname{Aut}(\mathfrak{n})\cap\mathsf{SL}(\mathfrak{n}))$-invariant.

###### Proof.

If $h_{1}=q_{1}\cdot\bar{h}$ then by assumption we have $h_{2}=(\varphi
q_{1})\cdot\bar{h}$. Thus, by definition of $v_{\beta^{+}}$ we have

$v_{\beta^{+}}(h_{2})={\det}_{-\beta^{+}}(\varphi
q_{1})={\det}_{-\beta^{+}}(\varphi)v_{\beta^{+}}(h_{1}).$

Also,

${\det}_{-\beta^{+}}(\varphi)=\left({\det}_{\beta}(\varphi)\right)^{-1/\operatorname{tr}\beta^{2}}\,{\det}_{-{\operatorname{Id}}}(\varphi)=\det(\varphi)^{-1},$

since $\operatorname{Aut}(\mathfrak{n})\subset\det_{\beta}^{-1}(1)$ [BL18,
$\S$7]. ∎

###  3.5. The weighted volume density of the orbits

In this section we finally define the weighted volume density that will play a
key role in the proof of Theorem A.

Consider a nilpotent Lie group $\mathsf{N}$ acting properly and isometrically
on a complete Riemannian manifold $(M,g)$, which we assume is also endowed
with a semi-definite metric $g_{s}$. Evaluating Killing fields, $g_{s}$ gives
rise to a smooth map

$h:M\to\operatorname{Sym}^{2}_{\geq 0}(\mathfrak{n}^{*}),\qquad p\mapsto
h_{p},\qquad h_{p}(U,V):=g_{s}(U^{*}_{p},V^{*}_{p}),\qquad
U,V\in\mathfrak{n},$

associating to each $p\in M$ a semi-definite inner product in $\mathfrak{n}$.
We define the $\beta^{+}$-weighted volume density of the $\mathsf{N}$-orbits
with respect to the semi-definite metric $g_{s}$, as the smooth function

(7) $v_{\beta^{+}}:M\to{\mathbb{R}},\qquad
v_{\beta^{+}}(p):=v_{\beta^{+}}(h_{p}),$

where the right-hand-side was defined in $\S$3.4.

######  Lemma 3.12.

Let $f\in\operatorname{Isom}(M,g)$ normalise $\mathsf{N}$ (as per the
discussion preceding Lemma 3.2). Conjugation by $f$ in $\mathsf{N}$ induces a
Lie algebra automorphism denoted by
$\operatorname{Ad}(f)|_{\mathfrak{n}}\in\operatorname{Aut}(\mathfrak{n})$.
Then,

$v_{\beta^{+}}(f(p))=\det\left(\operatorname{Ad}(f)|_{\mathfrak{n}}\right)^{-1}\,v_{\beta^{+}}(p).$

In particular, $v_{\beta^{+}}$ is $\mathsf{N}$-invariant.

###### Proof.

The inner products $h_{f(p)},h_{p}$ induced by $g_{s}$ on $\mathfrak{n}$ at
$f(p)$ and $p$, respectively, are such that

$\operatorname{Ad}(f)|_{\mathfrak{n}}:\left(\mathfrak{n},h_{p}\right)\to\big{(}\mathfrak{n},h_{f(p)}\big{)}$

is a linear isometry and an automorphism. The formula thus follows from Lemma
3.11. The last claim follows by applying the formula to $f\in\mathsf{N}$, and
the fact that $\mathsf{N}$ is unimodular. ∎

The above lemma is absolutely key for geometric applications, as it will imply
the $\mathsf{G}$-invariance of the function $v_{\beta^{+}}:M\to{\mathbb{R}}$
provided $\mathsf{N}\lhd\mathsf{G}$ is the nilradical of a unimodular Lie
group $\mathsf{G}$.

For the reader’s convenience, we connect the above definition (7) with the one
used in [BL23], and recall a useful estimate. To that end, let $M_{\sf reg}$
denote the regular part for the $\mathsf{N}$-action, and set
$P:=\mathsf{N}\backslash M_{\sf reg}$. By Lemma 3.12 we may write
$v_{\beta^{+}}={\underline{v}}_{\beta^{+}}\circ\pi^{\mathsf{N}}$, for some
smooth function ${\underline{v}}_{\beta^{+}}\in\mathcal{C}^{\infty}(P)$.

######  Lemma 3.13.

Let $v_{\beta}$ be defined as in [BL23, $\S$5]. Then,
$v_{\beta^{+}}=v_{\beta}v_{\mathsf{N}}$, and on $P$ we have

$\Delta_{P}\,\log{\underline{v}}_{\beta^{+}}\geq\sum_{j=1}^{d}\operatorname{tr}\left(\left(\nabla_{X_{j}}L\right)_{X_{j}}\cdot\beta^{+}\right),$

where $\\{X_{j}\\}$ is any local orthonormal frame for
$\mathcal{H}^{\mathsf{N}}$.

###### Proof.

As functions on $\operatorname{Sym}^{2}_{+}(\mathfrak{n}^{*})$, both
$v_{\beta^{+}}$ and $v_{\beta}v_{\mathsf{N}}$ evaluate to $1$ on the
background inner product $\bar{h}$, and their logarithmic first variations are
both given by

(8) $X(\log v_{\beta^{+}})=X(\log v_{\beta}+\log v_{\mathsf{N}})=\langle
L_{X},\beta^{+}\rangle,\qquad X\in\mathcal{H}^{\mathsf{N}},$

thanks to [BL23, Lemma 5.9] and (4). Thus,
$v_{\beta^{+}}=v_{\beta}v_{\mathsf{N}}$.

Regarding the estimate, we have
${\underline{v}}_{\beta^{+}}=\underline{v}_{\beta}\underline{v}_{\mathsf{N}}$,
where
$\underline{v}_{\beta}\circ\pi^{\mathsf{N}}=v_{\beta},\underline{v}_{\mathsf{N}}\circ\pi^{\mathsf{N}}=v_{\mathsf{N}}$.
By [BL23, Lemma 2.3] and (4) we have

$\Delta_{P}\log\underline{v}_{\mathsf{N}}=-\operatorname{div}_{P}(\pi^{\mathsf{N}}_{*}{\operatorname{H}}^{\mathsf{N}})=\sum_{j=1}^{d}\operatorname{tr}\left((\nabla_{X_{j}}L)_{X_{j}}\right).$

On the other hand, [BL23, Lemma 5.9] gives

$\Delta_{P}\,\log\underline{v}_{\beta}\geq\sum_{j=1}^{d}\operatorname{tr}\left((\nabla_{X_{j}}L)_{X_{j}}\cdot(\beta^{+}-{\operatorname{Id}}_{{\mathcal{V}}^{\mathsf{N}}}\right),$

and the claim follows from the fact that
$\Delta_{P}\log{\underline{v}}_{\beta^{+}}=\Delta_{P}\log\underline{v}_{\beta}+\Delta_{P}\log\underline{v}_{\mathsf{N}}$.
(Notice that the lemmas from [BL23] are concerned with local curvature
estimates, and thus can be applied in this more general situation provided we
are in the regular part.) ∎

##  4\. Negative Ricci curvature and unimodular symmetry

In this section our aim is to prove Theorem 4.1, which is the first step
towards A, and holds under the more general (partial) negative Ricci
assumption.

######  Theorem 4.1.

Let $\mathsf{G}$ be a connected unimodular Lie group acting cocompactly,
properly and isometrically on a Riemannian manifold $(M^{n},g)$. Assume that
the Ricci curvature of $g$ is strictly negative in directions tangent to the
$\mathsf{G}$-orbits. Then, $\mathsf{G}$ is semisimple.

Arguing by contradiction, let $(M^{n},g)$ and $\mathsf{G}$ be as in the
theorem’s statement, and assume that $\mathsf{G}$ is not semisimple. Then its
nilradical $\mathsf{N}$ has positive dimension. Consider the lifted action of
$\mathsf{N}$ on the frame bundle $FM$, which by the discussion in $\S$B.2, is
isometric and free. We endow $FM$ with the positive semi-definite metric
$\hat{g}_{s}:=\pi^{*}g$ which vanishes precisely on directions which are
tangent to $\mathsf{O}(n)$-orbits.

We now apply $\S$3.5 to the $\mathsf{N}$-action on $FM$, endowed with the
semi-definite metric $\hat{g}_{s}$. That is, after fixing a basis
$\\{e_{i}\\}$ for $\mathfrak{n}$, we consider the $\beta^{+}$-weigthed volume
density ${\hat{v}}_{\beta^{+}}:FM\to{\mathbb{R}}_{\geq 0}$ of the
$\mathsf{N}$-orbits in $FM$ with respect to $\hat{g}_{s}$, as defined in
$\S$3.5. This is a continuous function which vanishes precisely on those
frames $u\in FM_{\sf sing}$ for which $\mathsf{N}\cdot(\pi(u))\subset M$ is a
singular orbit.

######  Lemma 4.2.

The function ${\hat{v}}_{\beta^{+}}:FM_{\sf reg}\to{\mathbb{R}}_{>0}$ is
smooth, $\mathsf{G}$-invariant and $\mathsf{O}(n)$-invariant.

###### Proof.

Smoothness follows from Proposition 3.8 and the definition of
${\hat{v}}_{\beta^{+}}$.

Since the nilradical is normal, any isometry $f\in\mathsf{G}$ normalises
$\mathsf{N}$. Moreover, by unimodularity of $\mathsf{G}$ it follows that
$\det\operatorname{Ad}(f)|_{\mathfrak{n}}=1$ (see [LL14a, Lemma 2.6]). Hence,
${\hat{v}}_{\beta^{+}}$ is $\mathsf{G}$-invariant by Lemma 3.12.

Regarding $\mathsf{O}(n)$-invariance, let $r\in\mathsf{O}(n)$ and denote by
$R_{r}\in\operatorname{Isom}(FM,\hat{g}_{s})$, $u\mapsto ur$, the
corresponding isometry. Since the actions of $\mathsf{N}$ and $\mathsf{O}(n)$
commute, the corresponding subgroups of $\operatorname{Isom}(FM,\hat{g}_{s})$
commute as well. Hence, in the notation of Lemma 3.12 we have
$\operatorname{Ad}(R_{r})|_{\mathfrak{n}}={\operatorname{Id}}_{\mathfrak{n}}$
and thus ${\hat{v}}_{\beta^{+}}(ur)={\hat{v}}_{\beta^{+}}(u)$. ∎

By $\mathsf{O}(n)$-invariance, $\log{\hat{v}}_{\beta^{+}}:FM_{\sf
reg}\to{\mathbb{R}}$ induces a smooth function $\log v_{\beta^{+}}:M_{\sf
reg}\to{\mathbb{R}}$, which satisfies the following estimate:

######  Lemma 4.3.

On the regular part $M_{\sf reg}$, $\log v_{\beta^{+}}$ satisfies the
differential inequality

$\Delta_{M}\log
v_{\beta^{+}}+\langle\operatorname{Ric}_{M}|_{{\mathcal{V}}^{\mathsf{N}}},\beta^{+}\rangle\geq
0.$

###### Proof.

Lemma 3.13 gives that $v_{\beta^{+}}=v_{\beta}v_{\mathsf{N}}$ on $M_{\sf
reg}$. Thus, using the Riemannian submersion

$\pi^{\mathsf{N}}:M_{\sf reg}\to\mathsf{N}\backslash M_{\sf reg}=:P,$

we get

$\displaystyle\Delta_{M}\log v_{\beta^{+}}$
$\displaystyle=\Delta_{P}\log{\underline{v}}_{\beta^{+}}-g^{P}\left(\nabla\log{\underline{v}}_{\beta^{+}},\,\pi^{\mathsf{N}}_{*}\,{\operatorname{H}}^{\mathsf{N}}\right)$
$\displaystyle\geq\sum_{j=1}^{d}\operatorname{tr}\left(\left(\nabla_{X_{j}}L\right)_{X_{j}}\cdot\beta^{+}\right)-\operatorname{tr}(\beta^{+}L_{{\operatorname{H}}^{\mathsf{N}}})$
$\displaystyle=\langle\operatorname{Ric}^{\mathsf{N}}+A^{*}A,\beta^{+}\rangle-\langle\operatorname{Ric}_{M}|_{{\mathcal{V}}^{\mathsf{N}}},\beta^{+}\rangle$
$\displaystyle\geq-\langle\operatorname{Ric}_{M}|_{{\mathcal{V}}^{\mathsf{N}}},\beta^{+}\rangle.$

Here ${\underline{v}}_{\beta^{+}}\circ\pi^{\mathsf{N}}=v_{\beta^{+}}$,
$\underline{v}_{\beta}\circ\pi^{\mathsf{N}}=v_{\beta}$,
$\underline{v}_{\mathsf{N}}\circ\pi^{\mathsf{N}}=v_{\mathsf{N}}$. The first
equality is simply the formula for the divergence of invariant vector fields
under an isometric group action [BL23, Lemma B.3]; the estimate in the second
line uses Lemma 3.13 and (8); the third line is just (1); and the last
estimate follows from $A^{*}A\geq 0$, $\beta^{+}>0$, and the GIT Ricci
curvature estimate
$\langle\operatorname{Ric}^{\mathsf{N}},\beta^{+}\rangle\geq 0$. ∎

###### Proof of Theorem 4.1.

Assuming that the nilradical of $\mathsf{G}$ has positive dimension, we may
define the smooth function $\log{\hat{v}}_{\beta^{+}}:FM_{\sf
reg}\to{\mathbb{R}}$ as above, which by construction approaches $-\infty$ as
we approach $FM_{\sf sing}$. Since $\mathsf{G}\backslash FM$ is compact and
$\log{\hat{v}}_{\beta^{+}}$ is $\mathsf{G}$-invariant (Lemma 4.2),
$\log{\hat{v}}_{\beta^{+}}$ attains a global maximum somewhere in the non-
singular part $FM_{\sf reg}$. By $\mathsf{O}(n)$-invariance, this means that
$\log v_{\beta^{+}}$ also attains a maximum on $M_{\sf reg}$. Using Lemma 4.3
at that point and the fact that $\beta^{+}>0$ and $\operatorname{Ric}_{M}<0$
along orbits, we obtain a contradiction. Hence the nilradical must be trivial
and $\mathsf{G}$ is semisimple. ∎

##  5\. Einstein manifolds with semisimple symmetry

In this section we continue working towards a proof of Theorem A. The main
result is Proposition 5.4. From now on and for the rest of the paper, unless
otherwise stated, we assume that $(M^{n},g)$ is a complete Riemannian manifold
with $\operatorname{ric}_{g}=-g$, on which a connected unimodular Lie group
$\mathsf{L}$ acts properly, isometrically and cocompactly. After taking a
quotienti by the ineffective kernel —a normal subgroup—, we may and will also
assume that the action is effective. By Theorem 4.1, we know that $\mathsf{L}$
is semisimple. Moreover, if we write
$\mathsf{L}=\mathsf{L}_{c}\mathsf{L}_{nc}$, where $\mathsf{L}_{c}$ (resp.
$\mathsf{L}_{nc}$) is the product of all normal compact (resp. non-compact)
simple factors, then the action of $\mathsf{L}_{nc}$ on $(M^{n},g)$ is also
proper, isometric and cocompact. Thus, we may assume that
$\mathsf{L}=\mathsf{L}_{nc}$.

######  Lemma 5.1.

The center $\mathsf{Z}$ of $\mathsf{L}$ has a finite-index subgroup
$\mathsf{Z}_{f}\leq\mathsf{Z}$ acting freely on $(M^{n},g)$. In particular,
$M/{\mathsf{Z}_{f}}$ is locally isometric to $M$ and admits a proper,
isometric and cocompact action of the semisimple Lie group
$\mathsf{L}/{\mathsf{Z}_{f}}$, which has finite center.

###### Proof.

By an old result of P.A. Smith [Smi35] (see Proposition C.1), $\mathsf{Z}$ is
a finitely-generated abelian group. Let
$\mathsf{Z}=\mathsf{Z}_{f}\mathsf{Z}_{t}$ be the decomposition into its free
abelian part $\mathsf{Z}_{f}$ and the subgroup consisting of torsion elements
$\mathsf{Z}_{t}$. We claim that $\mathsf{Z}_{f}\leq\mathsf{L}$ acts freely on
$M^{n}$. To see that, recall that for each $p\in M$ the isotropy group
$\mathsf{L}_{p}$ is compact by properness. Hence,
$\mathsf{Z}_{f}\cap\mathsf{L}_{p}$ is compact and discrete, thus finite. Since
$\mathsf{Z}_{f}$ does not contain any non-trivial finite subgroups, it follows
that $\mathsf{Z}_{f}\cap\mathsf{L}_{p}=1$, that is, $\mathsf{Z}_{f}$ acts
freely. Clearly, the quotient group $\mathsf{L}/\mathsf{Z}_{f}$ acts freely,
isometrically and cocompactly on $M/\mathsf{Z}_{f}$, the latter endowed with
the unique metric so that the covering map $M\to M/\mathsf{Z}_{f}$ is a local
isometry. Finally, the center of $\mathsf{L}/\mathsf{Z}_{f}$ is isomorphic to
$\mathsf{Z}_{t}$, and is in particular finite. ∎

In summary, throughout the following sections we may and will assume that

######  Assumption 5.2.

$\mathsf{L}$ is a connected, semisimple Lie group without compact simple
factors and with finite center, acting properly, isometrically and cocompactly
on a complete Einstein manifold $(M^{n},g)$ with $\operatorname{ric}_{g}=-g$.

Consider now any Iwasawa decomposition

$\mathsf{L}=\mathsf{K}\mathsf{A}\mathsf{N},$

with corresponding decomposition
$\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}$ on Lie algebra
level. Recall that the latter is obtained from a Cartan involution
$\theta\in\operatorname{Aut}(\mathfrak{l})$ as follows: the $+1$ and $-1$
eigenspaces for $\theta$ give rise to a Cartan decomposition
$\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{p}$. One then takes a maximal
abelian subspace $\mathfrak{a}\subset\mathfrak{p}$, and after choosing a
notion of positivity for the restricted roots in $\mathfrak{a}^{*}$,
$\mathfrak{n}$ is the direct sum of all positive root spaces.

Since $\mathsf{L}$ has finite center, $\mathsf{K}$ is a maximal compact
subgroup of $\mathsf{L}$ [Kna02, Thm. 6.31]. The Lie subgroup
$\mathsf{G}:=\mathsf{A}\mathsf{N}\leq\mathsf{L}$ is solvable, simply-
connected, and we have a diffeomorphism
$\mathsf{L}=\mathsf{K}\mathsf{G}\simeq\mathsf{K}\times\mathsf{G}$; in
particular $\mathsf{G}$ is closed in $\mathsf{L}$ and
$\mathsf{G}\cap\mathsf{K}=1$. By analogy with the complex case, we call
$\mathsf{G}$ a _Borel subgroup_. (More precisely, $\mathsf{G}$ is simply the
radical of a minimal parabolic subgroup.) Let
$\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n}$ be the
corresponding decomposition on Lie algebra level. We say
$\mathfrak{g}:=\operatorname{Lie}(\mathsf{G})=\mathfrak{a}\oplus\mathfrak{n}$
is a Borel subalgebra of $\mathfrak{l}$, and $\mathfrak{n}$ (and also
$\mathsf{N}$) a _Borel nilradical_.

We denote by $\mathfrak{n}^{-}:=\theta(\mathfrak{n})$ the Borel nilradical
corresponding to the opposite choice of positive roots. Recall that
$\mathfrak{l}=\mathfrak{n}^{-}\oplus\mathfrak{m}\oplus\mathfrak{a}\oplus\mathfrak{n}$,
where $\mathfrak{m}=Z_{\mathfrak{k}}(\mathfrak{a})$. When $\mathfrak{m}=0$ the
Lie algebra $\mathfrak{l}$ is called split-semisimple (e.g.
$\mathfrak{l}=\mathfrak{sl}(n,{\mathbb{R}})$), however this is far from true
in general (e.g. for $\mathfrak{l}=\mathfrak{so}(2,n)$,
$\mathfrak{m}\simeq\mathfrak{so}(n-1)$).

######  Lemma 5.3.

The Borel subgroup $\mathsf{G}\leq\mathsf{L}$ acts on $(M,g)$ freely,
properly, isometrically and cocompactly.

###### Proof.

Since $\mathsf{L}$ acts effectively and isometrically, the same is true for
$\mathsf{G}$. Also, $\mathsf{G}\leq\mathsf{L}$ is closed, thus properness of
the $\mathsf{L}$-action implies that of $\mathsf{G}$ by Lemma A.1. Moreover,
$\mathsf{G}$ is solvable and simply-connected, hence it has no nontrivial
compact subgroups. It follows that $\mathsf{G}$ must act freely, and that
$\mathsf{G}\backslash M$ is a manifold.

Consider now the continuous and surjective map

$\mathsf{G}\backslash M\to\mathsf{L}\backslash M,\qquad\mathsf{G}\cdot
p\mapsto\mathsf{L}\cdot p.$

We claim that its fibers are the compact sets $\\{\mathsf{G}\cdot(k\cdot
p):k\in K\\}\subset\mathsf{G}\backslash M$. Indeed, since
$\mathsf{L}=\mathsf{K}\mathsf{G}$, two $\mathsf{L}$-orbits $\mathsf{L}\cdot
p$, $\mathsf{L}\cdot q$ coincide if and only if $p=ks\cdot q$ for some
$k\in\mathsf{K},s\in\mathsf{G}$, and the latter is in turn equivalent to
$\mathsf{G}\cdot(k^{-1}\cdot p)=\mathsf{G}\cdot q$. Compactness of
$\mathsf{G}\backslash M$ now easily follows from this claim and the
compactness of $\mathsf{L}\backslash M$. ∎

Combining this with the results from [BL23] and a Lie theoretic property of
semisimple Lie algebras (Appendix C), we deduce the following:

######  Proposition 5.4.

Let $(M^{n},g)$, $\mathsf{L}$ be as in $\S$5. Then, the action of $\mathsf{L}$
is polar.

###### Proof.

As remarked above, it suffices to show that

(9) $\langle\nabla_{X}U^{*},Y\rangle=0,$

for all $X,Y\in\Gamma(\mathcal{H}^{\mathsf{L}})$ and all $U\in\mathfrak{l}$.
By Lemma 5.3 we may apply Theorem 2.2, i and deduce the vanishing of the
integrability tensor $A^{\mathsf{N}}$ of the Riemannian submersion
$M\to\mathsf{N}\backslash M$. In other words, (9) holds for all
$X,Y\in\Gamma(\mathcal{H}^{\mathsf{N}})$ and all $U\in\mathfrak{n}$. Since
$\mathcal{H}^{\mathsf{L}}\subset\mathcal{H}^{\mathsf{N}}$, (9) holds for all
$X,Y\in\Gamma(\mathcal{H}^{\mathsf{L}})$ and all $U\in\mathfrak{n}$.

By varying the Iwasawa decomposition and applying Theorem 2.2, i to the action
of other Borel subgroups $\mathsf{G}$, we obtain that (9) holds for all $U$ in
_some_ Borel nilradical of $\mathfrak{l}$. Since (9) is linear in $U$, the
proposition follows from the fact that the Borel nilradicals linearly span all
of $\mathfrak{l}$ (Proposition C.3). ∎

From Lemma 5.3 we also obtain all the structure results for the Riemannian
submersion $M\to\mathsf{N}\backslash M$ arising from [BL23] and listed in
Theorem 2.2.

##  6\. The Lie-theoretic structure of the orbits

In this section our first goal is to prove Proposition 6.1, which is key for
our purposes, and to discuss a number of very important consequences of it.
Recall that if $\mathsf{M}:=N_{\mathsf{K}}(\mathfrak{a})$ denotes the
normaliser of $\mathfrak{a}$ in $\mathsf{K}$, one defines the (identity
component of the) _minimal parabolic subgroup_ via

(10) $\mathsf{Q}:=(\mathsf{M}\mathsf{A}\mathsf{N})_{0}.$

For details about these subgroups see [Kna02, Ch.VII, $\S$7].

######  Proposition 6.1.

Let $(M^{n},g),\mathsf{L}$ be as in $\S$5 and let $\mathsf{Q}$ be defined as
in (10), for any choice of Iwasawa decomposition. Then, $\mathsf{Q}$ acts
transitively on all $\mathsf{L}$-orbits: that is,
$\mathsf{Q}\mathsf{L}_{p}=\mathsf{L}$ for any $p\in M$.

We introduce the following ‘partial trace’ of a vertical endomorphism
$E\in\operatorname{End}({\mathcal{V}}^{\mathsf{L}})$:

$\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}E(p):=\sum_{r}g(EU_{r},U_{r})_{p}.$

Here, $\\{U_{r}\\}_{r=1}^{\dim\mathfrak{n}}$ is any vertical frame for
${\mathcal{V}}^{\mathsf{N}}$ which is orthonormal at $p$ (it is easy to see
that the definition is independent of such frame). The proof of Proposition
6.1 will also yield:

######  Corollary 6.2.

If $p\in M$ satisfies the condition (11) below, then
$\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}_{p}^{\mathsf{L}}=-\dim\mathfrak{n}$.

We now start working towards proving Proposition 6.1. Consider any identity
component of a minimal parabolic
$\mathsf{Q}=\mathsf{M}\mathsf{A}\mathsf{N}\leq\mathsf{L}$ associated to an
Iwasawa decomposition $\mathsf{L}=\mathsf{K}\mathsf{A}\mathsf{N}$, and let
$\mathsf{L}\cdot p^{\prime}\subset M$ be a principal $\mathsf{L}$-orbit.
Recall that, due to the isometric action of $\mathsf{L}$ on $M$, we may
identify any $X\in\mathfrak{l}$ with the Killing field $X^{*}\in\Gamma(TM)$,
and for $\mathfrak{s}\subset\mathfrak{l}$ we set

$\mathfrak{s}\cdot
p:=\\{X^{*}_{p}:X\in\mathfrak{s}\\}\subset{\mathcal{V}}^{\mathsf{L}}(p)=T_{p}(\mathsf{L}\cdot
p)\subset T_{p}M.$

We now show that we can change the point $p^{\prime}$ to another one on the
same $\mathsf{L}$-orbit at which the algebra and the geometry are compatible:

######  Lemma 6.3.

There exists $p\in\mathsf{N}\cdot p^{\prime}\subset\mathsf{L}\cdot p^{\prime}$
such that

(11) $\mathfrak{a}\cdot p\perp\mathfrak{n}\cdot p.$

Moreover, if $\sigma:\Sigma_{\mathsf{N}}\to M$ is a section through $p$ for
the polar action of $\mathsf{N}$, then (11) also holds along
$\sigma(\Sigma_{\mathsf{N}})$.

###### Proof.

Polarity of the $\mathsf{N}$-action (Theorem 2.2, i) implies that the simply-
connected solvmanifold $\mathsf{G}\cdot p^{\prime}$ is _standard_ , in the
sense of [Heb98]: that is, the subspace of Killing fields

$\mathfrak{a}_{p^{\prime}}:=\\{X\in\mathfrak{g}:X^{*}_{p^{\prime}}\perp\mathfrak{n}\cdot
p^{\prime}\\}$

is an abelian Lie subalgebra of $\mathfrak{g}$ complementary to
$\mathfrak{n}$. By [Heb98, Cor. 2.10] (which may be used since $\mathsf{G}$
comes from an Iwasawa decomposition), $\operatorname{Ad}(\mathsf{N})$ acts
transitively on the set of such abelian complements, thus
$\mathfrak{a}=\operatorname{Ad}_{x}(\mathfrak{a}_{p^{\prime}})$ for some
$x\in\mathsf{N}$. Then, for $p=x\cdot p^{\prime}$, we have
$\mathfrak{a}_{p}=\operatorname{Ad}_{x}(\mathfrak{a}_{p^{\prime}})=\mathfrak{a}$,
as desired.

For the second claim, let $a\in\mathsf{A}$, let $L_{a}:M\to M$ denote the
corresponding isometry, and set $q:=a\cdot p$. Since $\mathsf{A}$ normalises
$\mathsf{N}$, $a\cdot(\mathsf{N}\cdot p)=(a\mathsf{N}a^{-1})\cdot(a\cdot
p)=\mathsf{N}\cdot q$, thus $\mathsf{A}$ maps $\mathsf{N}$-orbits to
$\mathsf{N}$-orbits. In particular, $(L_{a})_{*}$ maps
${\mathcal{V}}^{\mathsf{N}}(p)$ to ${\mathcal{V}}^{\mathsf{N}}(q)$ and
$\mathcal{H}^{\mathsf{N}}(p)$ to $\mathcal{H}^{\mathsf{N}}(q)$. Since
$\mathfrak{a}\cdot p\subset\mathcal{H}^{\mathsf{N}}(p)$ and
$(L_{a})_{*}(\mathfrak{a}\cdot p)=\mathfrak{a}\cdot q$, we deduce that
$\mathfrak{a}\cdot q\subset\mathcal{H}^{\mathsf{N}}(q)$. This implies that
$\mathsf{A}\cdot p\subset\sigma(\Sigma_{\mathsf{N}})$.

We now claim that $\Sigma_{\mathsf{N}}$ is invariant under $\mathsf{A}$, from
which the second claim follows immediately. Let
$\gamma:I\to\sigma(\Sigma_{\mathsf{N}})\subset M$ be a geodesic with
$\gamma(0)=p$. For any $a\in\mathsf{A}$, $a\cdot\gamma$ is a geodesic through
$q\in\sigma(\Sigma_{\mathsf{N}})$, with velocity
$(a\cdot\gamma)^{\prime}(0)=(L_{a})_{*}(\gamma^{\prime}(0))\in\mathcal{H}^{\mathsf{N}}(q)$.
Hence, $a\cdot\gamma$ is a horizontal geodesic. It follows that
$\sigma(\Sigma_{\mathsf{N}})$ is $\mathsf{A}$-invariant. ∎

By Proposition 5.4, through each $p\in M$ there exists a _section_
$\sigma:\Sigma_{\mathsf{L}}\to M$ for the $\mathsf{L}$-action: an (immersed)
submanifold intersecting all $\mathsf{L}$-orbits orthogonally [HLO06, GZ12].
It is well-known that the section is a totally geodesic submanifold, hence, by
Lemma 6.3, Equation 11 is satisfied for all points on
$\sigma(\Sigma_{\mathsf{L}})$, provided it holds for $p$. We endow
$\Sigma_{\mathsf{L}}$ with the Riemannian metric $\sigma^{*}g$.

Consider the smooth function

$\log v_{\mathsf{N}}\circ\sigma:\Sigma_{\mathsf{L}}\to{\mathbb{R}},$

where $v_{\mathsf{N}}$ is the relative volume density of the
$\mathsf{N}$-orbits in $(M^{n},g)$ (see $\S$3.1). To prove Proposition 6.1, we
will establish a formula for the Laplacian of $\log v_{\mathsf{N}}\circ\sigma$
and an algebraic estimate for the zeroth-order term in said formula. The
potential non-compactness of $\Sigma_{\mathsf{L}}$ plays no role here, since
$\log v_{\mathsf{N}}\circ\sigma$ turns out to be constant by Theorem 2.2, Item
v.

######  Lemma 6.4.

The function $\log
v_{\mathsf{N}}\circ\sigma:\Sigma_{\mathsf{L}}\to{\mathbb{R}}$ satisfies the
following PDE on $\Sigma_{\mathsf{L}}$:

(12) $\Delta_{\Sigma_{\mathsf{L}}}(\log
v_{\mathsf{N}}\circ\sigma)-\big{\langle}{\operatorname{H}}^{\mathsf{L}},\nabla(\log
v_{\mathsf{N}}\circ\sigma)\big{\rangle}=\dim\mathfrak{n}+\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}^{\mathsf{L}}.$

###### Proof.

Since $\sigma$ is a local isometry and the computation is local, we may
perform it on $\sigma(\Sigma_{\mathsf{L}})\subset M$. Let
$p\in\sigma(\Sigma_{\mathsf{L}})$, let $\\{U_{r}\\}$ be a local orthonormal
frame for ${\mathcal{V}}^{\mathsf{N}}$, and choose a local orthonormal frame
$\\{X_{j}\\}$ for $\mathcal{H}^{\mathsf{L}}$ such that
$\nabla_{X_{j}}{X_{j}}\big{|}_{p}=0$ for all $j$. We compute:

$\displaystyle\Delta_{\sigma(\Sigma_{\mathsf{L}})}\log v_{\mathsf{N}}$
$\displaystyle=-\sum_{j}\langle\nabla_{X_{j}}{\operatorname{H}}^{\mathsf{N}},X_{j}\rangle=-\sum_{j,r}X_{j}\langle\nabla_{U_{r}}U_{r},X_{j}\rangle=\sum_{j,r}X_{j}\langle
L_{X_{j}}U_{r},U_{r}\rangle$
$\displaystyle=\sum_{j,r}\left(\langle(\nabla_{X_{j}}L)_{X_{j}}U_{r},U_{r}\rangle+\langle
L_{X_{j}}\nabla_{X_{j}}U_{r},U_{r}\rangle+\langle
L_{X_{j}}U_{r},\nabla_{X_{j}}U_{r}\rangle\right)$
$\displaystyle=\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\sum_{j}(\nabla_{X_{j}}L)_{X_{j}}+2\,\sum_{j,r,s}\langle
L_{X_{j}}U_{r},U_{s}\rangle\,\langle\nabla_{X_{j}}U_{r},U_{s}\rangle$
$\displaystyle=\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\sum_{j}(\nabla_{X_{j}}L)_{X_{j}},$

where $L_{X_{j}}$ are the shape operators of the $\mathsf{L}$-orbits. Notice
that in the first equality we should have used the
$\mathcal{H}^{\mathsf{L}}$-component of $\nabla\log
v_{\mathsf{N}}=-{\operatorname{H}}^{\mathsf{N}}$, but we may instead compute
with $-{\operatorname{H}}^{\mathsf{N}}$ thanks to $A^{\mathsf{L}}=0$. In the
second-to-last equality we have used that the $L_{X_{j}}$ preserve the
distribution ${\mathcal{V}}^{\mathsf{N}}$, because the $\mathsf{N}$-action is
polar (Theorem 2.2). Regarding the last equality, the sum over $j,r,s$
vanishes since the first factor is symmetric in $r,s$ while the second one is
alternating (notice also that $\nabla_{X_{j}}U_{r}$ is $\mathsf{L}$-vertical
due to Proposition 5.4). Also,

$-\langle{\operatorname{H}}^{\mathsf{L}},\nabla\log
v_{\mathsf{N}}\rangle=\langle{\operatorname{H}}^{\mathsf{L}},{\operatorname{H}}^{\mathsf{N}}\rangle=\sum_{r}\langle{\operatorname{H}}^{\mathsf{L}},\nabla_{U_{r}}U_{r}\rangle=-\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}L_{{\operatorname{H}}^{\mathsf{L}}}.$

Finally, since $A^{\mathsf{L}}=0$ by Proposition 5.4, the vertical part of the
Einstein equation for the $\mathsf{L}$-submersion (1) (in endomorphism form)
yields

$-{\operatorname{Id}}_{{\mathcal{V}}^{\mathsf{L}}}=\operatorname{Ric}^{\mathsf{L}}+L_{{\operatorname{H}}^{\mathsf{L}}}-\sum_{j}(\nabla_{X_{j}}L)_{X_{j}}.$

Tracing over ${\mathcal{V}}^{\mathsf{N}}$ gives

$\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\sum_{j}(\nabla_{X_{j}}L)_{X_{j}}-\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}L_{{\operatorname{H}}^{\mathsf{L}}}=\dim\mathfrak{n}+\operatorname{tr}_{{\mathcal{V}}^{\mathfrak{n}}}\operatorname{Ric}^{\mathsf{L}},$

and the lemma follows. ∎

Regarding the right-hand-side in (12), we have the following key algebraic
curvature estimate:

######  Lemma 6.5.

Let $p\in M_{\sf reg}$ satisfy (11). Then,

$\dim\mathfrak{n}+\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}^{\mathsf{L}}(p)\geq
0,$

with equality if and only if $\mathsf{L}\cdot p=\mathsf{Q}\cdot p$, where
$\mathsf{Q}$ is defined in (10).

###### Proof.

We apply [BL23, Prop. 12.4] to the orbit $\mathsf{L}\cdot p$. Notice that
$\mathsf{N}\leq\mathsf{L}$ is nilpotently embedded and acts freely and polarly
on $\mathsf{L}\cdot p$. Then, that result gives

(13)
$\operatorname{scal}^{\mathsf{N}}(p)-\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}^{\mathsf{L}}(p)=\sum_{i}\langle\nabla_{E^{*}_{i}}E^{*}_{i},{\operatorname{H}}^{\mathsf{N}}\rangle_{p}+\sum_{i,r}\langle[U^{*}_{r},[U^{*}_{r},E^{*}_{i}]],E^{*}_{i}\rangle_{p},$

where $\\{E_{i}\\}_{i=1}^{n}$ is any set in $\mathfrak{l}$ containing a basis
$\\{U_{r}\\}_{r=1}^{\dim\mathfrak{n}}$ for $\mathfrak{n}$, and so that
$\\{E_{i}^{*}\\}$ is an orthonormal basis of $T_{p}(\mathsf{L}\cdot p)$.
Choose now $A\in\mathfrak{l}$ with
$A^{*}_{p}=-{\operatorname{H}}^{\mathsf{N}}_{p}$. By Theorem 2.2, Item v, we
may assume $A\in\mathfrak{g}=\mathfrak{a}\oplus\mathfrak{n}$. Condition (11)
then gives $A\in\mathfrak{a}$. By definition of the Iwasawa decomposition,
this implies that
$D:=\operatorname{ad}_{\mathfrak{l}}A:\mathfrak{l}\to\mathfrak{l}$ is
diagonalisable with real eigenvalues. Arguing precisely as in the proof of
[BL23, Lemma 14.2], it follows from Theorem 2.2, (3) that
$D|_{\mathfrak{n}}>0$, $D|_{\theta\mathfrak{n}}<0$,
$D|_{\mathfrak{m}\oplus\mathfrak{a}}=0$, and moreover
$\operatorname{tr}D|_{\mathfrak{n}}=\operatorname{tr}\beta^{+}$. Then, [BL23,
Prop. 12.5] applies and gives a choice of Killing fields $\\{E_{i}^{*}\\}$ for
which (13) becomes

$\operatorname{scal}^{\mathsf{N}}(p)-\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}^{\mathsf{L}}(p)\leq\operatorname{tr}\beta^{+},$

with equality if and only if $\mathsf{Q}\cdot p=\mathsf{L}\cdot p$. Replacing
$\operatorname{scal}^{\mathsf{N}}$ using (3) gives the desired estimate. ∎

###### Proof of Proposition 6.1.

Given a principal $\mathsf{L}$-orbit, by Lemma 6.3 we may choose a point $p\in
M$ on it so that (11) holds. By Lemma 6.3, (11) will also hold along the
corresponding section $\Sigma_{\mathsf{L}}$ for the $\mathsf{L}$-action. Since
${\operatorname{H}}^{\mathsf{N}}=-\nabla\log v_{\mathsf{N}}$ is
$\mathsf{G}$-vertical by Theorem 2.2, (v), and in particular also
$\mathsf{L}$-vertical, $\log v_{\mathsf{N}}$ must be constant along
$\Sigma_{\mathsf{L}}$. Thus, the left-hand-side in (12) vanishes, giving
equality in Lemma 6.5, and hence $\mathsf{Q}\cdot p=\mathsf{L}\cdot p$. ∎

###### Proof of Corollary 6.2.

From the proof of Proposition 6.1 we have equality in Lemma 6.5. ∎

We now present some remarkable algebraic consequences of Proposition 6.1.

######  Corollary 6.6.

At a point $p\in M_{\sf reg}$ satisfying (11) the following hold:

  1. (1)

$\mathfrak{n}\cdot p\perp(\mathfrak{m}\oplus\mathfrak{a})\cdot p$;

  2. (2)

$\mathfrak{n}\cdot p\perp\mathfrak{k}\cdot p$;

  3. (3)

$\mathfrak{n}^{-}\cdot p\perp\mathfrak{k}\cdot p$;

  4. (4)

$\mathfrak{n}\cdot p\subset\mathfrak{p}\cdot p$.

###### Proof.

We first prove Item 1. Moreover, we claim that

$\mathfrak{m}\oplus\mathfrak{a}=\mathfrak{n}^{\perp}:=\\{X\in\mathfrak{q}:X^{*}_{p}\perp\mathfrak{n}\cdot
p\\}.$

Notice that
$\mathfrak{n}^{\perp}\cap\mathfrak{n}\subset\mathfrak{h}\cap\mathfrak{n}=0$,
since the $\mathsf{N}$-action is free. Hence,
$\dim\mathfrak{n}^{\perp}=\dim\mathfrak{q}-\dim\mathfrak{n}=\dim(\mathfrak{m}\oplus\mathfrak{a})$
(recall that $\mathfrak{q}=\mathfrak{m}\oplus\mathfrak{a}\oplus\mathfrak{n}$),
and it suffices to prove that
$\mathfrak{n}^{\perp}\subset\mathfrak{m}\oplus\mathfrak{a}$. To that end, let
$X=Y+U\in\mathfrak{n}^{\perp}$, with $Y\in\mathfrak{m}\oplus\mathfrak{a}$,
$U\in\mathfrak{n}$. Using that $\mathfrak{a}\subset\mathfrak{n}^{\perp}$ by
(11), $\mathfrak{m}\oplus\mathfrak{a}=Z_{\mathfrak{q}}(\mathfrak{a})$, and the
fact that $\mathfrak{n}^{\perp}$ is a Lie subalgebra of $\mathfrak{q}$ [BL23,
Cor. 8.7], for any $A\in\mathfrak{a}$ we obtain

$\mathfrak{n}^{\perp}\ni[X,A]=[U,A]\in[\mathfrak{n},\mathfrak{a}]\subset\mathfrak{n}.$

From this it follows that $[X,A]^{*}_{p}=0$, i.e.
$[X,A]\in\mathfrak{h}\cap\mathfrak{n}=0$, thus
$X\in\mathfrak{m}\oplus\mathfrak{a}$.

We now show Item 2. By Proposition 6.1 we know that
$\mathsf{L}=\mathsf{Q}\mathsf{H}$, $\mathsf{H}=\mathsf{L}_{p}$. Since
$\mathsf{L}=\mathsf{K}\mathsf{G}$ and $\mathsf{Q}=\mathsf{M}^{0}\mathsf{G}$,
it follows that

(14)
$\mathsf{K}=\mathsf{M}^{0}\mathsf{H},\qquad\mathfrak{k}=\mathfrak{m}+\mathfrak{h},$

where $\mathfrak{h}$ is the isotropy subalgebra at $p$ ($\mathfrak{h}\cdot
p=0$). Thus, $\mathfrak{k}\cdot p=\mathfrak{m}\cdot p\perp\mathfrak{n}\cdot p$
by Item 1.

Regarding Item 3, let $X\in\mathfrak{l}_{\lambda}\subset\mathfrak{n}$, for
some positive root $\lambda\in\mathfrak{a}^{*}$. Thanks to [Kna02, Prop.
6.52(c)], there exists $k\in N_{\mathsf{K}}(\mathfrak{a})$ such that
$\operatorname{Ad}(k)\mathfrak{l}_{\lambda}=\mathfrak{l}_{-\lambda}$. By (14),
we may write $k=hm$ for $h\in\mathsf{H}$, $m\in\mathsf{M}^{0}$. Using that
$\operatorname{Ad}(m)\mathfrak{n}\subset\mathfrak{n}$, for all
$Z\in\mathfrak{k}$ we have

$\langle(\operatorname{Ad}(k)X)^{*},Z^{*}\rangle_{p}=\langle(\operatorname{Ad}(m)X)^{*},(\operatorname{Ad}(h^{-1})Z)^{*}\rangle_{p}=0,$

by $\operatorname{Ad}(\mathsf{H})$-invariance of the metric and Item 2. Thus,
$\mathfrak{l}_{-\lambda}\cdot p\perp\mathfrak{k}\cdot p$ holds for all
positive roots $\lambda$, and Item 3 follows.

Finally, given $X\in\mathfrak{n}$, we have $\theta X\in\mathfrak{n}^{-}$ and
$X+\theta X\in\mathfrak{k}$. Items 2 and 3 imply that $(X+\theta
X)^{*}_{p}\perp X^{*}_{p}$, $(X+\theta X)^{*}_{p}\perp(\theta X)^{*}_{p}$,
thus $X+\theta X\in\mathfrak{h}$. Therefore,

$X^{*}_{p}=\tfrac{1}{2}(X-\theta X)^{*}_{p}+\tfrac{1}{2}(X+\theta
X)^{*}_{p}=\tfrac{1}{2}(X-\theta X)^{*}_{p},$

and $\tfrac{1}{2}(X-\theta X)\in\mathfrak{p}$, from which the lemma follows. ∎

If $\mathfrak{l}=\mathfrak{l}_{1}\oplus\cdots\oplus\mathfrak{l}_{r}$ where
each $\mathfrak{l}_{i}$ is a simple ideal, and we set
$\mathfrak{p}_{i}:=\mathfrak{l}_{i}\cap\mathfrak{p}$, then

$\mathfrak{p}=\mathfrak{p}_{1}\oplus\cdots\oplus\mathfrak{p}_{r}$

is the decomposition of $\mathfrak{p}$ as a sum of irreducible, pairwise
inequivalent, $\operatorname{Ad}(\mathsf{K})$-modules.

######  Corollary 6.7.

The modules $\mathfrak{p}_{i}$, $i=1,\ldots,r$, are irreducible and pairwise
inequivalent as $\operatorname{Ad}(\mathsf{H})$-modules.

###### Proof.

To prove irreducibility, it suffices to show that every
$\operatorname{Ad}(\mathsf{H})$-module is also an
$\operatorname{Ad}(\mathsf{K})$-module. To that end, let
$\mathfrak{v}\subset\mathfrak{p}$ be an
$\operatorname{Ad}(\mathsf{H})$-module, and let
$\mathfrak{a}_{\mathfrak{v}}:=\mathfrak{v}\cap\mathfrak{a}$. We claim that

$\mathfrak{v}=\bigcup_{k\in\mathsf{K}}\operatorname{Ad}(k)\mathfrak{a}_{\mathfrak{v}}.$

To see that, first observe that $\mathfrak{a}_{\mathfrak{v}}$ is a maximal
abelian subspace in $\mathfrak{v}$. Indeed, if
$\mathfrak{b}\subset\mathfrak{v}$ is abelian then by uniqueness of maximal
abelian subspaces in $\mathfrak{p}$ [Kna02, Thm. 6.51], there exists
$k\in\mathsf{K}$ such that
$\operatorname{Ad}(k)\mathfrak{b}\subset\mathfrak{a}$. Writing $k=mh$ with
$h\in\mathsf{H}$, $m\in\mathsf{M}$, and using that
$\operatorname{Ad}(\mathsf{M})$ acts trivially on $\mathfrak{a}$, we get
$\operatorname{Ad}(h)\mathfrak{b}\subset\mathfrak{a}$. But $\mathfrak{v}$ is
$\operatorname{Ad}(\mathsf{H})$-invariant, hence
$\operatorname{Ad}(h)\mathfrak{b}\subset\mathfrak{a}\cap\mathfrak{v}=\mathfrak{a}_{\mathfrak{v}}$.
This argument shows furthermore that

$\mathfrak{v}=\bigcup_{h\in\mathsf{H}}\operatorname{Ad}(h)\mathfrak{a}_{\mathfrak{v}}.$

Now any $k\in\mathsf{K}$ may be written as $k=hm$ with
$\operatorname{Ad}(m)|_{\mathfrak{a}}={\operatorname{Id}}_{\mathfrak{a}}$,
hence
$\operatorname{Ad}(k)\mathfrak{a}_{\mathfrak{v}}=\operatorname{Ad}(h)\mathfrak{a}_{\mathfrak{v}}$,
and the claim follows.

Finally, if we had $\mathfrak{p}_{i}\simeq\mathfrak{p}_{j}$ as
$\operatorname{Ad}(\mathsf{H})$-modules for some $i\neq j$, then there would
be more irreducible $\operatorname{Ad}(\mathsf{H})$-modules other than the
$\mathfrak{p}_{i}$’s. These would also be irreducible as
$\operatorname{Ad}(K)$-modules by argument in the previous paragraph, and this
is a contradiction because the $\mathfrak{p}_{i}$’s are inequivalent as
$\operatorname{Ad}(K)$-modules. ∎

##  7\. The orbits are symmetric spaces

The upshot of Proposition 6.1 is that the stabiliser
$\mathsf{H}=\mathsf{L}_{p}$ of a point $p$ satisfying (11) must be large
enough so that $\mathsf{Q}\mathsf{H}=\mathsf{L}$ holds. This condition
significantly constrains the geometry of the $\mathsf{L}$-orbits, as they are
diffeomorphic to $\mathsf{L}/\mathsf{H}$. For example, if $\mathsf{L}$ is
split semisimple (i.e. $\mathfrak{m}=0$, satisfied for instance when
$\mathsf{L}=\mathsf{SL}_{n}({\mathbb{R}})$), then the condition
$\mathfrak{q}+\mathfrak{h}=\mathfrak{l}$ forces $\mathsf{H}=\mathsf{K}$ to be
a maximal compact subgroup, and the $\mathsf{L}$-orbits must be symmetric
spaces. Our aim in this section is to show that this holds even without the
split semisimple assumption:

######  Proposition 7.1.

The $\mathsf{L}$-orbits are symmetric spaces.

To prove this, it is important to first deduce a geometric application of
Corollary 6.7. Recall the Riemannian submersion
$\pi^{\mathsf{N}}:M\to\mathsf{N}\backslash M$. Since $\mathsf{N}$ is a normal
subgroup of $\mathsf{Q}$, we have the corresponding Lie group and Lie algebra
morphisms

$\varphi:\mathsf{Q}\to\mathsf{Q}/\mathsf{N},\qquad\Phi:=d\varphi|_{e}:\mathfrak{q}\to\mathfrak{q}/\mathfrak{n}.$

There is also a natural action of $\mathsf{Q}$ on $\mathsf{N}\backslash M$:

$q\cdot(\mathsf{N}\cdot x):=\mathsf{N}\cdot(q\cdot x),\qquad
q\in\mathsf{Q},\quad x\in M,$

for which by definition $\pi^{\mathsf{N}}$ is $\mathsf{Q}$-equivariant. The
ineffective kernel of said action contains $\mathsf{N}$, thus we have an
isometric, cocompact action of $\mathsf{Q}/\mathsf{N}$ on
$\mathsf{N}\backslash M$, given by

$\varphi(q)\cdot(\mathsf{N}\cdot x):=\mathsf{N}\cdot(q\cdot x),\qquad\hbox{or
equivalently,}\qquad\varphi(q)\cdot\pi^{\mathsf{N}}(x)=\pi^{\mathsf{N}}(q\cdot
x).$

In order to simplify the notation, given $Z\in\mathfrak{q}$ we denote
$\check{Z}:=\Phi(Z)$.

######  Lemma 7.2.

Let $Z\in\mathfrak{q}$ and let $Z^{*}\in\Gamma(TM)$ be the corresponding
Killing field. Then,

$(\pi^{\mathsf{N}})_{*}Z^{*}=\check{Z}^{*},$

where $(\check{Z})^{*}$ is the Killing field on $\mathsf{N}\backslash M$
associated to $\check{Z}=\Phi(Z)\in\mathfrak{q}/\mathfrak{n}$.

###### Proof.

Let $p\in M$, $b:=\pi^{\mathsf{N}}(p)\in\mathsf{N}\backslash M$. If
$\exp_{\mathsf{Q}}:\mathfrak{q}\to\mathsf{Q}$,
$\exp_{\mathsf{Q}/\mathsf{N}}:\mathfrak{q}/\mathfrak{n}\to\mathsf{Q}/\mathsf{N}$
denote the corresponding Lie exponential maps, then we have
$\exp_{\mathsf{Q}/\mathsf{N}}\circ\,\Phi=\varphi\circ\exp_{\mathsf{Q}}$. Using
this, we obtain:

$\displaystyle(\check{Z})^{*}_{b}$ $\displaystyle=\tfrac{{\rm d}}{{\rm
d}t}\big{|}_{0}\exp_{\mathsf{Q}/\mathsf{N}}(t\check{Z})\cdot b=\tfrac{{\rm
d}}{{\rm d}t}\big{|}_{0}\varphi(\exp(tZ))\cdot b$ $\displaystyle=\tfrac{{\rm
d}}{{\rm d}t}\big{|}_{0}\pi^{\mathsf{N}}(\exp(tZ)\cdot
p)=(\pi^{\mathsf{N}})_{*}Z_{p}^{*}.$

∎

Notice that the above in particular implies that
$\mathcal{H}^{\mathsf{N}}Z^{*}$ is a basic vector field.

######  Lemma 7.3.

Let $Z\in\mathfrak{m}$ and let $Z^{*}\in\Gamma(TM)$ be the corresponding
Killing field. Then, the shape operator of the $\mathsf{N}$-orbits in $M$ in
the direction of $Z^{*}$ vanishes, that is, $L_{Z^{*}}=0$. In particular,

(15) $-\|\check{Z}^{*}\|^{2}=\operatorname{Ric}_{\mathsf{N}\backslash
M}(\check{Z}^{*},\check{Z}^{*}).$

###### Proof.

Let $p\in M$ satisfy (11). By Corollary 6.6, for each $Z\in\mathfrak{m}$,
$Z^{*}$ is $\mathsf{N}$-horizontal at $p$. We then have

$\langle
L_{Z^{*}}U^{*},U^{*}\rangle_{p}=-\langle\nabla_{U^{*}}U^{*},Z^{*}\rangle_{p}=\langle[Z^{*},U^{*}],U^{*}\rangle_{p},$

for all $U\in\mathfrak{n}$, where the last equality follows from [Bes87,
(7.27)]. We now claim that
$(\operatorname{ad}Z)|_{\mathfrak{n}}:\mathfrak{n}\to\mathfrak{n}$ is skew-
symmetric with respect to the inner product induced by the linear isomorphism
$\mathfrak{n}\simeq\mathfrak{n}\cdot p$. To see that, Corollary 6.7 implies
that the inner product induced by $g$ on $\mathfrak{p}\cdot p\subset
T_{p}(\mathsf{L}\cdot p)$ is $\operatorname{Ad}(\mathsf{K})$-invariant. Since
$\mathfrak{n}\cdot p\subset\mathfrak{p}\cdot p$ by Corollary 6.6, and
$\mathfrak{n}$ is $\operatorname{Ad}(\mathsf{M})$-stable, it follows that
$(\operatorname{ad}Z)|_{\mathfrak{n}}$ is skew-symmetric for each
$Z\in\mathfrak{m}$. In other words, $L_{Z^{*}}=0$. Plugging this into Theorem
2.2, iv yields Equation 15. ∎

###### Proof of Proposition 7.1.

Since $\mathsf{L}=\mathsf{K}\mathsf{A}\mathsf{N}$,
$\mathsf{Q}=\mathsf{M}_{0}\mathsf{A}\mathsf{N}$, $\mathsf{M}\leq\mathsf{K}$,
and $\mathsf{L}=\mathsf{Q}\mathsf{H}$ (Proposition 6.1), to prove that
$\mathsf{H}=\mathsf{K}$ is maximal compact we must show that
$\mathsf{M}\leq\mathsf{H}$, or equivalently,
$\mathsf{Q}/\mathsf{N}=\mathsf{G}/\mathsf{N}$. To that end, we let
$Z\in\mathfrak{m}$, and consider the smooth function

$f:=\tfrac{1}{2}\left\|\check{Z}^{*}\right\|^{2}:\mathsf{N}\backslash
M\to{\mathbb{R}},$

where $\check{Z}=\Phi(Z)\in\mathfrak{q}/\mathfrak{n}$. For all
$X\in\Gamma(T(\mathsf{N}\backslash M))$ we have the well-known Bochner
formulas

$\langle\nabla
f,X\rangle=-\left\langle\nabla_{\check{Z}^{*}}\check{Z}^{*},X\right\rangle,\qquad\Delta_{\mathsf{N}\backslash
M}f=\big{\|}\nabla^{\mathsf{N}\backslash
M}\check{Z}^{*}\big{\|}^{2}-\operatorname{Ric}_{\mathsf{N}\backslash
M}(\check{Z}^{*},\check{Z}^{*}).$

The first formula and the fact that $[\mathfrak{m},\mathfrak{a}]=0$ imply that
$f$ is constant along $\mathsf{G}/\mathsf{N}$-orbits. Since
$(\mathsf{G}/\mathsf{N})\backslash(\mathsf{N}\backslash
M)\simeq\mathsf{G}\backslash M$ is compact, $f$ must admit a global maximum at
$x\in\mathsf{N}\backslash M$, hence

$\left(\Delta_{\mathsf{N}\backslash M}f\right)|_{x}\leq 0.$

On the other hand, the second formula and Lemma 7.3 yield

$\Delta_{\mathsf{N}\backslash M}f=\|\nabla^{\mathsf{N}\backslash
M}\check{Z}^{*}\|^{2}+\|\check{Z}^{*}\|^{2}\geq 0.$

In particular, we have $(\nabla\check{Z}^{*})|_{x}=0$ and
$\check{Z}^{*}_{x}=0$. Since these determine a Killing field completely, we
conclude that $\check{Z}=0$. This also gives $Z=0$, as
$\Phi|_{\mathfrak{m}\oplus\mathfrak{a}}$ is injective. ∎

Next, recall the definition of the _polar group_ :
$\Pi:=\mathsf{L}_{\Sigma}/\mathsf{H}$. Here $\mathsf{L}_{\Sigma}$ is the
stabiliser of $\sigma(\Sigma_{\mathsf{L}})$ and $\mathsf{H}:=\mathsf{L}_{p}$
is the isotropy at $p$, which is also precisely the set of elements in
$\mathsf{L}$ fixing $\sigma(\Sigma_{\mathsf{L}})$ pointwise. Recall also that,
by [GZ12, Prop. 1.3], $\Pi$ is a discrete subgroup of
$N(\mathsf{H})/\mathsf{H}$ which acts on $\Sigma_{\mathsf{L}}$ isometrically
and properly discontinuously, making $\sigma$ equivariant, and so that the
metric spaces $\Sigma_{\mathsf{L}}/\Pi$ and $\mathsf{L}\backslash M$ are
isometric.

######  Corollary 7.4.

The polar group $\Pi$ is trivial and the section $\Sigma_{\mathsf{L}}$ is
compact.

###### Proof.

By Proposition 7.1 $\mathsf{H}=\mathsf{K}$ is a maximal compact subgroup of
the semisimple Lie group $\mathsf{L}$. Since $\mathsf{K}$ is self-normalising
(Proposition C.2), $\Pi\leq N(\mathsf{K})/\mathsf{K}$ must be trivial. In
particular, $\Sigma_{\mathsf{L}}\simeq\mathsf{L}\backslash M$ is compact. ∎

##  8\. Isometric splitting

We are finally in a position to prove Theorem A. We continue using the same
notation as in previous sections, with one slight change: since by Corollary
7.4 the submanifold $\Sigma_{\mathsf{L}}$ is embedded, we will identify it
with its image in $M$ and simply assume that
$\sigma:\Sigma_{\mathsf{L}}\hookrightarrow M$ is the inclusion map. As in
previous sections, we may and will also assume that the points in
$\Sigma_{\mathsf{L}}$ satisfy the technical condition (11).

Given $U\in\mathfrak{l}$, consider the smooth function

$f_{U}:=\tfrac{1}{2}\|U^{*}\|^{2}:\Sigma_{\mathsf{L}}\to{\mathbb{R}}.$

######  Lemma 8.1.

Along $\Sigma_{\mathsf{L}}$ we have that

$\Delta_{\Sigma_{\mathsf{L}}}f_{U}-\langle\nabla
f_{U},{\operatorname{H}}^{\mathsf{L}}\rangle=\operatorname{ric}^{\mathsf{L}}(U^{*},U^{*})+\|U^{*}\|^{2}+2\,\|LU^{*}\|^{2}.$

###### Proof.

For $X\in\Gamma(T\Sigma_{\mathsf{L}})$ we have

$\langle\nabla f_{U},X\rangle=\langle\nabla_{X}U^{*},U^{*}\rangle=-\langle
X,\nabla_{U^{*}}U^{*}\rangle.$

We now use an local, $\mathsf{L}$-basic (i.e. horizontal and projectable),
orthonormal frame $\\{X_{k}\\}$ for $\mathcal{H}^{\mathsf{L}}$ satisfying
$\nabla_{X_{k}}X_{k}=0$ at the point:

$\displaystyle\Delta_{\Sigma_{\mathsf{L}}}f_{U}=$
$\displaystyle-\sum_{k}\langle\nabla_{X_{k}}\mathcal{H}\nabla_{U^{*}}U^{*},X_{k}\rangle=-\sum_{k}X_{k}\langle\nabla_{U^{*}}U^{*},X_{k}\rangle$
$\displaystyle=$ $\displaystyle\sum_{k}X_{k}\langle
L_{X_{k}}U^{*},U^{*}\rangle$ $\displaystyle=$
$\displaystyle\sum_{k}\left(\left\langle\big{(}\nabla_{X_{k}}L\big{)}_{X_{k}}U^{*},U^{*}\right\rangle+\left\langle
L_{X_{k}}(\nabla_{X_{k}}U^{*}),U^{*}\right\rangle+\langle
L_{X_{k}}U^{*},\nabla_{X_{k}}U^{*}\rangle\right)$ $\displaystyle=$
$\displaystyle\sum_{k}\left\langle\big{(}\nabla_{X_{k}}L\big{)}_{X_{k}}U^{*},U^{*}\right\rangle+2\,\|LU^{*}\|^{2},$

where in the last step we have used that basic vector fields commute with
vertical Killing fields [BL23, Lemma 3.5]. The lemma now follows from the
vertical Einstein equation (1) with $\mathsf{L}$-symmetry, and the fact that
$A^{\mathsf{L}}=0$ (Proposition 5.4). ∎

###### Proof of Theorem A.

By Theorem 4.1, $\mathsf{L}$ is semisimple and, by Lemma 5.1, it may be
assumed to have finite center.

By Proposition 7.1, for each simple factor $\mathsf{L}_{i}\leq\mathsf{L}$
there exists $\lambda_{i}<0$ such that for any $U_{i}\in\mathfrak{l}_{i}$ we
have

(16)
$\operatorname{ric}^{\mathsf{L}}(U_{i}^{*},U_{i}^{*})=\lambda_{i}\,\|U_{i}^{*}\|^{2}.$

Since $\mathfrak{n}=\mathfrak{n}_{1}\oplus\cdots\oplus\mathfrak{n}_{r}$,
$\mathfrak{n}_{i}=\mathfrak{l}_{i}\cap\mathfrak{n}$, Corollary 6.2 gives

(17)
$\sum_{i}(\dim\mathfrak{n}_{i})\lambda_{i}=\operatorname{tr}_{{\mathcal{V}}^{\mathsf{N}}}\operatorname{Ric}^{\mathsf{L}}=-\dim\mathfrak{n}=-\sum_{i}(\dim\mathfrak{n}_{i}).$

On the other hand, using (16), Lemma 8.1 reads as

(18) $\Delta_{\Sigma_{\mathsf{L}}}f_{U_{i}}-\langle\nabla
f_{U_{i}},{\operatorname{H}}^{\mathsf{L}}\rangle=(1+\lambda_{i})\|U_{i}^{*}\|^{2}+2\,\|LU_{i}^{*}\|^{2}.$

Since $\Sigma_{\mathsf{L}}$ is compact (Corollary 7.4), by the maximum
principle applied to (18) we must have that $\lambda_{i}\leq-1$ for all $i$.
Equation 17 then yields $\lambda_{i}=-1$ for all $i=1,\ldots,r$.

Putting this back into (18), from the maximum principle we deduce that
$LU^{*}=0$ for all $U\in\mathfrak{l}$. That is, the $\mathsf{L}$-orbits are
totally geodesic. Since $A^{\mathsf{L}}=0$, at least locally $M$ splits
isometrically as the product of the compact Einstein manifold
$\Sigma_{\mathsf{L}}$ and the Einstein symmetric space
$\mathsf{L}/\mathsf{K}$, both with Einstein constant $-1$. Moreover, since the
polar group is trivial by Corollary 7.4, the section intersects each orbit
exactly once, and hence the splitting is global. ∎

##  9\. Ricci flat manifolds with cocompact symmetry

The following argument is essentially contained in the proof of [CG72, Thm.
3]. Recall that a _line_ is a geodesic $\gamma:{\mathbb{R}}\to M$ which
realises the distance between any pair of its points.

######  Lemma 9.1.

Let $(M^{n},g)$ be a complete, non-compact Riemannian manifold for which
$\operatorname{Isom}(M^{n},g)$ acts cocompactly. Then, $(M^{n},g)$ contains a
line.

###### Proof.

Let $p\in M$. Given a sequence $\\{q_{n}\\}_{n\in{\mathbb{N}}}$ such that
$d(p,q_{n})=n$ for all $n\in{\mathbb{N}}$, choose minimising geodesics
$\gamma_{n}$ with $\gamma_{n}(0)=p$ and $\gamma_{n}(n)=q_{n}$, for each
$n\in{\mathbb{N}}$. After passing to a subsequence we may assume that
$(\gamma^{\prime}_{n}(0))_{n\in{\mathbb{N}}}$ converges to a unit vector $v\in
T_{p}M$. It is then clear that the geodesic $\gamma$ with $\gamma(0)=p$,
$\gamma^{\prime}(0)=v$, is a _ray_ : it minimises the distance between
$\gamma(t)$ and $\gamma(s)$ for all $t,s\geq 0$.

Let $K\subset M$ be a compact fundamental domain for
$\operatorname{Isom}(M,g)$. For each $n\in{\mathbb{N}}$, there exists an
isometry $f_{n}\in\operatorname{Isom}(M,g)$ such that $f_{n}(\gamma(n))\in K$.
By compactness, after passing to a subsequence we may assume that
$f_{n}(\gamma(n))\to q\in K$ and $(f_{n})_{*}\gamma^{\prime}(n)\to w\in
T_{q}M$. Notice that for each $n\in{\mathbb{N}}$, the geodesic
$f_{n}\circ\gamma$ is minimising on $(-n,\infty)$. It follows that the
geodesic $\sigma:{\mathbb{R}}\to M$ with $\sigma(0)=q$, $\sigma^{\prime}(0)=w$
is a line. ∎

######  Theorem 9.2.

Let $(M^{n},g)$ be a complete, Ricci flat Riemannian manifold, and let
$\mathsf{G}$ be a Lie group acting properly, isometrically and cocompactly on
it. Then, the universal cover splits isometrically as a Riemannian product
$\tilde{M}=\bar{M}\times{\mathbb{R}}^{k}$, with $\bar{M}$ compact. In
particular, $\mathsf{G}_{0}\subset\operatorname{Isom}({\mathbb{R}}^{k})$ if
the action is effective.

###### Proof.

The universal cover $\tilde{M}$ is Ricci flat with respect to the pull-back
metric $\tilde{g}$, and the action of $\mathsf{G}$ lifts to a proper isometric
action on $\tilde{M}$. Also, the fundamental group $\pi_{1}(M)$ also acts –via
deck transformations– properly and isometrically on $\tilde{M}$. Since
$M/\mathsf{G}$ is compact, it follows that
$\operatorname{Isom}(\tilde{M},\tilde{g})$ acts cocompactly on $\tilde{M}$. If
$\tilde{M}$ is compact, then we are done. Otherwise, by Lemma 9.1, $\tilde{M}$
contains a line, thus by the Cheeger-Gromoll splitting theorem [CG72] we have
an isometric splitting $\tilde{M}\simeq\tilde{M}_{1}\times{\mathbb{R}}^{1}$.
In particular, $\tilde{M}_{1}$ is Ricci flat. If $\tilde{M}_{1}$ contains a
line, we split again. In this way, we may continue applying the splitting
theorem until we reach a point where
$\tilde{M}\simeq\tilde{M}_{k}\times{\mathbb{R}}^{k}$, with $\tilde{M}_{k}$
containing no lines.

We now claim that $\operatorname{Isom}(\tilde{M}_{k})$ acts cocompactly on
$\tilde{M}_{k}$. To see this, notice that all the lines in $\tilde{M}$ are
contained in the ${\mathbb{R}}^{k}$ factor. Since isometries map lines to
lines, it follows that any isometry of $\tilde{M}$ must map the
${\mathbb{R}}^{k}$ factor to itself, and hence also preserve $\tilde{M}_{k}$.
This implies that
$\operatorname{Isom}(\tilde{M})\simeq\operatorname{Isom}(\tilde{M}_{k})\times\operatorname{Isom}({\mathbb{R}}^{k})$.
The claim thus follows from the fact that $\operatorname{Isom}(\tilde{M})$
acts cocompactly on $\tilde{M}$. By Lemma 9.1 we conclude that $\tilde{M}_{k}$
is compact.

Regarding the claim about $\mathsf{G}_{0}$, it will follow once we prove that
$\operatorname{Isom}(\tilde{M}_{k})$ is discrete. But if $X$ is a non-zero
Killing field then by Bochner’s theorem, $X$ must be parallel ($\tilde{M}_{k}$
is compact and with non-positive Ricci curvature). Since $\tilde{M}_{k}$ is
also simply-connected, this would imply that it splits an ${\mathbb{R}}$
factor, thus contradicting compactness. ∎

######  Remark 9.3.

In the case of a cocompact isometric action by a _connected_ Lie group,
completeness follows. However, in Theorem 9.2 we do not assume that
$\mathsf{G}$ is connected, and we are thus forced to assume completeness.

##  Appendix A Proper isometric actions and polar actions

Let $\mathsf{G}$ be a Lie group acting isometrically on a complete Riemannian
manifold $(M^{n},g)$ via the Lie group homomorphism

$\tau:\mathsf{G}\to\operatorname{Isom}(M^{n},g).$

After changing $\mathsf{G}$ by $\mathsf{G}/\ker\tau$, we assume from now on
that the action is effective, i.e. $\tau$ is injective. If $\mathsf{G}$ is
non-compact, it is natural to assume that the action is _proper_ , that is,
the ‘shear map’

$\mathsf{G}\times M\to M\times M,\qquad(g,p)\mapsto(\tau(g)p,p),$

is a proper continuous function. (Since $M$ is locally compact and Hausdorff,
the above definition is equivalent to the notions of _Borel properness_ , see
[Pal61, Thm. 1.2.9 (5)], and to _Palais properness_ , see [Pal61, Def.
1.2.2].) For isometric actions there is a simple criterion:

######  Lemma A.1.

The action of $\mathsf{G}$ is proper if and only if $\tau(\mathsf{G})$ is a
closed subgroup of $\operatorname{Isom}(M^{n},g)$.

###### Proof.

Let $\\{(\tau(g_{k})p_{k},p_{k})\\}_{k}$ be a sequence contained in a compact
set in $M\times M$. Then up to passing to subsequences, as $k\to\infty$ we
have

(19) $p_{k}\to p_{\infty},\qquad\tau(g_{k})p_{k}\to q_{\infty}.$

The triangle inequality yields

$d(p_{k},p_{\infty})=d(\tau(g_{k})p_{k},\tau(g_{k})p_{\infty})\geq|d(q_{\infty},\tau(g_{k})p_{\infty})-d(\tau(g_{k})p_{k},q_{\infty})|\geq
0.$

By (19) it follows that $\tau(g_{k})p_{\infty}\to q_{\infty}$ as $k\to\infty$.
Thus, the isometries $\tau(g_{k})$ subconverge to some
$f\in\operatorname{Isom}(M^{n},g)$, since $(M^{n},g)$ is complete. By
assumption we have $f=\tau(g_{\infty})$ for some $g_{\infty}\in\mathsf{G}$,
and it follows that $(g_{k},p_{k})\to(g_{\infty},p_{\infty})$ as desired.

Conversely, if the action is proper and $\tau(g_{k})\to f$ for some sequence
$\\{g_{k}\\}_{k}$ in $\mathsf{G}$, then

$(\tau(g_{k})p,p)\to(f(p),p),$

hence $\\{g_{k}\\}_{k}$ subconverges to some $g_{\infty}\in\mathsf{G}$, and
$f=\tau(g_{\infty})$. ∎

In particular, for all $p\in M$ the isotropy group $\mathsf{G}_{p}$ is
compact. In addition, one of the main advantages of a proper Lie group action
is that the Slice Theorem holds [Pal61]. Recall that an orbit $\mathsf{G}\cdot
p$ is called:

  * •

_principal_ , if the isotropy group $\mathsf{G}_{p}$ is not properly contained
in any other isotropy group $\mathsf{G}_{q}$, $q\in M$; $\mathsf{G}_{p}$ is
then called a principal isotropy group;

  * •

_exceptional_ , if $\mathsf{G}_{p}\lneq\mathsf{G}_{q}$ for some principal
isotropy group $\mathsf{G}_{q}$, with $\mathsf{G}_{q}/\mathsf{G}_{p}$
discrete;

  * •

_singular_ , if $\dim\mathsf{G}_{p}<\dim\mathsf{G}_{q}$ for some principal
isotropy group $\mathsf{G}_{q}$.

In general, the orbit space $\mathsf{G}\backslash M$ is a possibly singular
metric space. An important consequence of the Slice Theorem is that there is,
up to conjugation, a unique principal isotropy group, and the union of all
principal orbits forms an open dense subset of full measure whose quotient is
a convex subset of $\mathsf{G}\backslash M$ and a smooth manifold. We set

$M_{\sf sing}:=\\{p\in M:\mathsf{G}\cdot p\hbox{ is a singular
orbit}\\},\qquad M_{\sf reg}:=M\setminus M_{\sf sing}.$

Notice that $M_{\sf reg}$ contains not only principal orbits but also the
exceptional ones. It will become clear after $\S$B.3 that $M_{\sf reg}$ is
also open in $M$.

In certain situations, there exists a globally-defined slice: a proper
isometric action is called _polar_ , if there exists an immersed submanifold
$\sigma:\Sigma\to M$ –called a _section_ – intersecting _all_ orbits
orthogonally. By a result of Boualem [Bou95] (see also [HLO06, GZ12]), this is
equivalent to the vanishing of the integrability tensor $A$ of the Riemannian
submersion defined on the regular part:

$\pi^{\mathsf{G}}:M_{\sf reg}\to\mathsf{G}\backslash M_{\sf reg}=:B,\qquad
p\mapsto\mathsf{G}\cdot p.$

Next, recall the definition of the _polar group_ :
$\Pi:=\mathsf{G}_{\Sigma}/\mathsf{H}$. Here,
$\mathsf{G}_{\Sigma}=\\{f\in\mathsf{G}:f\cdot\sigma(\Sigma)=\sigma(\Sigma)\\}$
is the stabiliser of $\sigma(\Sigma)$, and $\mathsf{H}:=\mathsf{G}_{p}$ is the
isotropy at $p$, which is also precisely the set of elements of $\mathsf{G}$
fixing $\sigma(\Sigma)$ pointwise. Recall also that, by [GZ12, Prop. 1.3],
$\Pi$ is a discrete subgroup of $N(\mathsf{H})/\mathsf{H}$ which acts on
$\Sigma$ isometrically and properly discontinuously, making $\sigma$
equivariant, and so that the metric spaces $\Pi\backslash\Sigma$ and
$\mathsf{G}\backslash M$ are isometric.

##  Appendix B The frame bundle

The material contained in this section is standard, but we have decided to
include it for the reader’s convenience, and to introduce our notation.

###  B.1. Basic definitions

Given a smooth Riemannian manifold $(M^{n},g^{M})$, its _frame bundle_ is the
set

$FM=\\{u_{p}\mid p\in M,u_{p}:{\mathbb{R}}^{n}\to T_{p}M\hbox{ is a linear
isometry}\\}.$

We refer to each $u_{p}$ as a frame at $p$. $FM$ has a natural structure of an
$\mathsf{O}(n)$-principal bundle over $M$ with the obvious projection map
$\pi:FM\to M$, $\pi(u_{p})=p$, and where the (right) action of
$h\in\mathsf{O}(n)$ on $FM$ is simply given by composition
$u_{p}\,h:=u_{p}\circ h$. The latter makes sense since by definition
$h:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is a linear isometry.

The Levi-Civita connection of $g^{M}$ is an affine connection on $TM$, thus it
gives rise to a principal connection on $FM$, that is, a decomposition

(20)
$T(FM)=\hat{\mathcal{V}}^{\mathsf{O}}\oplus\hat{\mathcal{H}}^{\mathsf{O}},\qquad\hat{\mathcal{V}}^{\mathsf{O}(n)}=\ker
d\pi.$

More explicitly, a curve $u(t)$ in $FM$ is horizontal (i.e. its velocity
always lies in $\hat{\mathcal{H}}^{\mathsf{O}}$) if and only if the
corresponding frames move by parallel transport along the curve $\pi(u(t))$ in
$M$.

After fixing a bi-invariant metric $g^{\mathsf{O}}$ on $\mathsf{O}(n)$,
defined by an $\operatorname{Ad}(\mathsf{O}(n))$-invariant inner product on
$\mathfrak{so}(n)$, we obtain an $\mathsf{O}(n)$-invariant metric
$\hat{g}^{\mathsf{O}}$ on the vertical distribution by means of identifying
$\hat{\mathcal{V}}^{\mathsf{O}}\simeq\mathfrak{so}(n)$ via fundamental vector
fields. We also have the important pull-back metric $\pi^{*}g^{M}$ on $FM$,
which is only positive semi-definite as it vanishes on
$\hat{\mathcal{V}}^{\mathsf{O}}$, but still highly relevant to the geometry on
$M$. From these, we construct the (positive-definite) lifted Riemannian metric
on $FM$:

$\hat{g}:=\hat{g}^{\mathsf{O}}+\pi^{*}g^{M},$

Notice that, by definition, $\hat{g}$ is constant on fundamental vector
fields. That is, if $U,V\in\mathfrak{so}(n)$ and $\hat{U}^{*},\hat{V}^{*}$
denote the corresponding fundamental vector fields on $FM$, then
$\hat{g}(\hat{U}^{*},\hat{V}^{*})_{u}=g^{\mathsf{O}}(U,V)$ is independent of
$u\in FM$. In particular,

$\hat{g}(\nabla_{\hat{U}^{*}}\hat{V}^{*},X)=-\tfrac{1}{2}X\hat{g}(\hat{U}^{*},\hat{V}^{*})=0$

for all $X\in\hat{\mathcal{H}}^{\mathsf{O}}$, from which it follows that the
$\mathsf{O}(n)$-orbits in $FM$ are totally geodesic.

###  B.2. Lifting isometries

Let $f:M\to M$ be an isometry. Then for each $p\in M$, $df_{p}:T_{p}M\to
T_{f(p)}M$ is a linear isometry. We define $\hat{f}:FM\to FM$ via

$\hat{f}(u_{p}):=df_{p}\circ u_{p}.$

This is clearly a smooth map, and it lifts $f$ in the sense that

$\pi\circ\hat{f}=f\circ\pi.$

Since isometries preserve parallel transport, it is also clear that
$\hat{f}_{*}$ preserves $\hat{\mathcal{H}}^{\mathsf{O}}$. Moreover, using that
$\hat{f}$ commutes with the principal $\mathsf{O}(n)$-action, it follows that
$\hat{f}_{*}$ also preserves $\hat{\mathcal{V}}^{\mathsf{O}}$ and that
$\hat{f}_{*}|_{\hat{\mathcal{V}}^{\mathsf{O}}}={\operatorname{Id}}_{\hat{\mathcal{V}}^{\mathsf{O}}}$.
Hence,

$\hat{f}^{*}(\hat{g})=\hat{g}^{\mathsf{O}}+\hat{f}^{*}\pi^{*}g^{M}=\hat{g}^{\mathsf{O}}+\pi^{*}f^{*}g^{M}=\hat{g}^{\mathsf{O}}+\pi^{*}g^{M}=\hat{g},$

that is, $\hat{f}$ is an isometry of $\hat{g}$.

The key observation at this point is that the lifted isometry $\hat{f}$ has no
fixed points, unless $f={\operatorname{Id}}_{M}$ (in which case also
$\hat{f}={\operatorname{Id}}_{FM}$). Indeed, if $\hat{f}(u_{p})=u_{p}$ then
$f(p)=p$ and $df_{p}\circ u_{p}=u_{p}$ which implies
$df_{p}={\operatorname{Id}}_{T_{p}M}$ since $u_{p}$ is invertible. But an
isometry is completely determined by its first jet, thus
$f={\operatorname{Id}}_{M}$.

###  B.3. Quotients and the frame bundle

The main reason to consider the frame bundle is that, given a proper isometric
action of a Lie group $\mathsf{N}$ on $(M^{n},g)$, the corresponding lifted
action on $(FM,\hat{g})$ is free and isometric, and hence
$\mathsf{N}\backslash FM$ is a smooth Riemannian manifold. Since the lifted
action of $\mathsf{N}$ and the principal $\mathsf{O}(n)$-action commute, the
latter sends $\mathsf{N}$-orbits to $\mathsf{N}$-orbits in $FM$, thus it
induces an action of $\mathsf{O}(n)$ on $\mathsf{N}\backslash FM$ which
happens to be by isometries, with orbit space $\mathsf{N}\backslash M$:

(21) ${FM}$${\mathsf{N}\backslash FM}$${M}$${\mathsf{N}\backslash
M}$$\scriptstyle{\hat{\pi}^{\mathsf{N}}}$$\scriptstyle{\pi}$$\scriptstyle{\check{\pi}}$$\scriptstyle{\pi^{\mathsf{N}}}$

Notice that for each $u_{p}\in FM$, $\mathsf{N}\cdot
u_{p}\cap\mathsf{O}(n)\cdot u_{p}=\mathsf{N}_{p}\cdot u_{p}$, where
$\mathsf{N}_{p}$ denotes the isotropy group at $p$ (for the action on $M$).
Thus, the $\mathsf{N}$\- and $\mathsf{O}(n)$-orbits through $u_{p}$ are
transversal if and only if $\mathsf{N}_{p}$ is discrete.

For each $U\in\mathfrak{n}$ we denoted by $U^{*}\in\mathfrak{X}(M)$ the
corresponding Killing field, and let us denote by
$\hat{U}^{*}\in\mathfrak{X}(FM)$ the Killing field for the lifted action.
Observe that $\hat{U}^{*}$ and $U^{*}$ are $\pi$-related, and hence

$(\pi^{*}g^{M})\left(\hat{U}^{*},\hat{U}^{*}\right)=g^{M}(U^{*},U^{*}).$

Let us assume that $\mathsf{N}$ has discrete principal isotropy groups. Let
$M_{\sf sing}$ denote the union of singular orbits, a closed, stratified
subset of $M$. If $\iota:M_{\sf sing}\to M$ is the inclusion, the
corresponding pull-back bundle $\iota^{*}FM$ is the $\mathsf{O}(n)$-bundle
$FM_{\sf sing}\to M_{\sf sing}$ given by

$FM_{\sf sing}=\\{u_{p}\in
FM:\hat{\mathcal{V}}^{\mathsf{N}}(u_{p})\cap\hat{\mathcal{V}}^{\mathsf{O}}(u_{p})\neq
0\\}.$

Since $\hat{\mathcal{V}}^{\mathsf{N}}$ and $\hat{\mathcal{V}}^{\mathsf{O}}$
are vector subbundles of $T(FM)$, it is clear that $FM_{\sf sing}$ is closed
in $FM$. Moreover, because the actions of $\mathsf{N}$ and $\mathsf{O}(n)$
commute, both vector subbundles are $\mathsf{O}(n)$-invariant, and hence so is
$FM_{\sf sing}$. Notice that $u\in FM_{\sf sing}$ if and only if the orbit
$\mathsf{N}\cdot\pi(u)$ has dimension strictly smaller than the principal
orbits. We also set

$FM_{\sf reg}:=FM\backslash FM_{\sf
sing}=\left\\{u_{p}:(\pi^{*}g^{M})|_{\hat{\mathcal{V}}^{\mathsf{N}}(u_{p})}>0\right\\},$

which is a principal $\mathsf{O}(n)$-bundle over $M_{\sf reg}$.

##  Appendix C Some results on semisimple Lie groups and algebras

In this appendix we prove three important properties of semisimple Lie groups
and algebras for which we were not able to find a direct reference in the Lie
theory literature.

######  Proposition C.1.

[Smi35] The center of a connected semisimple Lie group is a finitely-generated
abelian group.

###### Proof.

It suffices to prove this under the assumption that the Lie group $\mathsf{L}$
is simply-connected, for the center of a Lie group is a quotient of the center
of its universal cover, and quotients of finitely-generated groups are
finitely generated. The center $\mathsf{Z}$ of a simply-connected Lie group
$\mathsf{L}$ is isomorphic to the fundamental group of the quotient group
$\mathsf{L}/\mathsf{Z}$, and it is shown in [Smi35] that the latter is
finitely generated. ∎

Consider now a Cartan involution $\theta$ for the semisimple Lie algebra
$\mathfrak{l}$, with $\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{p}$ the
corresponding Cartan decomposition. There is a corresponding decomposition at
the group level, $\mathsf{L}=\mathsf{K}\exp(\mathfrak{p})$, where
$\mathsf{K}\leq\mathsf{L}$ is the connected Lie subgroup with Lie algebra
$\mathfrak{k}$.

######  Proposition C.2.

If $\mathsf{L}$ is a connected semisimple Lie group then
$N_{\mathsf{L}}(\mathsf{K})=\mathsf{K}$.

###### Proof.

Let $\exp(A)\in N_{\mathsf{L}}(\mathsf{K})\setminus\mathsf{K}$,
$A\in\mathfrak{p}$. For each $k\in\mathsf{K}$ we have

$\exp(A)k\exp(-A)=\tilde{k}\in\mathsf{K},$

from which

$\exp(\operatorname{Ad}(k^{-1})A)=k^{-1}\exp(A)\,k=k^{-1}\,\tilde{k}\exp(A)$

and hence $k=\tilde{k}$ and $\operatorname{Ad}(k)A=A$ by the fact that the
global Cartan decomposition gives a diffeomorphism
$\mathsf{L}\simeq\mathsf{K}\times\mathfrak{p}$. This implies that
$[\mathfrak{k},A]=0$. Using the inner product $B_{\theta}$ we deduce that for
all $C\in\mathfrak{p}$ and $Z\in\mathfrak{k}$ it holds that

$0=B_{\theta}([Z,A],C)=-B_{\theta}(Z,[A,C]),$

hence $[A,C]\in\mathfrak{k}$ vanishes. This yields
$A\in\mathfrak{z}(\mathfrak{l})$, from which $A=0$ and the proposition
follows. ∎

Given any maximal abelian subspace $\mathfrak{a}\subset\mathfrak{p}$ and a
choice of positive roots $\Lambda^{+}\subset\Lambda\subset\mathfrak{a}^{*}$,
we consider the corresponding Iwasawa decomposition

$\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{a}\oplus\mathfrak{n},\qquad\mathfrak{n}=\bigoplus_{\lambda\in\Lambda^{+}}\mathfrak{l}_{\lambda},$

where $\mathfrak{l}_{\lambda}$ denotes the root space corresponding to the
root $\lambda\in\Lambda$. We denote by $\mathfrak{m}$ the centraliser of
$\mathfrak{a}$ in $\mathfrak{k}$. Recall that
$\mathfrak{l}_{0}=\mathfrak{m}\oplus\mathfrak{a}$ normalises $\mathfrak{n}$.
Finally, we consider on $\mathfrak{l}$ the
$\operatorname{Ad}(\mathsf{L})$-invariant inner product
$B_{\theta}(\cdot,\cdot):=-B(\cdot,\theta\cdot)$, where $B$ denotes the
Killing form of $\mathfrak{l}$.

######  Proposition C.3.

Let $\mathfrak{l}$ be a semisimple Lie algebra without compact simple factors.
Then, $\mathfrak{l}$ is the linear span of all Borel nilradicals, when varying
over all possible Iwasawa decompositions.

###### Proof.

Recall that
$\mathfrak{l}=\theta\mathfrak{n}\oplus\mathfrak{m}\oplus\mathfrak{a}\oplus\mathfrak{n}$,
where $\theta\mathfrak{n}$ is the Borel nilradical corresponding to the
opposite choice of positivity for the roots. It remains to be shown that the
linear span of Borel nilradicals contains $\mathfrak{m}\oplus\mathfrak{a}$.

Given $x\in\mathsf{L}$, it is well-known that the decomposition

$\mathfrak{l}=\operatorname{Ad}(x)\mathfrak{k}\oplus\operatorname{Ad}(x)\mathfrak{a}\oplus\operatorname{Ad}(x)\mathfrak{n}$

is another Iwasawa decomposition. Thus, it suffices to show that the linear
span of $\\{\operatorname{Ad}(x)\mathfrak{n}:x\in\mathsf{L}\\}$ contains
$\mathfrak{m}\oplus\mathfrak{a}$. To that end, we will prove that

$\mathfrak{m}\oplus\mathfrak{a}\subset[\mathfrak{n},\theta\mathfrak{n}],$

which is clearly enough.

Suppose that $Z\in\mathfrak{m}\oplus\mathfrak{a}=\mathfrak{l}_{0}$ is such
that $Z\notin[\mathfrak{n},\theta\mathfrak{n}]$. By the Jacobi identity and
the fact that $\theta\in\operatorname{Aut}(\mathfrak{l})$, the subspace
$[\mathfrak{n},\theta\mathfrak{n}]$ is
$\operatorname{ad}(\mathfrak{a})$-stable and $\theta$-stable, hence it
decomposes as a direct sum

$[\mathfrak{n},\theta\mathfrak{n}]=\left([\mathfrak{n},\theta\mathfrak{n}]\cap\mathfrak{m}\right)\oplus\left([\mathfrak{n},\theta\mathfrak{n}]\cap\mathfrak{a}\right)\oplus\bigoplus_{\lambda\in\Lambda}[\mathfrak{n},\theta\mathfrak{n}]\cap\mathfrak{l}_{\lambda}.$

We may therefore assume without loss of generality that either
$Z\in\mathfrak{m}$ or $Z\in\mathfrak{a}$. Since the Killing form $B$ is
definite on each of $\mathfrak{m}$ and $\mathfrak{a}$, and it makes the root
spaces orthogonal, we may also assume that
$Z\perp_{B}[\mathfrak{n},\theta\mathfrak{n}]$. Thus, for each
$X,Y\in\mathfrak{n}$ we have

$0=B(Z,[X,\theta Y])=B([Z,X],\theta Y)=-B_{\theta}([Z,X],Y).$

This implies that $[Z,\mathfrak{n}]=0$ and hence $Z=0$ by Lemma C.4 below, a
contradiction. ∎

######  Lemma C.4.

Under the asumptions of Proposition C.3, we have that
$Z_{\mathfrak{m}}(\mathfrak{n})=Z_{\mathfrak{a}}(\mathfrak{n})=0$.

###### Proof.

Let $Z\in\mathfrak{m}$ with $[Z,\mathfrak{n}]=0$. Then
$[Z,\theta\mathfrak{n}]=[Z,\mathfrak{a}]=0$ as well. Since
$\mathfrak{p}\subset\mathfrak{a}\oplus\mathfrak{n}\oplus\theta\mathfrak{n}$,
we have $[Z,\mathfrak{p}]=0$. By the Jacobi identity this implies that
$[Z,\mathfrak{p}+[\mathfrak{p},\mathfrak{p}]]=0$. Now using the bracket
relations from the Cartan decomposition
$\mathfrak{l}=\mathfrak{k}\oplus\mathfrak{p}$, it is easy to see that
$\mathfrak{p}+[\mathfrak{p},\mathfrak{p}]$ is an ideal in $\mathfrak{l}$,
which intersects all non-compact simple factors. Since $\mathfrak{l}$ has no
compact simple factors,
$\mathfrak{l}=\mathfrak{p}+[\mathfrak{p},\mathfrak{p}]$, and hence
$Z\in\mathfrak{z}(\mathfrak{l})=0$. The proof for
$Z_{\mathfrak{a}}(\mathfrak{n})$ is completely analogous. ∎

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