# Positive Ricci curvature through Cheeger deformation

Leonardo F. Cavenaghi  Instituto de Matemática e Estatística – USP, R. do
Matão, Vila Universitária, 1010, 05508-090, São Paulo - SP, Brazil
<EMAIL_ADDRESS>,  Renato J.M. e Silva  Instituto de Matemática e
Estatística e Computação – UNICAMP, Rua Sérgio Buarque de Holanda,
651,13083-97 Campinas - SP, Brazil<EMAIL_ADDRESS>and  Llohann D.
Sperança  Instituto de Ciência e Tecnologia – Unifesp, Avenida Cesare Mansueto
Giulio Lattes, 1201, 12247-014, São José dos Campos, SP, Brazil
<EMAIL_ADDRESS>
###### Abstract.

Let $(M,g)$ be a Riemannian manifold with an isometric $G$-action. If a
principal orbit has finite fundamental group and
$\operatorname{Ric}_{M^{reg}/G}\geq 1$, Searle–Wilhelm proved that $M$ admits
a new metric $\tilde{g}$ of positive Ricci curvature. $\tilde{g}$ is obtained
after a conformal change followed by a Cheeger deformation. The question
remained on whether it is sufficient to consider only the Cheeger deformation
to attain positive Ricci curvature on the new metric $\tilde{g}$. Here we
approach this question by giving necessary and sufficient conditions on the
$G$-action. In particular, we construct an infinite family of manifolds
satisfying the hypothesis of Searle–Wilhelm that do not develop positive Ricci
curvature after Cheeger deformation. Further exploring the theory, we give a
alternative proofs for Lawson–Yau result on positive scalar curvature under
non-abelian symmetry, among others.

##  1\. Introduction

Producing metrics of positive sectional curvature is a very hard task.
Difficult which is expressed in the great discrepancy between the number of
examples of positive sectional curvature and of non-negative sectional
curvature. On the other hand, a lot of work has been done in what concerns the
existence of metrics with positive Ricci curvature. Nash [Nas79] and Poor
[Poo75] construct metrics of positive Ricci curvature on classes of principal,
sphere and vector bundles, applying the results to the very interesting exotic
spheres. They were generalized in [GW09, Theorem 2.7.3] (see also
Gilkey–Park–Tuschmann [GPT98]). Belegradek–Wei [BW04] construct examples of
vector bundles with positive Ricci curvature over compact manifolds with
almost non-negative Ricci curvature, even without positive Ricci in the base
space.

The main idea on the works above is to lift positive Ricci curvature from the
base to the total space. The same principle is applied by Searle–Wilhelm
[SW15] on the context of $G$-manifolds:

######  Theorem 1.1 (Searle–Wilhelm, [SW15]).

Let $G$ be a compact, connected Lie group acting isometrically and effectively
on a compact Riemannian manifold $(M,g)$. Suppose that:

  1. (SW1)

a principal orbit has finite fundamental group

  2. (SW2)

the orbital distance metric on $M/G$ has Ricci curvature $\geq 1$

Then $M$ admits a $G$-invariant metric $\tilde{g}$ with positive Ricci
curvature.

The metric $\tilde{g}$ is obtained in two steps: one first uses a conformal
change along the singular strata, then uses Cheeger deformation. It was left
the question on whether the Cheeger deformation alone is sufficient to produce
such a Ricci positively curved metric.

Here we explore this question by presenting counter-examples (manifolds
$(M,g)$ satisfying $(SW1)$ and $(SW2)$ such that $g$ has directions with non-
positive Ricci curvature after any Cheeger deformation) and sufficient
conditions on the $G$-action so that every metric satisfying the hypothesis on
Theorem 1.1 has positive Ricci curvature finite Cheeger deformation. In the
former, we say that the Cheeger deformation is effective. The existence of the
counter-examples reinforces the need for a further deformation to obtain
positive Ricci curvature, as in [SW15].

We recall that, given a $G$-invariant metric $g$, the Cheeger deformation is a
smooth family of $G$-invariant metrics $g_{t}$ defined in Cheeger [Che73],
with $g_{0}=g$ (see also [M8̈7, Zil] for further details). The Cheeger
deformation enjoys the property that a suitable reparametrized sectional
curvature is non-decreasing on $t$.

Assuming $(SW1),~{}(SW2)$, we show that the effectiveness of the Cheeger
deformation is solely related to the isotropic representation at singular
points (given a point $p\in M$, we call isotropy representation the
representation $\rho:G_{p}\to O(\mathcal{H}_{p})$, induced by the differential
of the action restricted to the isotropy subgroup at $p$ acting on the space
orthogonal to the orbit, $\mathcal{H}_{p}$), specifically, to the existence of
fixed axes, i.e., vectors $X\in\mathcal{H}_{p}$ satisfying
$\rho(G_{p}^{0})X=X$, where $G_{p}^{0}$ is the identity component of $G_{p}$.

To this aim, we estimate how the second fundamental form of singular fibers
blows up with the deformation (compare [SW15, section 4]). Schur’s Lemma
applied to the curvature tensor explains the algebraic nature of the results.

Next, we state a result that guarantees the effectiveness of the Cheeger
deformation.

######  Theorem 1.2.

Let $(M^{n},g)$ be a Riemannian manifold with an isometric $G$-action, $G$ a
connected Lie group. Assume that:

  1. (SW1)

a principal orbit has finite fundamental group;

  2. (SW2)

the orbital distance metric on $M/G$ has Ricci curvature $\geq 1$.

If $g$ has directions with negative Ricci curvature for every finite Cheeger
deformation, then

  1. (a)

there is a singular point $p\in M$ and a non-zero vector $X\in\mathcal{H}_{p}$
fixed by $\rho({G_{p}^{0}})$

  2. (b)

the restriction of $\rho$ to $X^{\bot}\cap\mathcal{H}_{p}$ is reducible

Furthermore, if $X^{\bot}\cap\mathcal{H}_{p}$ has exactly $n$
$\rho$-irreducible summands,
$X^{\bot}\cap\mathcal{H}_{p}=\mathcal{H}_{1}+\mathcal{H}_{2}+\dots+\mathcal{H}_{n}$,
then (up to changing the order of the summands), for every regular vector
$Y=Y_{1}+Y_{2}+\dots+Y_{n}\in\mathcal{H}_{1}+\mathcal{H}_{2}+\dots+\mathcal{H}_{n},$
there exist $j,i\in\\{1,\dots,n\\}$ such that

$\dim\mathcal{H}_{j}-\dim G_{p}Y_{j}>\dfrac{(k-1)\dim\sum_{l\neq
i}\mathcal{H}_{l}}{\dim\mathcal{H}_{p}-1},$

where $k$ is the codimension of a principal orbit.

In particular, if the Cheeger deformation is not effective on $M$, then $M$
has a geodesic which is totally contained in the singular stratum.

######  Corollary 1.3.

Let $(M,g)$ be a compact Riemannian manifold with a $G$-invariant metric
satisfying $(SW1),~{}(SW2)$, $G$ a connected Lie group. Then, $g$ develops
positive Ricci curvature after a finite Cheeger deformation if one of the
following conditions is true:

  1. $(a)$

The singular strata is composed of isolated orbits;

  2. $(b)$

For every singular point $p$, the linear isotropy has, at most, three
irreducible components;

  3. $(c)$

No isotropy representation have fixed axis;

  4. $(d)$

The induced action of $G_{p}$ on the unitary tangent $T_{1}M$ have no fixed
points for any $p\in M$.

We also conclude a partial local converse of Theorem 1.2:

######  Theorem 1.4.

Let $G$ be a compact connected Lie group acting on $M$. Assume that:

  1. (a’)

there is a point $p$ whose isotropy representation, $\rho:G\to
O(\mathcal{H}_{p})$, has a fixed non-zero vector $X$;

  2. (b’)

there is a $\rho$-invariant decomposition
${\mathcal{H}_{p}}=span\\{X\\}\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}$
where both $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are non-trivial and

$\dim\mathcal{H}_{1}-\dim\rho(G_{p})Y_{1}>\dfrac{(k-1)\dim\mathcal{H}_{1}}{\dim\mathcal{H}_{p}-1}$

for every $\rho$-regular vector $Y\in\mathcal{H}_{p}$, where $Y_{1}$ is the
$\mathcal{H}_{1}$-component of $Y$.

Then there is a $G$-invariant metric $g$ on a neighborhood $U$ of $p$
satisfying

  1. (1)

$\operatorname{Ric}_{U^{reg}/G}\geq 1$

  2. (2)

$\operatorname{Ric}_{g_{t}}(X)<0$ for every Cheeger deformation $g_{t}$ of $g$

Note that Theorem 1.4 does not assume $\mathcal{H}_{1}$ irreducible. The
algebraic nature of the results arises as follow: from [SW15, Proposition 3.1]
one knows that negatively Ricci vectors could only occur on singular points.
Moreover, the estimates on section 3 guarantee that such vectors are fixed by
the isotropy representation at their footpoint, thus, given such a vector
$X\in\mathcal{H}_{p}$, Schur’s Lemma implies that the restriction of
$R_{X}(Y)=R(Y,X)X$ to each $\rho$-irreducible subspace is a multiple of the
identity (recall that $R_{X}$ commutes with the isotropy representation). The
inequality in Theorems 1.2 and 1.4 follows from combinatorial arguments (see
Lemma 6).

In addition to Theorem 1.4, we construct a family of global examples where the
Cheeger deformation is not effective. Due to the algebraic characterization,
we are able to produce relatively simple examples such as doubly twisted
metrics on spheres:

######  Theorem 1.5.

Let $\mathbb{S}^{n}$ be the unit sphere in $\mathbb{R}^{n+1}$ endowed with the
standard $SO(n-2)$-action fixing the first 3 coordinates. Then, for each
$n\geq 5$, there is a $SO(n-2)$-invariant metric $g$ on $\mathbb{S}^{n}$
satisfying:

  * •

$\operatorname{Ric}_{M^{reg}/G}\geq 1$;

  * •

There is a vector $X\in\mathcal{H}_{p}$ such that
$\operatorname{Ric}_{g_{t}}(X)<0$ for every Cheeger deformation $g_{t}$.

In [M8̈7], [Zil] one can find the following formula for a reparametrization of
the sectional curvature:

(1)
$\kappa_{t}(\overline{X},\overline{Y})=R_{g}(\overline{X},\overline{Y},\overline{Y},\overline{X})+\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}+z_{t}(\overline{X},\overline{Y}),$

where $Q$ is a biinvariant metric on $G$, $P$ is the orbit metric tensor and
$z_{t}$ is nonnegative; $\overline{X}=X+U^{*},$ where $X$ is the horizontal
component of $\overline{X}$ and $U^{*}$ its vertical component. One readily
observes that $\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}$ is unbounded whenever
$\|[PU,PV]\|_{Q}^{2}\neq 0$, what guarantees positive Ricci curvature away
from singular points. Our idea is to understand the behavior of the $z_{t}$
term at singular points. We show that $z_{t}(X,Y)$ can be unbounded whenever
$X$ is not fixed by the isotropy representation (Proposition 3.2). However, if
$X$ is fixed by $\rho$, then $z_{t}(X,\mathcal{H}_{p})=0$ and the Ricci
curvature of $X$, for large $t$, is dominated by the horizontal Ricci
curvature $\operatorname{Ric}^{\mathcal{H}}(X)=\sum_{i}R(X,e_{i},e_{i},X)$,
where $\\{e_{i}\\}$ is an orthonormal base for $\mathcal{H}_{p}$ (compare
[SC17, Theorem 6.2]). Thus, essentially, $X$ is not affected by the
deformation and its Ricci curvature will remain non-positive if its horizontal
Ricci curvature is non-positive in the original metric.

Compared to Ricci curvature, scalar curvature is much more susceptible of
becoming positive. Since scalar curvature does not sense particular
directions, the blow-up of $z_{t}$ guarantees positive scalar curvature, as
far as the (connected component of the identity of the) group is not abelian,
providing a simpler proof for Lawson–Yau [LS74]. [LS74] proves that any
manifold $M$ with non-abelian symmetry (more precisely, with an action of a
non-abelian compact connected Lie group $G$) has a metric of positive scalar
curvature. The metric is obtained by shrinking the orbits by a scalar factor
(as the canonical variation in [Bes08]), taking in consideration delicate
estimates near the singular points. On the other hand, the symmetry group is
usually reduced to $S^{3}$ or $SO(3)$. Using the estimates in section 3, one
observes that positive scalar curvature is obtained by Cheeger deformation,
naturally preserving the symmetry group.

######  Theorem 1.6.

Let $(M,g)$ be a compact Riemannian manifold with an effective isometric
$G$-action, where $G$ is a compact connected non-abelin Lie group. Then $g$
develops positive scalar curvature after a finite Cheeger deformation.

We further explore our methods in section 7 by giving alternative proofs for
Theorem 1.1 and, partially, for Grove–Ziller’s result [GZ02, Theorem A]. A
condition for positive sectional curvature on $G$-manifolds, along the lines
of Chavez–Derdzinski–Rigas [CDR92], is presented in section 7.2.

In section 2 we revise the basics on Cheeger deformation, following Müter
[M8̈7] and Ziller [Zil]. In section 3 we study the tensor $z_{t}$ at singular
points and prove item $(a)$ of Theorem 1.2. Theorem 1.5 is proved in section
4, which is essentially independent of the other sections. Sections 5 and 6
translate the geometric criteria for effectiveness into algebraic invariants,
completing the proofs of the remaining results. Section 7 explores further
results.

##  2\. Cheeger deformations on $G$-manifolds

Here we quickly recall the procedure known as Cheeger deformation. We follow
M. Müter [M8̈7] (see also [Zil]).

Let $(M,g)$ be a Riemannian manifold endowed with an isometric $G$-action,
where $G$ is a compact Lie group, and let $Q$ be a biinvariant metric on
$\mathfrak{g}$, the Lie algebra of $G$. For each point $p\in M$, the map
$\phi_{p}(g)=gp$ induces a diffeomorphism of $G\big{/}G_{p}$ onto the orbit
$Gp$, where $G_{p}$ is the isotropy subgroup at $p$. It induces an orthogonal
decomposition $\mathfrak{g}=\mathfrak{g}_{p}\oplus\mathfrak{m}_{p}$, where
$\mathfrak{g}_{p}$ is the Lie algebra of $G_{p}$. In particular,
$\mathfrak{m}_{p}$ is isomorphic to $T_{p}Gp$, the isomorphism being induced
by computing action fields: given an element $U\in\mathfrak{g}$, define

$U^{*}_{p}=\frac{d}{dt}\Big{|}_{t=0}e^{tU}p.$

We denote $\mathcal{V}_{p}=T_{p}Gp$ and call it as _the vertical space at
$p$_.Its orthogonal complement, $\mathcal{H}_{p}$, is _the horizontal space at
$p$_. From now on, we denotoe tangent vectors as $\overline{X},\overline{Y}\in
T_{p}M$, $\overline{X}=X+U^{\ast}_{p},~{}\overline{Y}=Y+V^{\ast}_{p}$, where
$X,Y\in\mathcal{H}_{p}$ and $U,V\in\mathfrak{m}_{p}$. The index $p$ is
supressed from $U_{p}^{*},V_{p}^{*}$ whenever there is no risk of ambiguity.

The main idea in the Cheeger deformation is to consider the product manifold
$M\times G$, observing that the action

(2) $r(p,g):=(rp,gr^{-1})$

is free by isometries of the product metric $g\times\frac{1}{t}Q$, $t>0$. Its
quotient space is diffeomorphic to $M$, since the projection

(3) $\displaystyle\pi^{\prime}:M\times G$ $\displaystyle\to M$
$\displaystyle(p,g)$ $\displaystyle\mapsto gp$

is a principal bundle for action (2), thus inducing a family of submersion
metrics $g_{t}$ on $M$.

We define main tensors:

  * •

The _orbit tensor_ at $p$ is the linear map
$P:\mathfrak{m}_{p}\to\mathfrak{m}_{p}$ defined by

$g(U^{\ast},V^{\ast})=Q(PU,V),\quad\forall
U^{\ast},V^{\ast}\in\mathcal{V}_{p}.$

  * •

For each $t>0$, define $P_{t}:\mathfrak{m}_{p}\to\mathfrak{m}_{p}$ as

$g_{t}(U^{\ast},V^{\ast})=Q(P_{t}U,V),\quad\forall
U^{\ast},V^{\ast}\in\mathcal{V}_{p}.$

  * •

The _metric tensor of $g_{t}$_, $C_{t}:T_{p}M\to T_{p}M$ is defined as

$g_{t}(\overline{X},\overline{Y})=g(C_{t}\overline{X},\overline{Y}),\quad\forall\overline{X},\overline{Y}\in
T_{p}M.$

All three tensors above are symmetric and positive definite. They satisfy (see
[M8̈7] or [Zil]):

######  Proposition 2.1.

The tensors above satisfy:

  * •

$P_{t}=(P^{-1}+t1)^{-1}=P(1+tP)^{-1}$,

  * •

If $\overline{X}=X+U^{\ast}$ then
$C_{t}(\overline{X})=X+((1+tP)^{-1}U)^{\ast}$.

The reparametrization (1) is induced by the tensor $C_{t}^{-1}$:

(4)
$\kappa_{t}(\overline{X},\overline{Y}):=R_{g_{t}}(C_{t}^{-1}\overline{X},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{X})$

where $R_{g}$ stands for the (4,1) Riemannian curvature

$R_{g}(X,Y,Z,W)=g(\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,W).$

######  Theorem 2.2.

Let $\overline{X}=X+U^{\ast},~{}\overline{Y}=Y+V^{\ast}$ be tangent vectors.
Then,

(5)
$\kappa_{t}(\overline{X},\overline{Y})=R_{g}(\overline{X},\overline{Y},\overline{Y},\overline{X})+\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}+z_{t}(\overline{X},\overline{Y}),$

where $z_{t}$ is non-negative.

A proof can be found in [Zil, Proposition 1.3]. A precise definition for
$z_{t}$ is given in Lemma 3.

###  2.1. The Ricci curvature of a Cheeger deformation

To finish the introduction of Cheeger deformation, we present a formula for
the limiting Ricci curvature. Consider $\overline{X}=X+U^{*}\in T_{p}M$ and a
onrthonormal base $\\{e_{1},....,e_{n}\\}$ of $T_{p}M$, where
$\\{e_{k+1},...,e_{n}\\}$ is a base for $\mathcal{H}_{p}$. We consider the
horizontal Ricci curvature

(6)
$\operatorname{Ric}^{\mathcal{H}}(\overline{X}):=\sum_{i=k+1}^{n}R(\overline{X},e_{i},e_{i},\overline{X}).$

######  Lemma 1.

For $\overline{X}\in T_{p}M$,

$\lim_{t\to\infty}\operatorname{Ric}_{g_{t}}(\overline{X})=\operatorname{Ric}_{g}^{\mathcal{H}}(X)+\lim_{n\to\infty}\sum_{i=1}^{n}z_{n}(C_{t}\overline{X},C_{t}^{1/2}e_{i})+\frac{1}{4}\sum_{j}\|[v_{j},U]\|^{2}_{Q},$

where $\\{v_{1},...,v_{k}\\}$ is a $Q$-orthonormal base for
$\mathfrak{m}_{p}$.

###### Proof.

Let $\\{v_{1},\ldots,v_{k}\\}$ be a $Q$-orthonormal basis of eigenvectors of
$P$, with eigenvalues $\lambda_{1}\leq\ldots\leq\lambda_{k}$. Given a
$g$-orthonormal basis $\\{e_{k+1},..,e_{n}\\}$ of $\mathcal{H}_{p}$, consider
the $g$-orthonormal basis $\\{e_{1},\ldots,e_{k},e_{k+1},\ldots,e_{n}\\}$ of
$T_{p}M$, where $e_{i}=\lambda_{i}^{-1/2}v^{\ast}_{i}$ for $i\leq k$.

Using Proposition 2.1, we conclude that the set
$\\{C_{t}^{-1/2}e_{i}\\}_{i=1}^{n}$ is a $g_{t}$-orthonormal basis for
$T_{p}M$. Moreover, $C_{t}^{-1/2}e_{i}=(1+t\lambda_{i})^{1/2}e_{i}$ for $i\leq
k$ and $C_{t}^{-1/2}e_{i}=e_{i}$ for $i>k.$ We claim that the Ricci curvature
$g_{t}$ satisfies:

(7) $\displaystyle\operatorname{Ric}_{g_{t}}(\overline{X})=$
$\displaystyle\operatorname{Ric}_{g}^{\mathcal{H}}(C_{t}\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},C_{t}\overline{X})$
$\displaystyle+\sum_{i=1}^{k}\frac{1}{1+t\lambda_{i}}\Big{(}\kappa_{0}(e_{i},C_{t}\overline{X})+\frac{\lambda_{i}t}{4}\|[v_{i},tP(1+tP)^{-1}U]\|_{Q}^{2}\Big{)}$

To prove equation (7), note that equation (5) gives

$\operatorname{Ric}_{g_{t}}(C_{t}^{-1}\overline{X})=\sum_{i=1}^{n}R_{g_{t}}(C_{t}^{-1/2}e_{i},C_{t}^{-1}\overline{X},C_{t}^{-1}\overline{X},C_{t}^{-1/2}e_{i})=\sum_{i=1}^{n}\kappa_{t}(C_{t}^{1/2}e_{i},\overline{X})\\\
=\sum_{i=1}^{n}\kappa_{0}(C_{t}^{1/2}e_{i},\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},\overline{X})+\frac{t^{3}}{4}\sum_{i=1}^{k}\|[PC_{t}^{1/2}\lambda_{i}^{-1/2}v_{i},PU]\|_{Q}^{2}\\\
=\operatorname{Ric}_{g}^{\mathcal{H}}(\overline{X})+\sum_{i=1}^{n}z_{t}(C_{t}^{1/2}e_{i},\overline{X})+\sum_{i=1}^{k}\frac{1}{1+t\lambda_{i}}\Big{(}\kappa_{0}(e_{i},\overline{X})+\frac{\lambda_{i}t}{4}\|[v_{i},tPU]\|_{Q}^{2}\Big{)}.$

Equation (7) now follows by replacing $\overline{X}$ by $C_{t}\overline{X}$.
Note that $C_{t}\overline{X}\to X$. Therefore,
$\operatorname{Ric}_{g}^{\mathcal{H}}(C_{t}\overline{X})\to\operatorname{Ric}_{g}^{\mathcal{H}}(X)$
and

$\sum_{i=1}^{k}\frac{1}{1+t\lambda_{i}}\kappa_{0}(e_{i},C_{t}\overline{X})\to
0$

and

$\sum_{i=1}^{k}\frac{t\lambda_{i}}{1+t\lambda_{i}}\frac{1}{4}\|[v_{i},tP(1+tP)^{-1}U]\|_{Q}^{2}\to\sum_{i=1}^{k}\frac{1}{4}\|[v_{i},U]\|_{Q}^{2}\qed$

We further notice that scalar curvature behaves in a similar way:

(8)
$scal_{g_{t}}(p)=\sum_{i,j=1}^{n}K_{g}(C_{t}^{1/2}e_{i},C^{1/2}_{t}e_{j})+z_{t}(C_{t}^{1/2}e_{i},C_{t}^{1/2}e_{j})\\\
+\sum_{i,j=1}^{k}\frac{\lambda_{i}\lambda_{j}t^{3}}{(1+t\lambda_{i})(1+t\lambda_{j})}\frac{1}{4}\|[v_{i},v_{j}]\|_{Q}^{2}.$

##  3\. The curvature at singular points

Given a compact connected manifold $M$ endowed with a smooth $G$-action, $G$
compact, there is an open and dense subset $M^{reg}\subset M$, called the
regular part whose orbit type is maximum (see e.g. Bredon [Bre72, IV.3]).
Orbits of smaller dimension are called singular.

The tensor $P$ looses rank when we move from the regular part to a singular
orbit. Such lost of rank directly affects the term
$\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}$ in (1), leaving $z_{t}$ as the only hope
to increase the initial curvature. Here we provide lower estimates for $z_{t}$
on singular orbits, proving Theorem 1.3. We achieve the estimates by using the
continuity of $z_{t}$ along a geodesic leaving the singular part to the
regular part (which always exist, since the regular part is open and dense).

Let $\mathcal{O}\subset M$ be an (singular) orbit and consider a horizontal
geodesic $\gamma$, $\gamma(0)=p\in\mathcal{O}$. Fixed $\gamma$, we identify a
subspace of $\mathcal{H}_{p}$ whose vectors, in some sense, behave like
vertical vectors. This space depends on the choice of $\gamma$.

Note that $\mathcal{H}_{p}$ is an invariant subspace with respect to the
isotropy representation of $G$. Denote the restriction of the isotropy
representation to $\mathcal{H}_{p}$ as $\rho:G_{p}\to O(\mathcal{H}_{p})$.
Observe that the differential of $\rho$ defines a map
$d\rho:\mathfrak{g}_{p}\to\mathfrak{o}(\mathcal{H}_{p})$. Given
$U\in\mathfrak{g}_{p}$, we think of $d\rho(U)$ as a skew-symmetric
endomorphism of $\mathcal{H}_{p}$.

By recalling that the geodesic exponential defines a $G_{p}$-equivariant map
from $T_{p}M$ to $M$, one readily observes that
$d\rho(U)X=(\nabla_{X}U^{*})_{p}$. Note that $d\rho(U)$ is skew-symmetric,
since $U^{*}$ is Killing. More generally, define

$\displaystyle\tilde{S}_{X}:\mathfrak{g}$ $\displaystyle\to T_{p}M$
$\displaystyle U$ $\displaystyle\mapsto\nabla_{X}U^{*}(0).$

Also recall that $U^{*}(s):=U^{*}(\gamma(s))$ is a Jacobi field along
$\gamma$.

######  Lemma 2.

Let $X\in\mathcal{H}_{p}$ and $\gamma(s)=\exp_{p}(sX)$. Then,

  1. $(i)$

$\tilde{S}_{X}(\mathfrak{g}_{p})\subseteq\mathcal{H}_{p}$. Moreover, for
sufficiently small $\epsilon>0$,

$\tilde{\mathcal{H}}_{t}=\begin{cases}\mathcal{H}_{\gamma(t)},~{}&t\neq 0\\\
(\tilde{S}_{X}(\mathfrak{g}_{p}))^{\bot},&t=0\end{cases}$

is a smooth bundle on $\gamma(-\epsilon,\epsilon)$;

  2. $(ii)$

${\mathfrak{g}_{p}}\cap\ker\tilde{S}_{X}=\mathfrak{g}_{X}$, the Lie algebra of
$G_{X}=\\{g\in G_{p}~{}|~{}\rho(g)X=X\\}$.

###### Proof.

We follow Wilking [Wil07] or Gromoll–Walschap [GW09]. Let

$\mathcal{J}=span\\{U^{*}|_{\gamma}~{}|~{}U\in\mathfrak{g}\\}\cup\\{J~{}|~{}J(0)=0,~{}J^{\prime}(0)\in\mathcal{H}_{\gamma(0)}\\}.$

Given $J_{1},J_{2}\in\mathcal{J}$, we have

$g(J_{1}^{\prime}(s),J_{2}(s))=g(J_{1}(s),J_{2}^{\prime}(s))$

for all $s$. That is, $\mathcal{J}$ is a $(n-1)$-dimensional family of normal
Jacobi fields with self-adjoint Ricatti operator, as in Wilking [Wil07] or
Gromoll–Walschap [GW09]. It follows that

(9) $\dot{\gamma}(s)^{\bot}=span\\{J(s)~{}|~{}J\in\mathcal{J}\\}\oplus
span\\{J^{\prime}(s)~{}|~{}J\in\mathcal{J},~{}J(s)=0\\}$

is an orthogonal splitting of $\dot{\gamma}(s)^{\bot}\subset T_{\gamma(s)}M$.
If $U^{*}(0)=0$, for all $V\in\mathfrak{g}$ we have

$g(\nabla_{X}U^{*}(0),V^{*}(0))=g(U^{*}(0),\nabla_{X}V^{*}(0))=0$

In particular, $\nabla_{X}U^{*}(0)\in\mathcal{H}_{p}$. The second statement in
item $(i)$ follows from [GW09, Wil07]. Item $(ii)$ follows directly from the
identity $d\rho(U)X=\nabla_{X}U^{*}(0)$. ∎

Denote by $\mathfrak{p}_{X}:=\mathfrak{g}_{p}\cap(\mathfrak{g}_{X})^{\bot}$,
the $Q$-orthogonal complement of $\mathfrak{g}_{X}$ on $\mathfrak{g}_{p}$. It
follows that:

######  Corollary 3.1.

The restriction
$\tilde{S}_{X}|_{\mathfrak{p}_{X}}:\mathfrak{p}_{X}\to\mathcal{H}_{p}$ is
injective.

######  Definition 1.

Let $p$ be a singular point and $X\in\mathcal{H}_{p}$. Elements in the image
$\tilde{S}_{X}(\mathfrak{p}_{X})$ are called fake horizontal vectors with
respect to $X$. Given $Y\in\mathcal{H}_{p}$, we denote by
$Y_{\mathfrak{p}_{X}}$ the unique element of
$\mathfrak{p}_{X}=\mathfrak{g}_{p}\ominus\mathfrak{g}_{X}$ such that
$\tilde{S}_{X}Y_{\mathfrak{p}_{X}}$ is the orthogonal projection of $Y$ to
$\tilde{S}_{X}(\mathfrak{g}_{p})$.

######  Remark.

The idea of taking the limit of horizontal vectors along horizontal geodesics
is present in Searle-Wilhlelm [SW15]. Here we only provide straightforward
estimates for $z_{t}$. A deeper study can be found in [SW15, Section 4].

###  3.1. The $z_{t}$-term

Our main estimative is the following:

######  Proposition 3.2.

Given $X,Y\in\mathcal{H}_{p}$,

(10) $z_{t}(X,Y)\geq
3t\dfrac{\|\tilde{S}_{X}Y_{\mathfrak{p}_{X}}\|_{g}^{4}}{\|Y_{\mathfrak{p}_{X}}\|^{2}_{Q}}.$

Given $Z\in\mathfrak{g}$, we define the auxiliary $1$-form

(11) $\displaystyle w_{Z}:TM$ $\displaystyle\to\mathbb{R}$ (12)
$\displaystyle\overline{X}$
$\displaystyle\mapsto\textstyle\frac{1}{2}g(\overline{X},Z^{*}),$

where $Z^{*}$ is the action vector associated to $Z$.

To prove Proposition 3.2, we take advantage of Lemma 2 and the following
characterization of $z_{t}$, due to Mütter [M8̈7] (see [M8̈7, Page 23, Lemma
3.9] for a proof 111Contrary to [M8̈7, Zil], we use the convention
$d\omega(X,Y)=X\omega(Y)-Y\omega(X)-\omega([X,Y])$.):

######  Lemma 3.

For every $\overline{X}=X+U^{*},~{}\overline{Y}=Y+V^{*}$, $z_{t}$ satisfies

(13)
$z_{t}(\overline{X},\overline{Y})=3t\max_{\begin{subarray}{c}Z\in\mathfrak{g}\\\
\|Z\|_{Q}=1\end{subarray}}\dfrac{\\{dw_{Z}(\overline{X},\overline{Y})+\frac{t}{2}Q([PU,PV],Z)\\}^{2}}{tg(Z^{\ast},Z^{\ast})+1}.$

Moreover, at regular points,

(14) $\displaystyle dw_{Z}(V^{\ast},X)$
$\displaystyle=\frac{1}{2}Xg(V^{*},Z^{*})=-g(S_{X}V^{*},Z^{*}),$ (15)
$\displaystyle dw_{Z}(X,Y)$
$\displaystyle=-\frac{1}{2}g([X,Y]^{\mathcal{V}},Z^{*})=-g(A_{X}Y,Z^{*}),$

where $A_{X}Y=p_{\mathcal{V}}(\nabla_{X}Y)$,
$S_{X}V^{*}=-p_{\mathcal{V}}(\nabla_{X}V^{*})$ and $p_{\mathcal{V}}$ is the
orthogonal projection onto $\mathcal{V}=\mathcal{H}^{\bot}$.

Given $X\in\mathcal{H}_{p}$, $\gamma(s)=\exp(sX)$ and a fake horizontal vector
$U\in\tilde{S}_{X}(\mathfrak{p}_{X})$, we define

(16) $U(s):=\frac{1}{s}U^{\ast}_{\mathfrak{p}_{X}}(\gamma(s)).$

It is easy to see that $U(s)$ is well defined and that $U(0)=U$. In fact,

(17) $\lim_{s\to
0}U(s)=\nabla_{X}U^{\ast}_{\mathfrak{p}_{X}}(0)=d\rho(U_{\mathfrak{p}_{X}})X=U.$

Our next goal is to compute $dw_{Z}(U,X)$. For any $W\in\mathfrak{g}$, we
write $W^{*}(s):=W^{*}(\gamma(s))$.

######  Claim 1.

Given $U\in\tilde{S}_{X}(\mathfrak{p}_{X})$, $\underset{s\to
0^{+}}{\lim}dw_{Z}(U(s),X(s))=g\left(U,\nabla_{X}Z^{*}(0)\right).$

###### Proof.

For the proof, we assume that the geodesic $\gamma$ intersects a principal
orbit for some, and therefore a dense and open set of, $s>0$. Although this is
not necessarily true for every choice of horizontal vector $X$, there is an
open and dense set of directions with this property. By the continuity of
$z_{t}$, the estimates holds for all directions. We first note that equation
(14) gives

(18)
$\frac{1}{2}Xg(U_{\mathfrak{p}_{X}}^{*},Z^{*})|_{s=0}=dw_{Z}(U_{\mathfrak{p}_{X}}^{*}(0),X)=0,$

since $U_{\mathfrak{p}_{X}}\in\mathfrak{g}_{p}$ and therefore,
$U^{\ast}_{\mathfrak{p}_{X}}(0)=0.$

For $s>0$, again by equation (14) we have

$dw_{Z}(U(s),X(s))=\frac{1}{2s}Xg(U_{\mathfrak{p}_{X}}^{*}(s),Z^{*}(s)).$

Therefore, by equation (18)

$\lim_{s\to 0^{+}}dw_{Z}(U(s),X(s))=\frac{1}{2}\frac{\partial^{2}}{\partial
s^{2}}\Big{|}_{s=0}g(U_{\mathfrak{p}_{X}}^{*}(s),Z^{*}(s))\\\
=\frac{1}{2}\left\\{g\left(\frac{D^{2}}{ds^{2}}U_{\mathfrak{p}_{X}}^{*}(0),Z^{*}(0)\right)+g\left(U_{\mathfrak{p}_{X}}^{*}(0),\frac{D^{2}}{ds^{2}}Z^{*}(0)\right)+2g\left(\frac{D}{ds}U_{\mathfrak{p}_{X}}^{*}(0),\frac{D}{ds}Z^{*}(0)\right)\right\\}.$

Claim (1) follows since
$\frac{D^{2}}{ds^{2}}U_{\mathfrak{p}_{X}}^{*}(0)=-R(U_{\mathfrak{p}_{X}}^{*}(0),X(0))X(0)=0$
and $\frac{D}{ds}U_{\mathfrak{p}_{X}}^{*}(0)=U$ by equation (17). ∎

###### Proof of Proposition 3.2.

Since $z_{t}(X,Y)\geq 0$, Proposition 3.2 follows immediately for
$Y\bot\tilde{S}_{X}(\mathfrak{g}_{p})$ (in this case,
$Y_{\mathfrak{p}_{X}}=0$). Otherwise, take
$Z=Y_{\mathfrak{p}_{X}}/\|Y_{\mathfrak{p}_{X}}\|_{Q}$ in equation (13) and
apply Claim 1. We get

$\displaystyle z_{t}(X,Y)$ $\displaystyle\geq
3t\dfrac{g\\!\left(Y,\frac{D}{ds}Z^{\ast}(0)\right)^{2}}{tg\\!\left(Z^{\ast}(0),Z^{\ast}(0)\right)+1}=3tg\\!\left(Y,\frac{\tilde{S}_{X}Y_{\mathfrak{p}}}{\|Y_{\mathfrak{p}_{X}}\|_{Q}}\right)^{2}=3t\frac{\|\tilde{S}_{X}Y_{\mathfrak{p}_{X}}\|_{g}^{4}}{\|Y_{\mathfrak{p}_{X}}\|_{Q}^{2}},$

since $Z^{*}(0)=0$ and
$\frac{D}{ds}Z^{\ast}(0)=\frac{\tilde{S}_{X}Y_{\mathfrak{p}}}{\|Y_{\mathfrak{p}}\|_{Q}}.$
∎

###  3.2. Sufficient conditions for positive Ricci curvature: Proof of Theorem
1.2 item $(a)$

The goal of this section is to prove item $(a)$ of Theorem 1.2. Corollary 1.3
follows since, in every stated case, item $(a)$ of Theorem 1.2 is
contradicted.

Theorem 1.2 is proved using a convergent (sub)sequence. Assume that $M$
satisfies hypothesis $(1)$ and $(2)$ in Theorem 1.2. By hypothesis, for every
$n$ there is a $g$-unitary vector $\overline{X}_{n}=X_{n}+U^{*}_{n}\in
T_{p_{n}}M$ such that $\operatorname{Ric}_{g_{n}}(\overline{X}_{n})<0$. Taking
a convergent subsequence, there is an $L>0$ and a limiting $g$-unitary vector
$\overline{X}=\lim_{n\to\infty}\overline{X}_{n}$ such that
$\operatorname{Ric}_{g_{t}}(\overline{X})\leq 0$ for all $t>L$. We show that
$p=\lim_{n\to\infty}p_{n}$ lies on a singular orbit (Claim 2) and that
$\rho(G^{0}_{p})X=X$. Using the notation in section 2.1, Lemma 1 gives

(19)
$0\geq\operatorname{Ric}_{g}^{\mathcal{H}}(X)+\lim_{n\to\infty}\sum_{i=1}^{n}z_{n}(C_{t}\overline{X},C_{t}^{1/2}e_{i})+\frac{1}{4}\sum_{j}\|[v_{j},U]\|^{2}_{Q}.$

We conclude that $U=0$, since the last term is positive whenever $U\neq 0$:
recall that $\frac{1}{4}\sum_{j}\|[v_{j},U]\|^{2}_{Q}$ is the Ricci curvature
of the normal homogeneous space $G/G_{p}$, with respect to the metric $Q$.
Such space has positive Ricci curvature as long as $\pi_{1}(G/G_{p})$ is
finite. On the other hand, $\pi_{1}(G/H)$ is finite, where $H$ is a principal
isotropy group. Following Bredon [Bre72, section IV.3], we can assume (up to
conjugation) that $H<G_{p}$. The long homotopy sequence of
$G_{p}/H\hookrightarrow G/H\to G/G_{p}$ gives:

$\cdots\rightarrow\pi_{1}(G/H)\rightarrow\pi_{1}(G/G_{p})\to\pi_{0}(G_{p}/H)\to\cdots$

Therefore, $\pi_{1}(G/G_{p})$ is finite if and only if $G_{p}/H$ has a finite
number of connected components. However, $G_{p}$ is a compact manifold, since
it is a closed subgroup of the compact Lie group $G$, therefore it has only a
finite number of components.

Furthermore, since the $z_{t}$-term in (19) is non-negative, we conclude that
$\operatorname{Ric}_{g}^{\mathcal{H}}(X)\leq 0$. Keeping in mind that
$\operatorname{Ric}_{M^{reg}/G}\geq 1$, the next claim shows that $p$ is not
regular.

######  Claim 2.

Consider the Riemannian submersion
$\pi{:}~{}(M^{reg},g)\to(M^{reg}/G,\bar{g})$. If $p\in M^{reg}$, then,

(20)
$\lim_{t\to\infty}\operatorname{Ric}_{g_{t}}({X})=\operatorname{Ric}_{\bar{g}}(d\pi
X).$

###### Proof.

Claim 2 follows from Proposition 3.3 in [SW15], we provide a proof for the
sake of completeness. As in section 2.1, Observe that

$\lim_{t\to\infty}\operatorname{Ric}_{g_{t}}({X})=\operatorname{Ric}_{g}^{\mathcal{H}}({X})+\sum_{i=1}^{n}\lim_{t\to\infty}z(C_{t}^{1/2}e_{i},{X}),$

as long as $\lim z_{t}(C_{t}^{1/2}e_{i},X)$ exists for all $i$. Lemma 3 (see
also Müter [M8̈7] or Ziller [Zil]) gives:

(21) $\displaystyle z_{t}(X,Y)$
$\displaystyle={3}\max_{\begin{subarray}{c}Z\in\mathfrak{g}\\\
\|Z\|_{Q}=1\end{subarray}}\left\\{\dfrac{g(A_{X}Y,Z^{*})^{2}}{g(Z^{\ast},Z^{\ast})+t^{-1}}\right\\}$
(22) $\displaystyle z_{t}(X,W^{*})$
$\displaystyle=3\max_{\begin{subarray}{c}Z\in\mathfrak{g}\\\
\|Z\|_{Q}=1\end{subarray}}\left\\{\dfrac{g(S_{X}W^{*},Z^{*})^{2}}{g(Z^{\ast},Z^{\ast})+t^{-1}}\right\\}$

for all $Y\in\mathcal{H}_{p}$, $W\in\mathfrak{g}$. In particular,
$z_{t}(C_{t}^{1/2}e_{i},X)\to 0$ for $i\leq k$, since $C_{t}^{1/2}e_{i}\to 0$.
On the other hand, O’Neill’s submersion formula and (5) give

$K_{\bar{g}}(d\pi X,d\pi Y)-K_{g_{t}}(X,Y)=3\|A_{X}Y\|^{2}_{g}-z_{t}(X,Y).$

The claim follows since $z_{t}(X,Y)\to 3\|A_{X}Y\|_{g}^{2}$. ∎

To conclude Theorem 1.2 item $(a)$, note that $d\rho(\mathfrak{g}_{p})X=0$. If
not, there is $Y\in\mathcal{H}_{p}$ such that $\tilde{S}_{X}Y\neq 0$, thus
Proposition 3.2 and Lemma 1 gives $\operatorname{Ric}_{g_{t}}(X)\to+\infty$.

##  4\. Fixed axes and an algebraic description of Ricci curvature

As we see in section 3.2, the (possible) non-effectiveness of the deformation
is directly connected to the existence of fixed axes. In this section we
further explore the Ricci curvature on fixed axes and use it to provide
examples of closed manifolds where the deformation is not effective.

From now on, we consider a fixed Riemannian manifold $(M^{n},g)$ equipped with
an effective isometric $G$-action satisfying the hypothesis in Theorem 1.1.
The next proposition guarantees that the non-effectiveness of Cheeger
deformation is related to the horizontal Ricci curvature on fixed axes,
$\operatorname{Ric}^{\mathcal{H}}_{g}(X)$.

######  Proposition 4.1.

Let $X\in\mathcal{H}_{p}$ be a fixed axis. Then, for every
$Y\in\mathcal{H}_{p},$

(23) $z_{t}(X,Y)=0,~{}\forall t.$

In particular,
$\lim\limits_{t\to\infty}\operatorname{Ric}_{g_{t}}(X)=\operatorname{Ric}_{g}^{\mathcal{H}}(X)$.

###### Proof.

It is enough to prove that $dw_{Z}(X,Y)=0,~{}\forall Z\in\mathfrak{g}.$ Using
the definition of the exterior derivative and recalling that $Z^{*}$ is
Killing, we have

$2dw_{Z}(X,Y)=Xg(Z^{*},Y)-Yg(Z^{*},X)-g([X,Y],Z^{*})=2g(\nabla_{X}Z^{*},Y),$

which is zero since $\tilde{S}_{X}=0$. ∎

Combining Proposition 4.1 and section 3.2, one concludes that $(M,g_{t})$ has
positive Ricci curvature for big $t$ if and only if
$\operatorname{Ric}_{g}^{\mathcal{H}}(X)>0$ for all fixed axis $X$.

######  Corollary 4.2.

Suppose $(M,g)$ satisfies

  1. (1)

a principal orbit has finite fundamental group

  2. (2)

$\operatorname{Ric}_{M^{reg}/G}\geq 1$

Then, $(M,g_{t})$ has positive Ricci curvature for all big $t$ if and only if
$\operatorname{Ric}^{\mathcal{H}}_{g}(X)>0$ for all fixed axis
$X\in\mathcal{H}$.

Moreover, taking Lemma 1 into account, we conclude that
$\operatorname{Ric}_{g_{t}}(X)<0$ for all large $t$ (as in Claim 2,
$z_{t}(X,C^{1/2}_{t}V^{*})\to 0$ for all $V\in\mathfrak{g}$.)

###  4.1. The Ricci tensor on fixed axes

Let $R$ be the Riemannian tensor associated to the Levi-Civita connection of
$g$. Let $X$ be a fixed axis and consider the operator $R_{X}:=R(\cdot,X)X$.
Given $\overline{Y}\in T_{p}M$, we recall that

(24) $\tilde{\rho}(r)\circ
R_{X}(\overline{Y})=R(\tilde{\rho}(r)\overline{Y},\tilde{\rho}(r)X)\tilde{\rho}(r)X=R(\tilde{\rho}(r)\overline{Y},X)X=R_{X}(\tilde{\rho}(r)\overline{Y}),$

for all $r\in G_{p}$, where $\tilde{\rho}:G_{p}\to O(T_{p}M)$ is the full
isotropy representation (not its restriction to the horizontal space). In
particular, Schur’s Lemma ([SS96, Page 13, Proposition 4]) applies and $R_{X}$
is block diagonal on each irreducible subspace of $\tilde{\rho}$. We state it
as a Lemma:

######  Lemma 4.

Let $V\subset T_{p}M$ be $\tilde{\rho}$-irreducible. Then, $R_{X}|_{V}$ is a
multiple of the identity.

In particular, since $\mathcal{H}_{p}$ is $\tilde{\rho}$-invariant,
$R_{X}(\mathcal{H}_{p})\subseteq\mathcal{H}_{p}$. Moreover, since
$R_{X}(X)=0$, $\mathcal{H}_{p}\cap X^{\bot}$ is $R_{X}$-invariant. From now
on, we abuse notation and denote $R_{X}$ for the restriction
$R_{X}:\mathcal{H}_{p}\cap X^{\bot}\to\mathcal{H}_{p}\cap X^{\bot}$, unless
otherwise stated. In this notation,
$\operatorname{Ric}^{\mathcal{H}}_{g}(X)=\operatorname{tr}R_{X}$.

Corollary 4.2 guarantees that the Cheeger deformation is ineffective on $M$
once we construct a metric satisfying conditions (1)-(2) in Corollary 4.2 and
$\operatorname{Ric}^{\mathcal{H}}_{g}(X)<0$ for some fixed axis $X$. Although
it is reasonably simple to produce a metric such that
$\operatorname{Ric}^{\mathcal{H}}_{g}(X)<0$, condition (2) does impose
restrictions to $R_{X}$. Let $l$ be the codimension of a regular orbit. Since
$\lim_{t\to\infty}\operatorname{Ric}_{g_{t}}(X^{\prime})\geq 1$ for every
$X^{\prime}\in\mathcal{H}$ in the regular part (see Claim 2), whenever
$\mathcal{W}\subseteq\mathcal{H}_{p}$ is a $l$-subspace which is the limit of
regular horizontal subspaces,

(25)
$\operatorname{Ric}_{g}^{\mathcal{W}}(X):=\sum_{i=1}^{l-1}R_{g}(X,e_{i},e_{i},X)=\lim_{t\to\infty}\sum_{i=1}^{l-1}R_{g_{t}}(X,e_{i},e_{i},X)\geq
1,$

where $\\{e_{0}=X,e_{1},...,e_{l-1}\\}$ is an orthonormal base for
$\mathcal{W}$; the last equality follows since $z_{t}(X,e_{i})=0$ for all $i$
(Proposition 4.1). However, the set of such $\mathcal{W}$’s can be restricted
enough so that both (25) and $\operatorname{Ric}_{g}^{\mathcal{H}}(X)<0$ hold
together.

Let $c(s)$ be a smooth curve with $c(0)=p$ such that $c(s)\in M^{reg}$ for
$s>0$. For every $s$, $\mathcal{H}_{c(s)}$ defines a curve in $Gr_{l}(TM)$.
Any limiting subspace $\mathcal{W}$ of the curve $\mathcal{H}_{c(s)}\in
Gr_{l}(TM)$, $s\to 0$ must satisfy (25). One concludes that every such
$\mathcal{W}$ arises as the limit of the horizontal space along a curve. We
call such $\mathcal{W}$ as limiting horizontal spaces. Denote the set of all
limiting horizontal spaces at $p$ as $\tilde{\mathcal{W}}_{p}$. Next, we give
an algebraic drescription of such spaces.

######  Lemma 5.

Let $\mathcal{W}\in\tilde{\mathcal{W}}_{p}$. Then there is
$Y\in\mathcal{H}_{p}$ such that
$\mathcal{W}=(\tilde{S}_{Y}\mathfrak{g}_{p})^{\bot}=(d\rho(\mathfrak{g}_{p})Y)^{\bot}$.

###### Proof.

Let $c(s)$ be a smooth curve such that $\mathcal{W}$ is a limit of
$\mathcal{H}_{c(s)}$. A smooth base for $\mathcal{H}_{c(s)}$,
$s\in(0,\epsilon)$, $\epsilon$ small enough, is given by
$\\{\frac{1}{s}v_{1}^{*},...,\frac{1}{s}v_{d}^{*},v_{d+1}^{*},...,v_{k}^{*}\\}$,
where $\\{v_{1},...,v_{k}\\}$ is a basis for $\mathfrak{m}_{c(\epsilon)}$,
$v_{1}^{*},...,v_{d}^{*}\in\mathfrak{g}_{p}$ and $v_{d+1}^{*},...,v_{k}^{*}$
are $Q$-orthogonal to $\mathfrak{g}_{p}$. Since $v_{i}^{*}(c(s))\to 0$ for
$i\leq d$, we have

$\lim_{s\to
0}\frac{1}{s}v^{*}_{i}(c(s))=\nabla_{\dot{c}(0)}v_{i}^{*}=\tilde{S}_{\dot{c}(0)}v_{i}^{*}.$

The Lemma follows since $\\{v_{1},...,v_{d}\\}$ must span $\mathfrak{p}_{X}$.
∎

Given a fixed $X$, since $d\rho(\mathfrak{g}_{p})X=0$, one concludes that
$X\in\mathcal{W}$ for all $\mathcal{W}\in\tilde{\mathcal{W}}_{p}$. In
particular, the Cheeger deformation is not effective as far as
$X\in\mathcal{H}_{p}$ meets the following requirements:

  1. (1’)

$\operatorname{Ric}^{\mathcal{H}}_{g}(X)<0$

  2. (2’)

$\operatorname{Ric}_{g}^{\mathcal{W}}(X)\geq 1$ for all
$\mathcal{W}\in\tilde{\mathcal{W}}_{p}$

Observe that, if $\mathcal{H}_{p}\cap X^{\bot}$ is $\rho$-irreducible,
condition (2’) implies that $\operatorname{Ric}^{\mathcal{H}_{p}}(X)>0$. This
proves condition $(b)$ on Theorem 1.3.

###  4.2. Algebraic description of effectiveness

This section is dedicated to present a combinatorial description of conditions
(1’), (2’). Specifically, we look for algebraic conditions for the existence
of a symmetric operator $R_{X}$ satisfying (1’) and (2’), then, a posteriori,
construct a metric with such a curvature operator.

For clarity sake, we first assume
$\mathcal{H}_{p}=span\\{X\\}\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}$, where
$\mathcal{H}_{i}$ are $\rho$-irreducible subspaces. As it will be clear from
Lemma 6, in this case $R_{X}$ has exactly two distinct eigenvalues
$\lambda_{1},\lambda_{2}$, therefore $\mathcal{H}_{1}\bot\mathcal{H}_{2}$. We
provide an existence result for $\lambda_{1},\lambda_{2}$, such that (1’),
(2’) are met.

######  Remark.

Although the space $\mathcal{H}_{p}$ is metric dependent, the linear action of
$G_{p}$ in $\mathcal{H}_{p}$ is equivalent to the classical isotropy
representation of $G_{p}$ in the quotient $T_{p}M/T_{p}Gp$. Therefore, all
calculations can be done using a fixed complement of $T_{p}Gp$.

######  Remark.

The irreducibility of $\mathcal{H}_{i}$ is only needed to prove Theorem 1.2.
To produce examples of non-effective metrics, we just assume
$\mathcal{H}_{1},\mathcal{H}_{2}$ $\rho$-invariant. In either case, since the
eigenvalues of $R_{X}|_{\mathcal{H}_{1}}$ and $R_{X}|_{\mathcal{H}_{2}}$ must
have different signs, $\mathcal{H}_{1}\perp\mathcal{H}_{2}$.

Consider the complementary projections

$p_{i}:\mathcal{H}_{p}\to\mathcal{H}_{i}.$

Then, for any $G$-invariant metric $g$ and
$Y\in\mathcal{H}_{1}\oplus\mathcal{H}_{2}$,

$R_{X}(Y)=\lambda_{1}\|p_{1}Y\|_{g}^{2}+\lambda_{2}\|p_{2}Y\|_{g}^{2},$

where $\lambda_{i}$ are the constants determined by Schur’s Lemma. In
particular,
$\operatorname{Ric}^{\mathcal{H}}(X)=\lambda_{1}\dim\mathcal{H}_{1}+\lambda_{2}\dim\mathcal{H}_{2}$
and, for $\mathcal{W}\in\tilde{\mathcal{W}}_{p}$,

(26)
$\operatorname{Ric}^{\mathcal{W}}(X)_{g}=\sum_{i=1}^{l}\left(\lambda_{1}\|p_{1}e_{i}\|_{g}^{2}+\lambda_{2}\|p_{2}e_{i}\|_{g}^{2}\right)=\lambda_{1}\operatorname{tr}(p_{1}|_{\mathcal{W}})+\lambda_{2}\operatorname{tr}(p_{2}|_{\mathcal{W}}).$

######  Remark.

Although the definition of $\mathcal{W}=(d\rho(\mathfrak{g}_{p})Y)^{\bot}$ is
metric dependent,

$\operatorname{tr}(p_{1}|_{\mathcal{W}})=\operatorname{tr}(p_{1})-\operatorname{tr}(p_{1}|_{d\rho(\mathfrak{g}_{p})Y})$

is metric independent.

The existence of $\lambda_{1},\lambda_{2}$ satisfying (1’), (2’) is totally
translated as properties of the set

$\mathcal{A}=\\{(\operatorname{tr}(p_{1}|_{\mathcal{W}}),\operatorname{tr}(p_{2}|_{\mathcal{W}}))\in\mathbb{R}^{2}~{}~{}~{}|~{}~{}~{}\mathcal{W}\in\tilde{\mathcal{W}}_{p}\\}$

We arrive at:

######  Problem 1.

Let $\mathcal{A}\subset\mathbb{R}^{2}$ be some given collection of real
numbers satisfying $a+b=l-1~{}\forall(a,b)\in\mathcal{A}$. Find
$\lambda_{1},\lambda_{2}\in\mathbb{R}$ such that

(27) $a\lambda_{1}+b\lambda_{2}\geq 1,~{}\forall(a,b)\in\mathcal{A}$

and,

(28) $\lambda_{1}\dim\mathcal{H}_{1}+\lambda_{2}\dim\mathcal{H}_{2}<0.$

Lemma 6 gives necessary and sufficient conditions to solve Problem 1.

######  Lemma 6.

Denote by $A:=\dim\mathcal{H}_{1}$ and $B:=\dim\mathcal{H}_{2}$. Then, Problem
1 has an affirmative answer if, and only if, either

(29) $\inf_{(a,b)\in\mathcal{A}}\\{a\\}>\frac{A(l-1)}{A+B},$

or

(30) $\inf_{(a,b)\in\mathcal{A}}\\{b\\}>\frac{B(l-1)}{A+B},$

###### Proof.

From conditions (27),(28), it is clear that $\lambda_{1}\lambda_{2}<0$.
Moreover, if $\lambda_{1},\lambda_{2}$ is a solution to Problem 1, so it is
$-\lambda_{1},-\lambda_{2}$, if you interchange the roles of $\mathcal{H}_{1}$
and $\mathcal{H}_{2}$. The two conditions stated on Lemma 6 differs by the
sign of $\lambda_{1}$. Without loss of generality, we assume $\lambda_{1}>0$
and prove (29).

To see that (29) is necessary, suppose Problem 1 has an affirmative answer
with $\lambda_{1}>0$. Let $(a,b)\in\mathcal{A}$. Equation (27) gives
$a>-\frac{\lambda_{2}}{\lambda_{1}}b+\frac{\epsilon}{\lambda_{1}},$ for some
$0<\epsilon<1$, so
$a>(a-(l-1))\frac{\lambda_{2}}{\lambda_{1}}+\frac{\epsilon}{\lambda_{1}}$,
since $a+b=l-1$. We get

$a>(l-1)\frac{-\frac{\lambda_{2}}{\lambda_{1}}}{(1-\frac{\lambda_{2}}{\lambda_{1}})}+\frac{\epsilon}{\lambda_{1}-\lambda_{2}}.$

On the other hand, Equation (28) gives
$\frac{A}{B}<-\frac{\lambda_{2}}{\lambda_{1}}$. Since the function
$f(x)=\frac{x}{x+1}$ is increasing in $]0,\infty[$, we conclude that

$a>(l-1)\frac{\frac{A}{B}}{(\frac{A}{B}+1)}+\frac{\epsilon}{\lambda_{1}-\lambda_{2}}=\frac{(l-1)A}{A+B}+\frac{\epsilon}{\lambda_{1}-\lambda_{2}},$

for every $(a,b)\in\mathcal{A}$, proving condition (29).

Conversely, suppose that there is $\epsilon>0$ such that

$a\geq\frac{A(l-1)+2\epsilon B}{A+B}$

for every $(a,b)\in\mathcal{A}$. Since $a+b=l-1$, we have $a(A+B)-2\epsilon
B\geq A(a+b)$, thus $\frac{a-2\epsilon}{b}\geq\frac{A}{B}$ whenever $b\neq 0$.
Choose $\lambda_{1},\lambda_{2}$ such that $\lambda_{1}>0$ and
$\frac{a-\epsilon}{b}\geq-\frac{\lambda_{2}}{\lambda_{1}}>\frac{A}{B}$. We
obtain $0>\lambda_{1}A+\lambda_{2}B$ and
$\lambda_{1}a+\lambda_{2}b\geq\epsilon$ for every $(a,b)\in\mathcal{A}$,
$b\neq 0$. The Lemma follows by rescalling $\lambda_{1},\lambda_{2}$ since,
when $\lambda_{1}>0$, equation (27) is automatically satisfied for $b=0$. ∎

We conclude an intermediate version of Theorem 1.4:

######  Proposition 4.3.

Suppose $\mathcal{H}_{p}$ has a $\rho$-invariant subspace $\mathcal{H}_{1}$,
$X\notin\mathcal{H}_{1}$, such that

(31)
$\inf_{\mathcal{W}\in\tilde{\mathcal{W}}_{p}}\\{\operatorname{tr}(p_{1}|_{\mathcal{W}})\\}>\frac{(l-1)\dim\mathcal{H}_{1}}{\dim\mathcal{H}_{p}-1}.$

Then, there is a $G$-invariant metric $g$ on a neighborhood $U$ of $p$
satisfying

  1. (1)

$\operatorname{Ric}_{U^{reg}/G}\geq 1$,

  2. (2)

$\operatorname{Ric}_{g_{t}}(X)<0$ for every Cheeger deformation $g_{t}$ of
$g$.

###### Proof.

Once (31) is verified, a direct application of Lemma 6 gives
$\lambda_{1},\lambda_{2}$ such that $R_{X}=\lambda_{1}p_{1}+\lambda_{2}p_{2}$
satisfies $(1),(2)$. This completes the proof, since classical theory should
give a $G$-invariant metric with such prescribed curvature. Nevertheless, it
is worth observing that a double warped product provides an explicit metric.

Consider a biinvariant metric $Q$ on $G$ and recall that there is a
$G$-invariant neighborhood of $p$ equivariantly diffeomorphic to
$G\times_{\rho}\mathcal{H}_{p}$. It is sufficient to define a
$\rho(G_{p})$-invariant metric on $\mathcal{H}_{p}$ such that
$R_{X}=\lambda_{1}p_{1}+\lambda_{2}p_{2}$ and, for some $c>0$,
$\operatorname{Ric}^{\mathcal{W}}\geq c$ for all
$\mathcal{W}\in\tilde{\mathcal{W}}_{p}$.

Write
$\mathcal{H}_{p}\cong\mathbb{R}\times\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$,
where $\mathbb{R}$ stands for the fixed axis and
$\mathbb{R}^{n_{1}},\mathbb{R}^{n_{2}}$ for the two $\rho(G_{p})$-invariant
subspaces $\mathcal{H}_{1},\mathcal{H}_{2}$, respectively. Consider the
metric:

(32)
$\bar{g}=dt^{2}+\phi^{2}\left(t\right)ds^{2}_{\mathbb{R}^{n_{1}}}+\psi^{2}\left(t\right)ds^{2}_{\mathbb{R}^{n_{2}}},$

where $ds^{2}_{\mathbb{R}^{n_{i}}}$ is the standard flat metric of
$\mathbb{R}^{n_{i}}$ and

(33) $\displaystyle\phi(t)$
$\displaystyle=\textstyle\frac{1}{\sqrt{\lambda_{1}}}\sin\left(\sqrt{\lambda_{1}}t\right),$
(34) $\displaystyle\psi(t)$
$\displaystyle=\textstyle\frac{1}{\sqrt{-\lambda_{2}}}\exp\left(\sqrt{-\lambda_{2}}t-b\right),$

for some $b>0$ to be fixed later. In the notation of [Pet06, page 71], given
$~{}V,V^{\prime}\in T\mathbb{R}^{n_{1}},~{}W,W^{\prime}\in
T\mathbb{R}^{n_{2}}$, we have

(35) $\displaystyle R_{\bar{g}}(X,V)$ $\displaystyle=\lambda_{1}X\wedge V,$
(36) $\displaystyle R_{\bar{g}}(X,W)$ $\displaystyle=\lambda_{2}X\wedge W,$
(37) $\displaystyle R_{\bar{g}}(V,V^{\prime})$
$\displaystyle=\lambda_{1}V\wedge V^{\prime},$ (38) $\displaystyle
R_{\bar{g}}(W,W^{\prime})$
$\displaystyle=-\lambda_{2}\left(\exp(b-\sqrt{-\lambda_{2}}t)^{2}-1\right)W\wedge
W^{\prime},$ (39) $\displaystyle R_{\bar{g}}(V,W)$
$\displaystyle=-\sqrt{-\lambda_{1}\lambda_{2}}\cot(\sqrt{\lambda_{1}}t)V\wedge
W,$

where $X\wedge Y(v)=g({Y,v})X-g({X,v})Y$. In particular,
$R_{X}=\lambda_{1}p_{1}+\lambda_{2}p_{2}$. To verify that
$\operatorname{Ric}_{\bar{g}}^{\mathcal{W}}\geq c$ at $p$, identify
$\mathbb{R}\times\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$ with
$\mathcal{H}_{p}$ in such a way that $p$ is the point $(t_{0},0,0)$, where

(40)
$-\sqrt{-\lambda_{1}\lambda_{2}}\cot(\sqrt{\lambda_{1}}t_{0})\geq\max\left\\{\lambda_{1},(1-\lambda_{2})\frac{\dim\mathcal{H}_{p}-1}{(l-1)\dim\mathcal{H}_{1}}\right\\}.$

Choose $b>\sqrt{-\lambda_{2}}t_{0}$. Using (35)-(39) we conclude that

$\operatorname{Ric}^{\mathcal{W}}(\alpha
X+V+W)=\alpha^{2}\operatorname{Ric}^{\mathcal{W}}(X)+\operatorname{Ric}^{\mathcal{W}}(V)+\operatorname{Ric}^{\mathcal{W}}(W),$

for all $\alpha\in\mathbb{R}$. Using $\lambda_{1},\lambda_{2}$ given in Lemma
6, we guarantee that $\operatorname{Ric}_{\bar{g}}^{\mathcal{W}}(X)\geq 1$ and
$\operatorname{Ric}_{\bar{g}}^{\mathcal{W}}(V)\geq\lambda_{1}\|V\|^{2}$
(equations (35),(37) and (39)).
$\operatorname{Ric}_{\bar{g}}^{\mathcal{W}}(W)$, on its turn, satisfies

$\displaystyle\operatorname{Ric}_{\bar{g}}^{\mathcal{W}}(W)\geq
K(X,W)+\operatorname{tr}(p_{1}|_{\mathcal{W}})(-\sqrt{-\lambda_{1}\lambda_{2}}\cot(\sqrt{\lambda_{1}}t_{0}))\|W\|^{2}\geq\lambda_{1}\|W\|^{2},$

where the last inequality follows from (40), since
$\operatorname{tr}(p_{1}|_{\mathcal{W}})\geq\frac{(l-1)\dim\mathcal{H}_{1}}{\dim\mathcal{H}_{p}-1}$.
A suitable rescaling of (32) completes the proof. ∎

Next, we provide global examples where the Cheeger deformation is not
effective. The proof of Theorem 1.4 (in the case of two subspaces) is
completed in section 5, where we prove:

$\displaystyle\inf_{\mathcal{W}\in\tilde{\mathcal{W}}_{p}}\\{\operatorname{tr}(p_{1}|_{\mathcal{W}})\\}=\dim\mathcal{H}_{1}-\dim\rho(G_{p})Y_{1}.$

###  4.3. A family of counter examples

Here we apply Lemma 6 to prove Theorem 1.5. Our model example consists in a
doubly warped metric in the sphere $\mathbb{S}^{5}\subset\mathbb{R}^{6}$, with
the usual mono-axial $SO(3)$-action.

Consider $\mathbb{S}^{5}\subset\mathbb{R}^{6}$ with the standard linear
$SO(3)$-action given by the inclusion $SO(3)\ni A\mapsto diag(1,1,1,A)\in
SO(6)$. Take $p=(1,0,0,0,0,0)$ and $X=(0,1,0,0,0,0)$. Note that
$G_{p}=G_{X}=SO(3).$ Moreover, regular orbits are diffeomorphic to 2-spheres.
Therefore $l$, the dimension of the horizontal space on regular points, is
$3$.

Denote
$\mathcal{H}_{p}=span\\{X\\}\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}$, the
$\rho$-invariant decomposition given by
$\mathcal{H}_{1}=span\\{(0,0,1,0,0,0)\\}$ and $\mathcal{H}_{2}$ as the space
spanned by the last three coordinates. Thus, $A=1$ and $B=3$ in Lemma 6. Let
us show that $\operatorname{tr}(p_{1}|_{\mathcal{W}})>\frac{1}{2}$ for all
$\mathcal{W}\in\tilde{\mathcal{W}}_{p}$:

Given $Y\in\mathcal{H}_{1}\oplus\mathcal{H}_{2}$,
$d\rho(\mathfrak{so}(3))Y\subset\mathcal{H}_{2}$, therefore
$(0,0,1,0,0,0)\in(d\rho(\mathfrak{so}(3))Y)^{\bot}$ for every $Y$. Using Lemma
5, we conclude that $tr(p_{1}|_{\mathcal{W}})=1$ for all $\mathcal{W}$. Lemma
6 guarantees that there are $\lambda_{1},\lambda_{2}$ such that, if $g$
satisfies $R_{X}|_{\mathcal{H}_{i}}=\lambda_{i}\operatorname{id}$, then
$\operatorname{Ric}_{g_{t}}(X)<0$ for all big $t$. To make explicit a global
metric with such a prescribed $R_{X}$, consider the doubly warped metric

(41)
$g=dt^{2}+\phi^{2}\left(t\right)ds^{2}_{\mathbb{S}^{2}}+\psi^{2}\left(t\right)ds^{2}_{\mathbb{S}^{2}},$

where
$t\in\Big{(}\frac{\pi}{2\sqrt{\lambda_{1}}},\frac{\pi}{\sqrt{\lambda_{1}}}\Big{)}$;
$\phi$ as in (33);
$\psi(t)=\frac{1}{\sqrt{-\lambda_{2}}}\sinh(\sqrt{-\lambda_{2}}t)$ for
$t\in\Big{(}\frac{\pi}{2\sqrt{\lambda_{1}}},\frac{\pi}{2\sqrt{\lambda_{1}}}+\epsilon\Big{)}$
and smoothly extended so that (41) defines a smooth metric on
$\mathbb{S}^{5}$. Here the first $ds^{2}_{\mathbb{S}^{2}}$ corresponds to the
first three coordinates in $\mathbb{S}^{5}$ and the second to the last three
coordinates. The fixed points corresponds to $t=\pi/2\sqrt{\lambda_{1}}$
which, by continuity of the metric, have
$R_{X}=\lambda_{1}p_{1}+\lambda_{2}p_{2}$. Moreover, the action is polar and
transitive in the second $\mathbb{S}^{2}$. Thus, its quotient space is a disc
with the warped metric

$\tilde{g}=\textstyle
dt^{2}+\frac{1}{\lambda_{1}}\sin(\sqrt{\lambda_{1}}t)^{2}ds_{\mathbb{S}^{2}}^{2},$

which has constant positive curvature. The sphere $\mathbb{S}^{5}$ with (a
rescaling of) (41) is our first global example satisfying $(SW1),(SW2)$ where
the Cheeger deformation is ineffective.

For higher dimensions, consider $\mathbb{S}^{n}$ with the standard mono-axial
$SO(n-2)$-action fixing the first three coordinates. Take $p=(1,0,0,...,0)$,
$X=(0,1,0,...,0)$, $\mathcal{H}_{1}=span\\{(0,0,1,0,...,0)\\}$ and
$\mathcal{H}_{2}$ its orthogonal complement (in the standard metric in
$\mathbb{R}^{n+1}$). Note that regular orbits have dimension $n-3$. Following
along the same lines, one proves that

$\operatorname{tr}(p_{1}|_{\mathcal{W}})=1>\frac{(l-1)\dim\mathcal{H}_{1}}{\dim\mathcal{H}_{p}-1}=\frac{2}{n-1}$

for every $\mathcal{W}\in\tilde{\mathcal{W}}_{p}$. The analogous doubly warped
metric has all the desired properties.

##  5\. An algebraic criterium for non-positive Ricci curvature

The purpose of this section is to complete the proofs of Theorems 1.2 and 1.4,
by computing $\inf\\{\operatorname{tr}(p_{1}|_{\mathcal{W}})\\}$ as an
invariant of the action. To this aim, fix a $\rho$-invariant inner product on
$\mathcal{H}_{p}$ and let $\mathcal{R}$ be the regular part of
$\mathcal{H}_{p}$ with respect to $\rho$. We prove:

######  Proposition 5.1.

Suppose $d\rho(\mathfrak{g}_{p})X=0$ and
$\mathcal{H}_{p}=span\\{X\\}\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}$ is a
$\rho$-invariant decomposition. For every $Y\in\mathcal{R}$, write $Y_{i}$ as
its $\mathcal{H}_{i}$-component. Then,

$\displaystyle\inf_{\mathcal{W}\in\tilde{\mathcal{W}}_{p}}\\{\operatorname{tr}(p_{1}|_{\mathcal{W}})\\}=\inf_{Y\in\mathcal{R}}\\{\dim\mathcal{H}_{1}-\dim\rho(G_{p})Y_{1}\\}.$

###### Proof.

Let us fix $Y=Y_{1}+Y_{2}\in\mathcal{R}$ and, for every $\alpha>0$, define
$\mathcal{W}^{\alpha}{=}\left(d\rho(\mathfrak{g}_{p})(\alpha
Y_{1}+Y_{2})\right)^{\perp}$. It is sufficient to show that

(42)
$\inf_{\alpha>0}\\{\operatorname{tr}(p_{1}|_{\mathcal{W}^{\alpha}})\\}=\dim\mathcal{H}_{1}-\dim\rho(G_{p})Y_{1}.$

Write
$\mathcal{W}^{\alpha}=\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{12}^{\alpha},$
where

(43) $\displaystyle\mathcal{W}_{1}$
$\displaystyle:=\ker~{}p_{2}\cap\mathcal{W}^{\alpha},$ (44)
$\displaystyle\mathcal{W}_{2}$
$\displaystyle:=\ker~{}p_{1}\cap\mathcal{W}^{\alpha},$ (45)
$\displaystyle\mathcal{W}_{12}^{\alpha}$
$\displaystyle:=\mathcal{W}^{\alpha}\cap\left(\mathcal{W}_{1}\oplus\mathcal{W}_{2}\right)^{\bot},$

where $p_{i}$ is the $\mathcal{H}_{i}$-projection. Note that
$\mathcal{W}_{1},\mathcal{W}_{2}$ do not depend on $\alpha$, since
$\mathcal{W}_{i}=\\{Z_{i}\in\mathcal{H}_{i}~{}|~{}Z_{i}\perp
d\rho(\mathfrak{g}_{p})Y_{i}\\}$. Observe also that the dimension of
$\mathcal{W}^{\alpha}_{12}$ does not depend on $\alpha$. Moreover:

######  Lemma 7.

There are constants $c,C>0$, not depending on $\alpha$, such that

$\frac{1}{{1+c\alpha^{2}}}\leq\frac{\|p_{1}(v_{\alpha})\|^{2}}{\|v_{\alpha}\|^{2}}\leq\frac{1}{{1+C\alpha^{2}}}$

for every $v_{\alpha}\in\mathcal{W}_{12}^{\alpha}$.

###### Proof.

Denote $\mathcal{M}=d\rho(\mathfrak{g}_{p})Y_{1}\oplus
d\rho(\mathfrak{g}_{p})Y_{2}$ and note that
$\mathcal{W}_{12}^{\alpha}\in\mathcal{M}$. Let $\mathcal{M}_{12}^{\alpha}$ be
the orthogonal complement of $\mathcal{W}_{12}^{\alpha}$ in $\mathcal{M}$ and
note that both
$p_{1}|_{\mathcal{M}_{12}^{\alpha}},p_{2}|_{\mathcal{M}_{12}^{\alpha}}$ are
isomorphisms onto their image: an element in $\ker
p_{1}|_{\mathcal{M}_{12}^{\alpha}}$ lies in $\mathcal{H}_{2}$ and is
orthogonal to both $\mathcal{W}_{2}$ and $\mathcal{W}_{12}^{\alpha}$, however
$p_{2}(\mathcal{W}_{2}+\mathcal{W}_{12}^{\alpha})=\mathcal{H}_{2}$ for every
$\alpha>0$. Analogously, $\ker p_{2}|_{\mathcal{M}_{12}^{\alpha}}=\\{0\\}$.
Moreover, given $Z_{1}+Z_{2}\in\mathcal{M}_{12}^{1}$, $\alpha
Z_{1}+Z_{2}\in\mathcal{M}_{12}^{\alpha}$. In particular, there is an
invertible linear map $T:p_{1}(\mathcal{M})\to p_{2}(\mathcal{M})$ such that

$Z\in\mathcal{M}_{12}^{\alpha}\Longleftrightarrow Z=\alpha
Z_{1}+T(Z_{1}),~{}\text{for some }Z_{1}\in p_{1}(\mathcal{M}).$

We conclude that:

$\displaystyle W\in\mathcal{W}_{12}^{\alpha}$
$\displaystyle\Longleftrightarrow W=W_{1}-\alpha(T^{*})^{-1}W_{1},~{}\text{for
some }W_{1}\in p_{1}(\mathcal{M}).$

Using that $\|T\|^{-1}\leq\|T^{-1}\|$ and $\|T^{*}\|=\|T\|$, we have

$\|W_{1}\|^{2}+\alpha^{2}\|T\|^{-2}\|W_{1}\|^{2}\leq\|W\|^{2}\leq\|W_{1}\|^{2}+\alpha^{2}\|(T^{-1})\|^{2}\|W_{1}\|^{2}.\qed$

Now we estimate $\operatorname{tr}(p_{1}|_{\mathcal{W}^{\alpha}}).$ Since
$\mathcal{W}_{2}\subset\ker p_{1}$, we take orthonormal basis
$\\{e_{1},...,e_{d_{1}}\\}$ and $\\{e_{1}^{\alpha},...,e_{d}^{\alpha}\\}$ for
$\mathcal{W}_{1}$ and $\mathcal{W}_{12}^{\alpha}$, respectively, and consider:

$\operatorname{tr}(p_{1}|_{\mathcal{W}^{\alpha}})=\sum_{j}^{d_{1}}\langle
p_{1}e_{j},e_{j}\rangle+\sum_{k}^{d}\langle
p_{1}e_{k}^{\alpha},e_{k}^{\alpha}\rangle=\dim\mathcal{W}_{1}+\sum_{k}^{d}\|p_{1}e_{k}^{\alpha}\|^{2}.$

Lemma 7 gives:

(46)
$\dim\mathcal{W}_{1}+\frac{1}{1+c\alpha}\dim\mathcal{W}_{12}^{\alpha}\leq\mathrm{tr}(p_{1}|_{\mathcal{W}^{\alpha}})\leq\dim\mathcal{W}_{1}+\frac{1}{1+C\alpha}\dim\mathcal{W}_{12}^{\alpha}.$

By taking $\alpha\to\infty$, we conclude that
$\inf_{\alpha}\\{\operatorname{tr}(p_{1}|_{\mathcal{W}^{\alpha}})\\}=\dim\mathcal{W}_{1}$.
On the other hand,
$\mathcal{H}_{1}=\mathcal{W}_{1}+d\rho(\mathfrak{g}_{p})Y_{1}$, therefore
$\dim\mathcal{W}_{1}=\dim\mathcal{H}_{1}-\dim\rho(G_{p})Y_{1}$, completing the
proof.∎

The proof of proposition 5.1 can be adapted when $\mathcal{H}$ is decomposed
in $n$ invariant components:

######  Proposition 5.2.

Suppose $d\rho(\mathfrak{g}_{p})X=0$ and
$\mathcal{H}_{p}=span\\{X\\}\oplus\mathcal{H}_{1}\oplus\mathcal{H}_{2}\oplus\dots\oplus\mathcal{H}_{n}$
is a $\rho$-invariant decomposition. For every $Y\in\mathcal{R}$, write
$Y_{i}$ as its $\mathcal{H}_{i}$-component. Then, there exists
$j_{0}\in\\{1,2,\dots,n\\}$ for which

$\displaystyle\inf_{\mathcal{W}\in\tilde{\mathcal{W}}_{p}}\\{\operatorname{tr}(p_{j_{0}}|_{\mathcal{W}})\\}=\inf_{Y\in\mathcal{R}}\\{\dim\mathcal{H}_{j_{0}}-\dim\rho(G_{p})Y_{j_{0}}\\}.$

###### Proof.

Following the proof of 5.1, fix $Y=Y_{1}+\dots+Y_{n}\in\mathcal{R}$ and for
every $\alpha>0$, define
$\mathcal{W}^{\alpha}{=}\left(d\rho(\mathfrak{g}_{p})(\alpha
Y_{1}+Y_{2}+\dots+Y_{n})\right)^{\perp}$, where $Y_{i}\in\mathcal{H}_{i}$.

Now, take $j_{0}$ given by Lemma 6 and let
$\mathcal{W}^{\alpha}=\mathcal{W}_{1}\oplus\mathcal{W}_{2}\oplus\mathcal{W}_{12}^{\alpha},$
where

(47) $\displaystyle\mathcal{W}_{1}$ $\displaystyle:=(\bigcap_{i\neq
j_{0}}\ker~{}p_{i})\cap\mathcal{W}^{\alpha},$ (48)
$\displaystyle\mathcal{W}_{2}$
$\displaystyle:=\ker~{}p_{j_{0}}\cap\mathcal{W}^{\alpha},$ (49)
$\displaystyle\mathcal{W}_{12}^{\alpha}$
$\displaystyle:=\mathcal{W}^{\alpha}\cap\left(\mathcal{W}_{1}\oplus\mathcal{W}_{2}\right)^{\bot},$

where $p_{i}$ is the $\mathcal{H}_{i}$ projection. As in Proposition 5.1,
$\mathcal{W}_{1},\mathcal{W}_{2}$ and the dimension of
$\mathcal{W}_{12}^{\alpha}$ does not depend on $\alpha$. Moreover, Lemma 7
remains valid in this case and we can estimate
$\operatorname{tr}(p_{j_{0}}|_{\mathcal{W}^{\alpha}})$ as well.

The same calculation did in Proposition 5.1 shows that
$\inf_{\alpha}\\{\operatorname{tr}(p_{j_{0}}|_{\mathcal{W}^{\alpha}})\\}=\dim\mathcal{W}_{1}$
and since the dimensions are finite, it is clear that
$\mathcal{W}_{1}{=}\mathcal{H}_{j_{0}}\cap\left(d\rho(\mathfrak{g}_{p})Y_{j_{0}}\right)^{\perp}$
which completes the proof.

∎

##  6\. The case of $n$ invariant subspaces

If there are a decomposition in $n$ invariant subespaces
$\mathcal{H}=\mathcal{H}_{1}\oplus\mathcal{H}_{2}\oplus\dots\oplus\mathcal{H}_{n}$,
a similar formulation for the existence of a metric satisfying (1’) and (2’)
can be given.

######  Problem 2.

Let $\mathcal{A}\subset\mathbb{R}^{n}$ be some given collection of real
numbers satisfying
$a_{1}+a_{2}+\dots+a_{n}=l-1~{}\forall(a_{1},\dots,a_{n})\in\mathcal{A}$. Find
$\lambda_{1},\lambda_{2},\dots,\lambda_{n}\in\mathbb{R}$ such that

(50) $\sum_{i=1}^{n}a_{i}\lambda_{i}\geq
1,~{}\forall(a_{1},\dots,a_{n})\in\mathcal{A}$

and,

(51) $\sum_{i=1}^{n}\lambda_{1}\dim\mathcal{H}_{1}<0.$

Since the geometric data plays no role in the proof of Lemma 6, we can solve
Problem 2:

######  Lemma 8.

Let $A_{1},\dots,A_{n}$ be positive real numbers. Then, Problem 2 has an
affirmative answer if, and olny if, there exist $i_{0}\neq
j_{0}\in\\{1,\dots,n\\}$ such that

(52)
$\inf_{(a_{1},\dots,a_{n})\in\mathcal{A}}\\{a_{j_{0}}\\}>\frac{(l-1)\sum_{j\neq
i_{0}}A_{j}}{\sum_{k=1}^{n}A_{k}},$

###### Proof.

If Problem 2 has an affirmative answer with solution
$\lambda_{1},\dots,\lambda_{n}$, define
$\lambda_{i_{0}}=\max_{i}\\{\lambda_{i}\\}$ and
$\lambda_{j_{0}}=\min_{j}\\{\lambda_{j}\\}$. Now, inequality (52) follows from
Lemma 6 applied to the problem

(53) $\displaystyle\Big{(}\sum_{i\neq
j_{0}}a_{i}\Big{)}\lambda_{i_{0}}+a_{j_{0}}\lambda_{j_{0}}$ $\displaystyle\geq
1,$ (54) $\displaystyle A_{i_{0}}\lambda_{i_{0}}+\Big{(}\sum_{j\neq
i_{0}}A_{j}\Big{)}\lambda_{j_{0}}$ $\displaystyle<0,$

where $(a_{1},\dots,a_{n})\in\tilde{\mathcal{A}}=\\{(\sum_{i\neq
j_{0}}a_{i},a_{j_{0}}),(a_{1},\dots,a_{n})\in\mathcal{A}\\}\subset\mathbb{R}^{2}$.

The converse follows straightforward from Lemma 6 by choosing $i_{0}\neq
j_{0}$ arbitrarily and setting $\lambda=\lambda_{j}=\lambda_{i}$ for all
$j,i\neq j_{0}$. ∎

##  7\. Further results

###  7.1. The Lawson–Yau Theorem

As an application of Proposition 3.2, we provide a simpler proof222C. Searle
and F. Wilehlm were already aware that Cheeger deformation yields this result.
for Lawson–Yau Theorem [LS74]:

######  Theorem 7.1.

Let $(M,g)$ be a compact Riemannian manifold with an effective isometric
$G$-action, where $G$ is a compact connected non-abelian Lie group. Then
$g_{t}$ has positive scalar curvature, for a finite Cheeger deformation
$g_{t}$.

###### Proof.

We argue by contradiction. Assume that for each $n\in\mathbb{N}$ there is
$p_{n}\in M$ such that $scal_{g_{n}}(p_{n})\leq 0$. Possibly passing to a
convergent subsequence, we obtain a limit point $p\in M$ such that
$scal_{g_{n}}(p)\leq 0$ for all big $n$.

By taking $t\to\infty$, observe that the rightmost term on equation (8)
diverges to $+\infty$, unless $p$ is singular. However, if $p$ is a singular
point, there is a pair of horizontal vector $X,Y$ such that $Y$ is a fake
horizontal with respect to $X$. Taking $X=e_{k+1},~{}Y=e_{k+2}$, Proposition
3.2 gives $scal_{g_{n}}(p)\geq scal_{g}(p)+z_{n}(e_{k+1},e_{k+2})\to+\infty$,
a contradiction with the choice of $p$. ∎

###  7.2. The curvature limit of a Cheeger deformation

Here we explore the behavior of the sectional curvature of a Cheeger
deformation when $t\to\infty$. We do so by providing a
Cheeger–Derdzinski–Rigas type of condition for positive sectional curvature
(compare [CDR92]).

As accounted in [SSW15], the Cheeger deformation ‘renormalizes’ the foliation:
as $t\to\infty$, $g_{t}$ gets closer to a metric with totally geodesic orbits.
So much so that (recalling that Cheeger deformation does not produce negative
curvature) one is tempted to conjecture the following:

######  Conjecture 7.2.

Suppose that $G$ is compact and acts freely by isometries on a compact $M$. If
$M$ has positive sectional curvature, then $M$ admits a metric with
nonnegative sectional curvature and totally geodesic orbits.

For instance, this conjecture implies the dimension restriction conjectured by
Petersen–Wilhelm conjecture (see [GÁR17], for instance) in the case of
principal bundles (see [Spe17]).

However the Conjecture does not follow from Cheeger deformations, although one
could proceed in the following way: define $\tilde{g}_{t}$ through the
following three conditions: the original horizontal space, $\mathcal{H}$, is
$\tilde{g}_{t}$-orthogonal to $\mathcal{V}$;
$\tilde{g}_{t}|_{\mathcal{H}}=g|_{\mathcal{H}}$; and
$\tilde{g}_{t}(U^{*},V^{*})=t^{-1}Q(U,V)$ for all $U,V\in\mathfrak{g}$. As a
bilinear forms, $g_{t}$ approaches $\tilde{g}_{t}$, roughly, as $1/t^{2}$. One
could try to use this proximity to control the curvature of $g_{t}$. However
the (vertizonthal) curvatures of $g_{t}$ decays as $1/t^{2}$ and the distance
between the $R_{g_{t}}$ and $R_{\tilde{g}_{t}}$ seems to go as $1/t$. This
‘renormalization’ resembles the procedure in [CSW15].

This behavior of the curvature becomes explicit when we take its limit, as a
Cheeger-Derdzinski-Rigas type condition, [CDR92]. Here we present a necessary
and sufficient condition for a $G$-manifold to have positive sectional
curvature after finite Cheeger deformation. Although seemingly hard to apply,
this condition enjoys the property to have the intrinsic geometry of the fiber
(decoded as the tensor $P$) decoupled from the integrability tensor.

Fixed $Q$, for every $p$ in the regular part of $M$, we denote by
$\Omega:\mathcal{H}_{p}\times\mathcal{H}_{p}\to\mathfrak{m}_{p}$ the bundle
curvature of $M^{reg}\to M^{reg}/G$, associated with the distribution
$\mathcal{H}$. Specifically, we define $\Omega(X,Y)$ as the unique element in
$\mathfrak{m}_{p}$ such that $\Omega(X,Y)^{*}=-2A_{X}Y$ (see [GW09, page 70]).
Analogously, we set:

$g(\Omega_{X}^{*}V^{*},Y)=Q(\Omega(X,Y),V)=-2g(A^{*}_{X}P^{-1}V^{*},Y).$

In particular, $A^{*}P^{-1}V^{*}=-2\Omega^{*}V$. Denote by $B$ the quotient
$M^{reg}/G$, endowed with the quotient metric.

######  Theorem 7.3.

Suppose $G$ compact and that $G/G_{p}$ has positive sectional curvature, as a
normal homogeneous manifold, for every regular $p\in M$. Then
$\sec_{g_{t}}>0$, for some $t$, if and only if there is $k>0$ such that

(55) $(R_{B}(X,Y)-k\|X\wedge
Y\|^{2}_{g})(\tfrac{1}{2}Q(\operatorname{Hess}P^{-1}(X)V,V)+\tfrac{1}{4}\|\Omega^{*}_{X}V\|_{g}^{2}-k\|X\|^{2}_{g}Q(P^{-1}V,V))\\\
\geq\tfrac{1}{4}Q((\nabla_{X}\Omega)_{X}Y,V)^{2}$

for every $X,Y$ horizontal, $V\in\mathfrak{m}_{p}$ and $p$ in the regular part
of $M$.

Here we think of $P$ as a function from $M$ to the linear endomorpisms of
$\mathfrak{g}$ (setting $P\mathfrak{g}_{p}=\\{0\\}$). Given a basis of
$\mathfrak{g}$, $\operatorname{Hess}$ is computed as the Hessian of each entry
in the matrix of $P$ on that basis.

######  Remark.

###### Proof.

From (5) and the continuity of $\kappa$, one notes that $\sec_{g_{t}}>0$ for
some $t$ if and only if $\lim_{t\to\infty}\kappa_{t}(X,Y+V^{*})\geq
k\|X+(Y+V^{*})\|^{2}_{g}$ for some $k>0$. Moreover, it is sufficient to prove
that

(56) $\lim_{t\to\infty}\kappa_{t}(L_{t}X,L_{t}(Y+V^{*}))\geq
k\|X+(Y+V^{*})\|^{2}_{g},$

for any continuous family of linear isomorphisms $L_{t}$. We choose
$L_{t}(Y+V^{*})=Y+t^{\frac{1}{2}}C_{t}^{\frac{1}{2}}P^{-\frac{1}{2}}V^{*}$.

######  Remark.

We motivate the choice of $L_{t}$: given $\overline{X},\overline{Y}$
$g$-orthonormal, recall that
$C^{-\frac{1}{2}}_{t}\overline{X},C^{-\frac{1}{2}}_{t}\overline{Y}$ are
$g_{t}$-orthonormal. Thus, it makes sense to compute

$R_{g_{t}}(C_{t}^{-\frac{1}{2}}X,C_{t}^{-\frac{1}{2}}(Y+V^{*}),C_{t}^{-\frac{1}{2}}(Y+V^{*}),C_{t}^{-\frac{1}{2}}X)=\kappa_{t}(C_{t}^{\frac{1}{2}}X,C_{t}^{\frac{1}{2}}(Y+V^{*})).$

Moreover, the right-hand-side decays as $t^{-2}$, thus $t^{\frac{1}{2}}$ in
$L_{t}$ corrects the decayment of $C_{t}^{\frac{1}{2}}$. The term
$P^{-\frac{1}{2}}$ is a time-independent reparametrization that ‘renormalize’
the foliation: note that
$g(P^{-\frac{1}{2}}V^{*},P^{-\frac{1}{2}}U^{*})=Q(V,U)$. I.e, the
reparametrization $P^{-\frac{1}{2}}$ returns (a multiple of) the natural
metric one would expect if the fibers were totally geodesic (compare with
[SSW15]).

The much simpler choice $L_{t}=P^{-1}$ also gives the desired result, however
the operator $t^{\frac{1}{2}}C^{\frac{1}{2}}P^{-\frac{1}{2}}$ shows up
naturally in section 7.3 and is, somehow, consistent with the choices around
(7).

Noting that $L_{t}\to P^{-1}$, we separate the limit of
$\kappa_{t}(L_{t}X,L_{t}(Y+V^{*}))$ in four terms:

(57)
$\lim_{t\to\infty}\kappa_{t}(X,Y+t^{\frac{1}{2}}C_{t}^{\frac{1}{2}}P^{-\frac{1}{2}}V^{*})=\\\
=\kappa_{0}(X,Y)+2R_{g}(X,Y,P^{-1}V^{*},X)+\kappa_{0}(X,P^{-1}V^{*})+z_{\infty}(X,Y+P^{-1}V^{*}),$

observing that (see Lemma 3):

(58) $z_{\infty}(X,Y+V^{*})=3\|A_{X}Y+S_{X}V^{*}\|^{2}_{g}.$

Considering (57) as a quadratic equation in $V^{*}$, one concludes that
$\lim_{t\to\infty}\kappa_{t}(X,Y+t^{\frac{1}{2}}C_{t}^{\frac{1}{2}}P^{-\frac{1}{2}}V^{*})\geq
k\|X\wedge(Y+V^{*})\|_{g}^{2}$ if and only if

$(R_{B}(X,Y)-k\|X\wedge
Y\|^{2}_{g})(\kappa_{0}(X,P^{-1}V^{*})-3\|S_{X}P^{-1}V^{*}\|_{g}^{2}-k\|X\|^{2}_{g}Q(P^{-1}V,V))\\\
\geq(R_{g}(X,Y,P^{-1}V^{*},X)+3\left\langle
S_{X}P^{-1}V^{*},A_{X}Y\right\rangle)^{2}.$

Let $\gamma$ be the geodesic defined by $X$. By considering $V^{*}$ as a
Killing field restricted to $\gamma$, we conclude that $V^{*}$ is a holonomy
Jacobi field along $\gamma$ (see [GW09, Definition 1.4.3]), i.e.,
$(\nabla_{X}V^{*})^{v}=-S_{X}V^{*}$. Moreover,

$g(W^{*},P^{-1}_{s}V^{*})=Q(P_{s}W,P_{s}^{-1}V)=Q(W,V)$

is constant on for every $W\in\mathfrak{g}$. In order to identify

(59)
$\kappa_{0}(X,P^{-1}V^{*})-3\|S_{X}P^{-1}V^{*}\|^{2}=\tfrac{1}{2}Q(\operatorname{Hess}P^{-1}(X)V,V)+\tfrac{1}{4}\|\Omega^{*}_{X}V\|_{g}^{2},$

one observes that

(60) $\nabla_{X}^{v}(P^{-1}_{s}V)=S_{X}(P^{-1}_{s}V),$

then uses O’Neill equations and a straightforward computation (equivalently,
$P_{s}^{-1}V^{*}$ is a dual holonomy field, as in [Spe11, Proposition 4.1,
item (1)]; the desired inequality is [Spe11, Proposition 4.2].)

The term $R_{g}(X,Y,P^{-1}V^{*},X)+3g({S_{X}P^{-1}V^{*},A_{X}Y})$ appears in
[Spe17] and can be identified with
$\tfrac{1}{4}\tfrac{d}{dt}Q(\Omega(X,Y),V)$: recall from [GW09] that

(61)
$R_{g}(X,Y,P^{-1}V^{*},X)=g((\nabla_{X}A)_{X}Y,P^{-1}V^{*})-2g(S_{X}P^{-1}V,A_{X}Y).$

For the next calculation, consider $\nabla_{X}^{h}X=\nabla_{X}^{h}Y=0$.
Dualizing and applying (60) gives

$R_{g}(X,Y,P^{-1}V^{*},X)+3g({S_{X}P^{-1}V^{*},A_{X}Y})=\\\
-\tfrac{1}{4}XQ(\Omega_{X}Y,V)-g(A_{X}Y,\nabla_{X}(P^{-1}V^{*}))+g(A^{*}_{X}{S_{X}P^{-1}V^{*},Y})\\\
=\tfrac{1}{4}Q((\nabla_{X}\Omega)_{X}Y,V).\qed$

###  7.3. Positive Ricci on cohomogeneity one manifolds

In the spirit of the limit deformation, we sketch a simpler proof for a
special case of Theorem A in Grove–Ziller [GZ02]:

######  Theorem 7.4 (Grove–Ziller, [GZ02]).

A compact cohomogeneity one manifold admits an invariant metric with positive
Ricci curvature if and only if its fundamental group is finite.

Precisely, we prove:

######  Theorem 7.5.

Suppose $M$ is a compact cohomogeneity one manifold with two singular orbits
and whose principal orbits have finite fundamental group. Then there is an
invariant metric $g$ on $M$ such that $g_{t}$ has positive Ricci curvature for
all large $t$.

We deal with the case where the quotient space is a closed interval and write
$M/G=[0,R]$. Let $\pi:M\to[0,R]$ be the quotient projection and
$\gamma:[0,R]\to M$ be a horizontal geodesic satisfying $\pi\circ\gamma(s)=s$.
Denote $\dot{\gamma}=X$. It is sufficient to show that
$\operatorname{Ric}_{g_{t}}>0$ at the points on $\gamma$. For the proof we
still use diagonal metrics, as in [GZ02], and a disc bundle argument to
provide suitable smoothness conditions at the singular orbits. In the process
we also show that not every invariant metric has positive Ricci curvature,
even after Cheeger deformation. This emphasizes the fact that $M/G$ is flat,
contrasting with Theorem 1.1.

The natural steps would be to produce an initial metric $g$ and then use a
sequential argument as in section 7.1. To motivate the initial metric we
invert the order of the steps, making explicit whatever conditions $g$ must
satisfy before producing $g$. We keep the notation from section 2.1 and denote
$P_{s}$ as the orbit tensor at $\gamma(s)$. We prove:

######  Proposition 7.6.

Suppose $g$ is an invariant metric on $M$, a cohomogeneity one manifold with
finite fundamental group. Then $g_{t}$ has positive Ricci curvature for all
large $t$ if and only if, there is $c>0$ such that, for all $s\in(0,R)$,

$\frac{d^{2}}{ds^{2}}\operatorname{tr}P^{-1}_{s}\geq c.$

###### Proof.

Suppose $g$ satisfies the hypothesis and consider the function
$F(\overline{Y},t)=t\operatorname{Ric}_{g_{t}}(\overline{Y})$, where
$\overline{Y}$ has unit $g$-norm (the $t$-term is used to renormalize decaying
curvature). Arguing by contradiction, we suppose that for every
$n\in\mathbb{N}$ there is $\overline{Y}_{n}$ such that
$F(\overline{Y}_{n},n)\leq 0$. Taking limits (possibly of a subsequence) we
arrive at a $g$-unit $\overline{Y}$ such that $\lim F(\overline{Y},t)\leq 0$.

Denote by $p\in\gamma([0,R])$ the footpoint of $\overline{Y}$ and
$\overline{Y}=Y+V^{*}$ the horizontal-vertical decomposition of
$\overline{Y}$. Note that $p$ must be regular, since
$d\rho(\mathfrak{g}_{p})Y\neq\\{0\\}$ whenever $p$ is a singular point (recall
for instance that $G_{\gamma(0)},G_{\gamma(R)}$ act transitively on their
respective horizontal spheres, as noted in [GZ00, GZ02]), therefore, the
$z_{t}$-term in Lemma 1 is unbounded whenever $Y\neq 0$ (Proposition 3.2). On
the other hand, if $\overline{Y}$ has no horizontal component,
$\lim\operatorname{Ric}_{g_{t}}(\overline{Y})>0$, since the orbits have finite
fundamental group (as in section 3.2).

If $p$ is regular, $\operatorname{Ric}_{g}^{\mathcal{H}}(Y)=0$. In particular
$V^{*}=0$, since
$\lim\limits_{t\to\infty}\operatorname{Ric}_{g_{t}}(\overline{Y})\geq\frac{1}{4}\sum\|[v_{i},V]\|^{2}_{Q}$,
which is positive whenever $V^{*}\neq 0$, since principal orbits have finite
fundamental group. On the other hand, (7) gives

(62)
$\lim_{t\to\infty}F(X,t)=\lim_{t\to\infty}\sum_{i=1}^{k}\frac{t}{1+t\lambda_{i}}\Big{(}\kappa_{0}(e_{i},X)+z_{t}(e_{i},X)\Big{)}\\\
=\sum_{i=1}^{k}\Big{(}\kappa_{0}({\lambda_{i}^{-1}}v_{i}^{*},X)+3\|S_{X}({\lambda_{i}^{-1}}v_{i}^{*})\|^{2}_{g}\Big{)}=\sum_{i=1}^{k}\Big{(}\kappa_{0}(P^{-1}_{s}v_{i}^{*},X)+3\|S_{X}(P^{-1}_{s}v_{i}^{*})\|^{2}_{g}\Big{)}.$

The Proposition follows since, as in (59),

(63)
$\kappa_{0}(P^{-1}_{s}v_{i}^{*},X)+3\|S_{X}(P^{-1}_{s}v_{i}^{*})\|^{2}_{g}=\frac{1}{2}\frac{d^{2}}{ds^{2}}Q(P^{-1}_{s}v_{i},v_{i}).\qed$

Recall that both $\mathfrak{m}_{\gamma(s)}$ and $G_{\gamma(s)}$ do not depend
on $s$, for $s\in(0,R)$. Denote $\mathfrak{m}_{\gamma(R/2)}=\mathfrak{n}$ and
$G_{\gamma(R/2)}=H$. A diagonal metric is a metric satisfying
$P_{s}|_{\mathfrak{n}_{i}}=f_{i}(s)^{2}\operatorname{id}$, where
$\mathfrak{n}=\oplus\mathfrak{n}_{i}$ is a $Ad(H)$-invariant decomposition of
$\mathfrak{n}$.

One concludes the proof of Theorem 7.5 if one guarantees that all functions
$f_{i}$ are concave and, for each $s\in(0,R)$, there is at least one $i$ such
that $f_{i}^{\prime\prime}<0$. But such functions are easily produced by first
considering bundle metrics on the tubular neighborhoods of each singular
orbit, then manipulating the behavior of the functions in the regular part (so
that the second condition is met.

For instance, we choose the decomposition
$\mathfrak{n}=\mathfrak{n}_{-}+\mathfrak{n}_{0}+\mathfrak{n}_{R}+\mathfrak{n}_{+}$
where: $\mathfrak{n}_{-}=\mathfrak{g}_{\gamma(0)}\cap\mathfrak{g}_{\gamma(R)}$
is composed by vectors whose action fields vanish on both $0$ and $R$;
$\mathfrak{n}_{0}=\mathfrak{g}_{\gamma(0)}\cap(\mathfrak{n}_{-})^{\bot}$
(respectively,
$\mathfrak{n}_{R}=\mathfrak{g}_{\gamma(R)}\cap(\mathfrak{n}_{-})^{\bot}$),
composed by vectors whose fields vanish only at $0$ (respectively, at $R$);
$\mathfrak{n}_{+}$, composed by fields that do not vanish at all. One then
imposes that: $f_{-},f_{0}$ coincide with $A^{-1}\sin(At)$ near $0$;
$f_{-},f_{R}$ coincide with $A^{-1}\sin(A(R-t))$ near $R$; and $f_{i}$ is
constant near an extreme where it does not vanish (see [GZ02].) Fixed these
conditions, one can easily manipulate the functions $f_{i}$ in such way that
the concavity conditions are verified.

###  7.4. Lifting positive Ricci curvature

In this section we present a simplified proof for Searle–Wilhelm theorem. The
simplification comes from Corollary 4.2, which shows that it is just left to
control the horizontal Ricci curvature at singular points. We complete an
alternative proof for [SW15, Theorem A] with:

######  Theorem 7.7.

Let $(M,g)$ be a compact connected Riemannian manifold with an isometric
$G$–action by a compact and connected Lie group. Assume that a principal orbit
has finite fundamental group and that $\operatorname{Ric}_{M/G}\geq 1.$ Then,
there exists a $G$–invariant function $f:M\to\mathbb{R}$ such that the metric
$\tilde{g}=ge^{2f}$ has positive Ricci curvature after a finite Cheeger
deformation.

###### Proof.

Here we proof that there is a smooth $G$-invariant function $f$ such that
$e^{2f}g$ has positive horizontal Ricci curvature on a neighborhood of the
singular set.

Using the Slice Theorem, we recall that every singular point $q$ has a
neighborhood $U_{q}$ such that, for every $q^{\prime}\in U_{q}$,
$G_{q^{\prime}}$ is conjugate to a subgroup of $G_{q}$. In particular
$U_{q}\subseteq GS_{q}$, where:

$S_{q}:=\mathrm{Fix}(G_{q}):=\\{q^{\prime}\in M~{}|~{}G_{q}\subseteq
G_{q^{\prime}}\\}$

is a totally geodesic submanifold of $M$ whenever $q$ is singular. Since the
singular strata $S$ is also compact, there are finitely many points
$\Lambda\subset M$ such that

(64) $\underset{q\in\Lambda}{\bigcup}GS_{q}=S.$

For each $q$, consider the function

$\psi_{q}:=\mathrm{dist}^{2}(\cdot,S_{q}),$

and let $\rho$ be a function whose second derivative is given by:

(65)
$\rho^{\prime\prime}:=\begin{cases}\frac{2}{\epsilon}-\frac{2}{\epsilon^{2}}t,~{}t\in[0,\epsilon],\\\
0,~{}t\geq\epsilon,\end{cases}$

where $\rho^{\prime}(0)=-1$; $\rho(0)$ is such that $\rho(\epsilon)=0$ and
$\epsilon$ is to be set later. Note that the integral of $\rho^{\prime\prime}$
is 1, so $\rho^{\prime}\leq 0$ for all $t$ and $\rho$ is constant for
$t\geq\epsilon$. We claim that there is a $b$ such that

(66) $f(p)=b\sum_{q}\int_{G}\rho\circ\psi_{q}(gp)dg$

has all desired properties.

To prove that, we first recall that the sectional curvature of the conformal
metric $\tilde{g}=e^{2f}g$ is given by (see [Wal04, page 144]):

$\sec_{\tilde{g}}(X,Y)=e^{-2f}(\sec_{g}(X,Y)-h_{f}(X)-h_{f}(Y)-|\nabla
f|^{2}+df(X)^{2}+df(Y)^{2}),$

where $X,Y$ are $g$-orthonormal. Fixing a $g$-orthonormal bases
$\\{e_{1},...,e_{n}\\}$ for $\mathcal{H}$, we obtain:

$\operatorname{Ric}^{\mathcal{H}}_{\tilde{g}}(Z)=e^{-2f}\Big{(}\operatorname{Ric}^{\mathcal{H}}_{g}(Z)-(n-1)h_{f}(Z)-\sum_{i=1}^{n-1}h_{f}(e_{i})\\\
+(n-1)\left\\{-|\nabla
f|^{2}+df(Z)^{2}\right\\}+\sum_{i=1}^{n-1}df(e_{i})^{2}\Big{)}.$

We further note that we can always take $e_{1}=Z=\alpha e^{\prime}_{1}+\beta
e^{\prime}_{2},~{}\alpha^{2}+\beta^{2}=1$, where $e^{\prime}_{1}=\nabla
f/|\nabla f|$, $e_{2}^{\prime}$ is orthonormal to $e_{1}^{\prime}$ and
$\\{e_{1}=Z,e_{2}:=-\beta e^{\prime}_{1}+\alpha
e^{\prime}_{2},e_{3},\ldots,e_{n}\\}$ is orthonormal. Note that
$df(e_{2}^{\prime})=df(e_{i})=0$ for $i\geq 3.$ Therefore,

$-|\nabla f|^{2}+df(Z)^{2}=-\sum_{i=2}^{n}df(e_{i})^{2}=-\beta^{2}|\nabla
f|^{2}.$

So,

$\operatorname{Ric}^{\mathcal{H}}_{\tilde{g}}(Z)=e^{-2f}\Big{(}\operatorname{Ric}_{g}^{\mathcal{H}}(Z)-(n-1)h_{f}(Z)-\sum_{i=2}^{n}h_{f}(e_{i})-(n-2)\beta^{2}|\nabla
f|^{2}\Big{)}.$

On the other hand,

$h_{f}(W)=\sum_{q}\int_{G}h_{\rho\circ\psi_{q}}(gW)dg=\sum_{q}\int_{G}\left\\{\rho^{\prime\prime}d\psi_{q}(gW)^{2}+\rho^{\prime}h_{\psi_{q}}(gW)\right\\}dg.$

Therefore,

(67)
$e^{2f}\operatorname{Ric}^{\mathcal{H}}_{\tilde{g}}(Z)=\operatorname{Ric}_{g}^{\mathcal{H}}(Z)-(n-2)\beta^{2}|\nabla
f|^{2}\\\
-\int_{G}\sum_{q}b\rho^{\prime\prime}\Big{(}(n-1)d\psi_{q}(gZ)^{2}+\sum_{i=1}^{n-1}d\psi_{q}(ge_{i})^{2}\Big{)}dg\\\
-\int_{G}\sum_{q}b\rho^{\prime}\Big{(}(n-1)h_{\psi_{q}}(gZ)+\sum_{i=1}^{n-1}h_{\psi_{q}}(ge_{i})\Big{)}dg,$

Since $d\psi_{q}|_{S_{q}}=0$ and $S_{q}$ is totally geodesic, there is a
neighborhood of $S_{q}$ such that $d\psi_{q}(X)^{2}\leq
4\|X\|^{2}_{g}\psi_{q}(x)$. Since $\rho^{\prime\prime}\leq 2/\epsilon$, we
have

(68)
$\rho^{\prime\prime}\left((n-1)d\psi_{q}(gZ)^{2}+\sum_{i=1}^{n-1}d\psi_{q}(ge_{i})^{2}\right)\leq\rho^{\prime\prime}(n-1)4\psi_{q}\leq
8(n-1)\epsilon.$

Furthermore,

$|\nabla
f|^{2}=\sum_{q\in\Lambda}\int_{G}b^{2}(\rho^{\prime})^{2}|\nabla\psi_{q}|^{2}\leq\epsilon^{2}b^{2}|\Lambda|.$

Since $S_{q}$ is totally geodesic, $h_{\psi_{q}}(X)=|X|^{2}$ for all $X$
normal to $S_{q}$ and $h_{\psi_{q}}(X)=0$ for $X$ tangent to $S_{q}$.
Therefore, given $c>0$, we can choose $\epsilon$ and $e_{3},...,e_{n}$ such
that either $h_{\psi_{q}}(e_{i})\geq 1-c\psi_{q}$ or
$h_{\psi_{q}}(e_{i})\geq-c\psi_{q}$. Moreover, $h_{\psi_{q}}(\nabla
f)\geq\|\nabla f\|^{2}(1-c)$. We have

$\displaystyle(n-1)h_{\psi_{q}}(Z)+\sum_{i=2}^{n}h_{\psi_{q}}(e_{i})\geq$
$\displaystyle(n-1)\alpha^{2}+\beta^{2}-(2n+2)c$ (69) $\displaystyle\geq$
$\displaystyle 1-(2n+2)c.$

Applying (68)-(69) to (67) gives

(70)
$e^{2f}\operatorname{Ric}^{\mathcal{H}}_{\tilde{g}}(Z)\geq\operatorname{Ric}_{g}^{\mathcal{H}}(Z)-(n-2)\beta^{2}b^{2}|\Lambda|\epsilon^{2}-8b|\Lambda|(n-1)\epsilon-b\int_{G}\sum_{q}\rho^{\prime}dg(1-(2n+2)c).$

Since $\rho^{\prime}$ is a non-positive function and $\rho^{\prime}(0)=-1$,
then $\int_{G}\sum_{q}\rho^{\prime}\circ\psi_{q}(gp)<0$ for all $p\in S$.
Therefore, for each $\delta>0$ one can choose $b$ and $\epsilon$ such that
(70) is positive for all unitary $Z\in\mathcal{H}|_{S}$ and

$(n-2)\beta^{2}b^{2}|\Lambda|\epsilon^{2}+8b|\Lambda|(n-1)\epsilon\leq\delta.$

The proof is concluded by recalling that
$\operatorname{Ric}^{\mathcal{H}}_{g}(Z)\geq 1$ for every
$Z\in\mathcal{H}|_{M\backslash S}$. ∎

## Ackonwldgement

The last author thanks C. Searle and F. Wilhelm for making him aware of this
problem. The authors thank M. Alexandrino, L. Gomes and M. Mazatto for helpful
discussions. The last author also thanks the hospitality of Universität zu
Köln. This work was supported by Fundação de Amparo a Pesquisa do Estado de
São Paulo [2017/24680-1 to L.C., 2017/10892-7 and 2017/19657-0 to L.S.];
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
[88882.329041/2019-01 to R.S]; and Conselho Nacional de Pesquisa
[404266/2016-9 to L.S.].

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