# Gromoll–Meyer’s actions and the geometry of (exotic) spacetimes

Leonardo F. Cavenaghi  Instituto de Matemática, Estatística e Computação
Científica – Unicamp, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas,
SP, Brazil  and  Lino Grama  Instituto de Matemática, Estatística e Computação
Científica – Unicamp, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas,
SP, Brazil

###### Abstract.

Since the advent of new pairwise non-diffeomorphic structures on smooth
manifolds, it has been questioned whether two topologically identical
manifolds could admit different geometries. Not surprisingly, physicists have
wondered whether a smooth structure assumption different from some classical
known models could produce different physical meanings. Motivated by the works
[Wit85, BL91], [Bra94, Sła96], in this paper, we inaugurate a very
computational manner to produce physical models on classical and exotic
spheres that can be built equivariantly, such as the classical Gromoll–Meyer
exotic spheres. As first applications, we produce Lorentzian metrics on
homeomorphic but not diffeomorphic manifolds that enjoy the same physical
properties, such as geodesic completeness, positive Ricci curvature, and
compatible time orientation. These constructions can be pulled back to higher
models, such as exotic ten spheres bounding spin manifolds, to be approached
in forthcoming papers.

###### Key words and phrases:

exotic manifolds, exotic spacetimes, Lorentzian manifolds, Lorentzian
completeness, Cheeger deformations

###### 2020 Mathematics Subject Classification:

53C18, 53C21, 53C25, 53C80, 83C05, 83C20

##  1\. Introduction

The notion of differentiation in a spacetime model is usually inherited from
the setup considered: observe that whenever we fix a background to study
differential geometry (and to physical content, especially in general
relativity and field theory, the advantage is taken from this mathematical
field), physicists have constantly assumed that smoothness is more a necessary
tool to the computations than it contains a hidden physical meaning. However,
due to the seminal work of John Milnor [Mil56] leading to the discovery of
exotic structures keeping the (point set) topology unchanged, it is natural to
ask whether smoothness can obstruct global descriptions of spacetime models.

The exotic $7$ spheres constructed by Milnor 1956 are all examples of fiber
bundles over the $4$ sphere with structure group $\mathrm{SO}(4)$. By
classifying bundles on spheres via the _clutching construction_ 111The
clutching construction recovers $G$-principal bundles from a cocycle in
$G$-Cech cohomology. For sphere bundles, it is given by the covering of the
two $n$-dimensional hemispheres that overlap at the equator and one single
transition function on that equator ($\mathrm{S}^{n-1}\rightarrow G$). These
correspond to homotopy classes of maps
$\mathrm{S}^{3}\rightarrow\mathrm{SO}(4)$; see [Mil56, Mil59, Sta74] for
clarifications and additional results. In this paper, we describe explicitly
three models for compact Lorentzian spacetimes, which are base of principal
bundles with structure group $\mathrm{S}^{1}\subset\mathrm{SU}(2)$, endowed
with Lorentzian metrics for which the total space has a semi-Riemannian
submersion metric. Moreover, on the base manifolds, the Ricci curvature is
shown to be strictly positive in all the cases despite the signature of the
metric being mixed.

The main motivation in our constructions comes from trying to understand if
two non-compatible smooth structures on the same “spacetime” could enjoy
different physical rules. In this direction, we remark on the works of [Wit85,
BL91, Bra94, Sła96]. For the constructions, we base ourselves on the following
(see [CS18, CS22]): Consider a compact connected principal bundle
$G\hookrightarrow P\to M$ with a principal action $\bullet.$ Assume that there
is another action (which we denote by $\star$) on $P$ that commutes with
$\bullet.$ This makes $P$ a $G\times G$–manifold. If one assumes that $\star$
is free, one gets a $\star$-diagram of bundles:

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{M^{\prime}}$$\textstyle{M}$

In the former diagram, $M$ is the quotient of $P$ by the $\bullet$-action, and
$M^{\prime}$ is the quotient of $P$ by the $\star$-action. Since $\bullet$ and
$\star$ commute, $\bullet$ descends to an action on $M^{\prime}$ and $\star$
descends to an action on $M$. We denote the orbit spaces of these actions by
$M^{\prime}/\bullet$ and $M/\star$, respectively. It is worth pointing
recalling that such diagrams were first introduced in [Dur01] and later
generalized in [Spe16, CS18, CS22].

In Section 3.1, we use Cheeger deformations to recover a construction for the
classical Minkowski spacetime. Based on this, we pass to consider the manifold
$\mathrm{SU}(2)$ with an _almost semi-free_ isometric action by
$\mathrm{S}^{1}$ and build an (almost) Lorentzian metric on $\mathrm{SU}(2)$
with positive Ricci curvature. More precisely, we consider a $\mathrm{S}^{1}$
action on $\mathrm{SU}(2)$, which is semi-free outside some fixed points.
After taking care of extending a Cheeger deformation of a fixed bi-invariant
metric on $\mathrm{SU}(2)$, we construct a semi-Riemannian submersion

$(\mathrm{SU}(2)\times\mathbb{R},\mathrm{h}_{-r^{2}})\rightarrow(\mathrm{SU}(2),\textsl{g}_{-r^{2}})$

where $\textsl{g}_{-r^{2}},\mathrm{h}_{-r^{2}}$ are Lorentzian metrics. An
interesting aspect of this construction is that the classical _de Sitter
spacetime_ has the same topology as $\mathrm{SU}(2)\times\mathbb{R}$, but
neither $\textsl{g}_{-r^{2}}$ nor $\mathrm{h}_{-r^{2}}$ are Einstein metrics.

Given the physical interest on $7$-dimensional spheres (classical and exotic),
for instance, as described in [Wit85], we built explicit Lorentzian metrics on
$\mathrm{S}^{7}$ and $\Sigma_{GM}^{7}$ (the Gromoll–Meyer exotic sphere
([GM74])) which have equivalent (in some sense), physical properties related
to the Lorentzian metrics there built. We list our results:

######  Theorem A.

Let $M^{\prime}\leftarrow P\rightarrow M$ be a star diagram with
$\mathrm{S}^{1}$ as its structure group. Then there is a Lorentzian metric
$\textsl{g}_{-r^{2}}$ in the principal stratum of $M^{reg}$ of the
$\star$-action $M$, and a $\textsl{g}^{\prime}_{-r^{2}}$ Lorentzian metric in
the principal stratum ${M^{\prime}}^{reg}$ of the $\bullet$-action on
$M^{\prime}$ such that the orbit spaces $M/\mathrm{S}^{1}$ and
$M^{\prime}/\mathrm{S}^{1}$ are isometric. Moreover, these metrics can be
smoothly extended to Lorentzian metrics on $M,M^{\prime}$ with positive Ricci
curvature.

The following description of vectors guarantees the extension of the metric in
the former theorem in the fixed points.

######  Theorem B.

Let $(M,\textsl{g})$ be a path-connected semi-Riemannian manifold with an
effective isometric action by $\mathrm{S}^{1}$ and $x$ be a
$\mathrm{S}^{1}$-fixed point. Assuming that $\textsl{g}_{-r^{2}}$ is a
Lorentzian metric in the principal stratum of $M$ induced by a
$-r^{2}$-Cheeger deformation of g, then given $X\in T_{x}M$, the geodesic
generated by $s\mapsto\gamma(s)$ is time-like for any $s>0$ if, and only if,
$\mathfrak{g}_{X}=\mathfrak{s}^{1}$, where $\mathfrak{g}_{X}$ is the Lie
algebra of the isotropy subgroup at $X$ of the isotropy representation in
$T_{x}M$.

######  Corollary C.

The classical sphere $\mathrm{S}^{7}$ and the Gromoll–Meyer exotic sphere
$\Sigma_{GM}^{7}$ admit time-oriented Lorentzian metrics of positive Ricci
curvature with isometric semi-free actions of $\mathrm{S}^{1}$ on an open and
dense subset (regular stratum). Moreover, these actions fix some points out of
the regular stratum, and the orbit-spaces $\mathrm{S}^{7}/\mathrm{S}^{1}$ and
$\Sigma_{GM}^{7}/\mathrm{S}^{1}$ are isometric as metric spaces.

A possible conclusion for the former corollary is that Einstein’s equations
recover classical and exotic differential topology on spacetimes. We want to
thank Prof. Ernesto Lupercio for pointing it out.

A further result is proved to ensure completeness of the built metrics on
exotic and classical spheres:

######  Theorem D.

Let $M^{\prime}\leftarrow P\rightarrow M$ be a star diagram with
$\mathrm{S}^{1}$ as its structure group. Then there is a Lorentzian metric
$\widetilde{\textsl{g}}_{-r^{2}}$ in $M$, and a Lorentzian metric
${\widetilde{\textsl{g}}}^{\prime}_{-r^{2}}$ in $M^{\prime}$, with complete
space and time-like geodesics.

##  2\. Fixing the notation and a minimum of Lorentzian geometry

Every aspect of this section is deeply based in [VC10, O’N97].

###  2.1. Recordatory aspects of Linear algebra of indefinite metrics

A semi-Euclidean space $V$ is a finite-dimensional real $n$-dimensional vector
space with a _non-degenerate_ quadratic form $q$. Such a quadratic form can,
via a choice of basis $\mathcal{B}:=\\{e_{1},\ldots,e_{n}\\}$, be computed in
any vector $v\in V$ furnishing

$q(v)=v_{1}^{2}+\ldots v_{p}^{2}-(v_{p+1}^{2}+\ldots v_{p+q}^{2}).$

The quadratic form $q$ gives rise to a symmetric bilinear g form via the
classical polarization formula. Given $v,w\in V$ then $v\perp w$ (i.e., $v$
and $w$ are orthogonal) if $\textsl{g}(v,w)=0$. Two vector subspaces
$A,B\subset V$ are said to be orthogonal, and we denote $A\perp B$ if $v\perp
w\forall v\in A,~{}w\in B$. Fixed any vector subspaces $A\subset V$ we define
$A^{\perp}:=\\{w\in V:\textsl{g}(v,w)=0,\forall v\in A\\}.$

######  Definition 1.

Let $g$ be a symmetric bilinear form. Given $v\in V$,
$q_{\textsl{g}}(v):=\textsl{g}(v,v)$, we say that $v$ is

  1. (i)

time-like if $q_{\textsl{g}}(v)<0$,

  2. (ii)

light-like if $q_{\textsl{g}}(v)=0$ and $v\neq 0$,

  3. (iii)

space-like if $q_{\textsl{g}}(v)>0$,

  4. (iv)

causal if $v$ is time-like or light-like.

######  Definition 2.

A scalar product g on $V$ is a non-degenerate symmetric bilinear form.

Let $\\{e_{1},\ldots,e_{n}\\}$ be a basis of $V$. We say that this basis is
_orthonormal_ if $|e_{i}|:=\sqrt{|\textsl{g}(e_{i},e_{i})|}=1$ and
$\textsl{g}(e_{i},e_{j})=0$ if $i\neq j$. A crucial result for the linear
theory of symmetric bilinear forms is based upon the following:

######  Proposition 2.1.

The number $\nu$ of time-like vectors in a basis $\mathcal{B}$ of
$(V,\textsl{g})$ does not depend on the basis but only on $(V,\textsl{g})$,
and it is called the _index_ of $(V,\textsl{g})$. Moreover, $(V,\textsl{g})$
admits an orthonormal basis.

######  Definition 3.

A scalar product g in a real vector space $V$ is

  1. (i)

Euclidean if $\nu=0$,

  2. (ii)

Lorentzian if $\nu=1$ and $n\geq 2$,

  3. (iii)

It is indefinite if it is in a symmetric bilinear form.

######  Proposition 2.2.

The subset of the time-like vectors (resp., causal; light-like if $n>2$) has
two connected components. Each one of these parts will be called a time-like
cone (resp. causal cone; light-like cone). Moreover, two time-like vectors $v$
and $w$ lie in the same time-like cone if, and only if, $\textsl{g}(v,w)<0$.
In particular, each time-like cone is convex.

######  Definition 4.

Let $(V,\textsl{g})$ be a Lorentzian vector space. We will say that a subspace
of $W\subset V$ is

  1. (1)

space-like, if $\textsl{g}|_{W}$ is Euclidean,

  2. (2)

time-like, if $\textsl{g}|_{W}$ is non-degenerated with index $1$ (that is, it
is Lorentzian whenever $\dim W\geq 2$),

  3. (3)

light-like if $g|_{W}$ is degenerated

The following proposition plays an essential (implicit though) role in this
paper:

######  Proposition 2.3.

A subspace $W\subset V$ is time-like if, and only if, $W^{\perp}$ is space-
like.

###  2.2. The basics of semi-Riemannian Geometry

A connected semi-Riemannian manifold $(M,\textsl{g})$ is a differentiable
$n$-dimensional manifold $M$ equipped with an everywhere non-degenerate,
smooth, symmetric metric tensor g of constant index $\nu\in\\{1,\ldots,n\\}$.
Such a metric is called a semi-Riemannian metric. The pair $(M,\textsl{g})$
will be called _Riemannian_ manifold if $\nu=0$ and _Lorentzian_ if $\nu=1$.

####  2.2.1. Temporal orientability

######  Definition 5.

A _time orientation_ of a Lorentzian vector space is a choice of one of the
two time-like cones (or, equivalently, of one of the causal or light-like
cones). The chosen cone will be called _future_ , and the other one, _past_.

Let $(M,\textsl{g})$ be a Lorentzian manifold. Any tangent space
$(T_{x}M,\textsl{g}_{x}),~{}x\in M$ is a Lorentzian vector space with two
time-like cones, and one can choose one of them as a time orientation. The
question of how this choice can be carried out continuously is the equivalent
(in affirmative) answer to define a _temporal orientation_ on $M$. More
precisely,

######  Definition 6.

A _time orientation_ in a Lorentzian manifold $(M,\textsl{g})$ is a map which
assigns, to each $x\in M$, a time-like cone $\tau_{x}\subset T_{x}M$ such that
there exists an open neighborhood $U\ni x$ and a time-like vector field $X$ in
$U$ such that $X_{y}\in\tau_{y}~{}\forall y\in U$. Any Lorentzian manifold
assuming a time orientation is said _time orientable_.

It can be proved that

######  Theorem 2.4.

Let $M$ be a connected differentiable manifold. The following conditions are
equivalent:

  1. (i)

$M$ admits a Lorentzian metric;

  2. (ii)

$M$ admit a time-orientable Lorentzian metric;

  3. (iii)

$M$ admits a vector field $X$ without zeros;

  4. (iv)

Either $M$ is non-compact or its Euler characteristic is $0$.

As we shall see, every example in this paper admits a Lorentzian metric.
Moreover, the built metrics shall be time-oriented.

####  2.2.2. Computational aspects

Let $(M,\textsl{g})$ be a Lorentzian manifold. Then, just as in the Riemannian
case, it is possible to consider the Levi-Civita connection $\nabla$ of g.
Moreover, for such a connection is well defined, the Riemann curvature tensor
$R_{\textsl{g}}(X,Y)Z:=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$
for every vector fields $X,Y,Z$ in $M$. It also enjoys the Riemann curvature
tensor symmetry and anti-symmetry property of the Riemannian case.

From the Riemannian curvature tensor $R_{\textsl{g}}$ can be derived the Ricci
tensor (that is well defined even for light-like vectors) via

$\mathrm{Ric}(X,Y)(x):=\mathrm{tr}\left(\mathrm{R}(\cdot,X)Y\right),$

and the scalar curvature
$\mathrm{scal}_{\textsl{g}}:=\mathrm{tr}\mathrm{Ric}$.

However, the sectional curvature of g is only defined equally as in the
Riemannian case for a plane $\sigma\in T_{x}M,~{}x\in M$ if, and only if,
$\sigma$ is non-degenerated: It has a positive or negative area. If
$\sigma=\mathrm{span}_{\mathbb{R}}\\{X,Y\\}$ then
$|X|^{2}|Y|^{2}-\textsl{g}(X,Y)^{2}\neq 0$.

###  2.3. General Notation

We denote by $R_{\textsl{g}}$ the Riemannian tensor of the metric g:

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

where $\nabla$ stands for the Levi-Civita connection of g. We denote either by
$K_{\textsl{g}}(X,Y)=\textsl{g}(R_{\textsl{g}}(X,Y)Y,X)$ or by
$R_{\textsl{g}}(X,Y)=R_{\textsl{g}}(X,Y,Y,X),$ making it clear in the context,
the unreduced sectional curvature of g. We adopt the standard decomposition
notation $TM=\mathcal{V}\oplus\mathcal{H}$ for any foliated manifold $M$,
being $\mathcal{V}$ the subbundle collecting tangent spaces to the leaves of
the foliation $\mathcal{F}$ and $\mathcal{H}$ to some subbundle complementary
choice in $TM$.

The superscripts $\mathbf{v}$ or $\mathbf{h}$ denote the projection of the
underlined quantities on such subbundles, $\mathcal{V}$ and $\mathcal{H}$,
respectively. Whenever we say we have a _Riemannian principal bundle_ , we
mean that the principal bundle is considered with a Riemannian submersion
metric.

##  3\. A Cheeger approach to the Minkowski space and a first compact model

We first recall the aspects of Cheeger deformations needed to motivate our
constructions well. Though the main formulae come from classical references,
such as [Zil06] and [M8̈7], we follow [CGS22] for the presentation. The reason
for that is to make the constructions of what we call “compact models of
exotic spacetimes” more natural once the concept of $\star$-diagrams is
established.

Take the product manifold $M\times G$ with the product metric
$\textsl{g}+t^{-1}Q$, where $G$ acts on $M$ via isometries and $Q$ is a bi-
invariant metric on $G$. We, therefore, see ourselves with two possibilities
of free (and commuting) actions:

(1)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{M\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{M}$$\textstyle{M}$

In (1) the action $\bullet$ stands to

(2) $r\bullet(m,g):=(m,rg),$

while the action $\star$ is nothing but the associated bundle action on
$M\times G$, that is

(3) $r\star(m,g):=(rm,rg).$

Therefore, $\pi((m,g)):=m$ meanwhile $\pi^{\prime}((m,g)):=g^{-1}m$. Since
$\pi$ and $\pi^{\prime}$ define principal bundles, the metric
$\textsl{g}+t^{-1}Q$ induces via $\pi$ and $\pi^{\prime}$, respectively, the
metrics g (the original one) and $\textsl{g}_{t}$, a $t$-_Cheeger deformation_
of g.

Throughout the paper, it shall be denoted by $\mathfrak{m}_{p}$, the
$Q$-orthogonal complement of $\mathfrak{g}_{p}$, the Lie algebra of the
isotropy subgroup at $G_{p}$. We recall that $\mathfrak{m}_{p}$ is isomorphic
to the tangent space to the orbit $Gp$ via action fields: For any
$U\in\mathfrak{g}$ the corresponding action field out of $U$ is defined by the
rule

$U^{*}_{p}=\frac{d}{dt}\Big{|}_{t=0}\mathrm{Exp}\left(tU\right)p$

where $\mathrm{Exp}:\mathfrak{g}\rightarrow G$ is the Lie group exponential
map. It is straightforward to check that the map $U\mapsto U^{*}_{p}$ is a
linear morphism whose kernel is $\mathfrak{g}_{p}$. In this manner, any vector
tangent to $T_{p}Gp$ is said to be _vertical_. Hence, such a space is named
the _vertical space_ at $p$, denoted by $\mathcal{V}_{p}$. For each $p\in M$,
its orthogonal complement, denoted by $\mathcal{H}_{p}$, is named _horizontal
space_. A tangent vector $\overline{X}\in T_{p}M$ can be uniquely decomposed
as $\overline{X}=X+U^{\ast}_{p}$, where $X$ is horizontal and
$U\in\mathfrak{m}_{p}$.

There are symmetric positive definite (whenever $t\geq 0$) tensors relating
original metric quantities with deformed ones:

  1. (1)

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

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

  2. (2)

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

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

  3. (3)

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

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

The former tensors are related as

######  Proposition 3.1 (Proposition 1.1 in [Zil06]).

  1. (1)

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

  2. (2)

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

In the field of non-negative/positive curvatures, the metric tensor
$C_{t}^{-1}$ can be used to define a reparametrization of $2$-planes, not only
making it easier to compute sectional curvature of a Cheeger deformed metric
but allowing us to observe that Cheeger deformations (for non-negative $t$) do
not create ‘new’ planes with zero sectional curvature, meaning that

######  Theorem 3.2.

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

$\kappa_{t}(\overline{X},\overline{Y}):=R_{\textsl{g}_{t}}(C_{t}^{-1}\overline{X},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{Y},C_{t}^{-1}\overline{X})$

satisfies

(4)
$\kappa_{t}(\overline{X},\overline{Y})=\kappa_{0}(\overline{X},\overline{Y})+\frac{t^{3}}{4}\|[PU,PV]\|_{Q}^{2}+z_{t}(\overline{X},\overline{Y}),$

where $z_{t}$ is non-negative if $t\geq 0$. Moreover, for points in regular
orbits, we have

$z_{t}(\overline{X},\overline{Y})=3t\left\|(1+tP)^{-1/2}P\nabla^{\mathbf{v}}_{\overline{X}}\overline{Y}-(1+tP)^{-1/2}t\frac{1}{2}[PU,PV]\right\|_{Q}^{2}.$

We refer to either [Zil06, Proposition 1.3] or [CeSS23, Lemma 3.5] for the
details on the proof and more references.

We will use Cheeger deformations with group actions by $\mathbb{R}$ or
$\mathrm{S}^{1}$ to produce Lorentzian metrics on some manifolds. As a first
step, we show that such a procedure allows us to recover the classic
_Minkowski spacetime_. Later, we proceed to study what we see as an “analogous
compact model” to the Minkowski spacetime since the same tool and the
practical computations using quaternionic algebra allow us to recover the
classical _de Sitter spacetime_ as some universal covering.

Before proceeding, however, we make some remarks.

######  Remark 1.

  1. (a)

if the $G$-action on $M$ is free and effective and the orbits of $G$ are
totally geodesic for g, then the orbit tensor $P$ is constant along the whole
manifold once seen as a section of the bundle (over $M$) of endomorphisms of
$\mathfrak{g}$. In this manner, it is convenient sometimes to take $P=1$ (the
identity operator) under the assumption of totally geodesic leaves.

  2. (b)

for effective free $G$-actions, we can always assume that the orbits of $G$
are totally geodesic for some Riemannian metric g, which preserves the initial
horizontal distribution. This is due to Searle–Solorzano–Wilhelm, [SSW15]. We
often use this implicitly, combined with the previous item.

  3. (c)

the metric tensor $C_{t}=(1+tP)^{-1}$ contains the needed control to produce
(semi)-Riemannian metrics via Cheeger deformations. Observe that, as
operators, if one takes $t<0$, then $\textsl{g}_{t}$ has mixed signature if,
and only if, $1<-tP$. For most of this paper, $P=1$, and we choose $t=-r^{2}$
assuming that $r^{2}>1$, aiming to produce Lorentzian metrics.

###  3.1. The standard Minkowski spacetime

The description of the so-called _Minkowski spacetime_ $\mathbb{R}^{1,3}$ is
obtained from the topological space $\mathbb{R}^{4}$, providing it a semi-
Riemannian metric.

Let $\mathbb{H}$ be the quaternionic space with
$\mathrm{dim}_{\mathbb{R}}\mathbb{H}=4$. In what follows, we shall use the
inherited algebraic structure of $\mathbb{H}$ to perform all the
$\mathbb{R}^{4}$ computations. In this manner, throughout this section, we
identify

$\mathbb{R}^{4}\cong\mathbb{H}$
$\vec{q}=(x^{0},x^{1},x^{2},x^{3})\leftrightarrow
x^{0}+x^{1}\mathbf{i}+x^{2}\mathbf{j}+x^{3}\mathbf{k}=q$

Let

$t\cdot\vec{q}:=(t+x^{0})+x^{1}\mathbf{i}+x^{2}\mathbf{j}+x^{3}\mathbf{k}.$

be an effective free action of $\mathbb{R}\cong\mathrm{Re}(\mathbb{H})$ on
$\mathbb{H}$ and consider the product $\mathbb{H}\times\mathbb{R}$ with the
following _twisted_ $\mathbb{R}$-action:

(5) $t\star(\vec{q},t^{\prime}):=(\vec{q}\cdot t,t+t^{\prime}).$

Observe that the action (5) works as the analogous $\star$-action defined in
equation (3). In this manner, to introduce a family of Lorentzian metrics on
$\mathbb{H}$, we take $r>0$ and consider the semi-Riemannian metric

(6)
$\widetilde{\textsl{g}}_{-r^{2}}:=\langle\cdot,\cdot\rangle_{\mathbb{H}}-r^{-2}d\theta^{2}$

in $\mathbb{H}\times\mathbb{R}$, where
$\langle\cdot,\cdot\rangle_{\mathbb{H}}$ is the quaternionic inner product
defined by

$\langle\vec{q},\vec{r}\rangle_{\mathbb{H}}:=\mathrm{Re}(\vec{q}\overline{\vec{r}}).$

Moreover, since we can see $\mathbb{R}$ inside $\mathbb{H}$ as
$\mathbb{R}\cong\mathrm{Re}~{}\mathbb{H}$ then $d\theta^{2}$ is nothing but
standard multiplication of real numbers.

It is straightforward to check that the vertical (one-dimensional) space at
$(q,t^{\prime})$ coincides with

(7)
$\mathcal{V}_{(\vec{q},t^{\prime})}:=\mathrm{span}_{\mathbb{R}}\\{(1+0\mathbf{i}+0\mathbf{j}+0\mathbf{k},1)\\}.$

We proceed to compute the horizontal space that is
$\widetilde{\textsl{g}}_{-r^{2}}$ complementary to the vertical space (7).
With this aim, we must take $\vec{X}=(X,t^{\prime})\in
T_{(q,t^{\prime})}\left(\mathbb{H}\oplus\mathbb{R}\right)\cong\mathbb{H}\oplus\mathbb{R}$
such that

$\langle X,1\rangle-r^{-2}t^{\prime}=0.$

Since $\langle X,1\rangle=\mathrm{Re}(X\overline{1})=\mathrm{Re}(X)$ one gets
that $\mathrm{Re}(X)=r^{-2}t^{\prime}$ so

(8)
$\mathcal{H}_{(q,t^{\prime})}=\left\\{\vec{X}=(X,t^{\prime})\in\mathbb{H}\times\\{t^{\prime}\\}:\vec{X}=(r^{-2}t^{\prime}+X^{1}\mathbf{i}+X^{2}\mathbf{j}+X^{3}\mathbf{k},t^{\prime})\right\\}$

Furthermore, note that

$\displaystyle|\vec{X}|^{2}_{-r^{2}}$
$\displaystyle=r^{-4}(t^{\prime})^{2}+|\mathrm{Im}~{}X|^{2}-r^{-2}(t^{\prime})^{2}$
$\displaystyle=-(t^{\prime})^{2}r^{-4}(r^{2}-1)+|\mathrm{Im}~{}X|^{2},$

so for any $r>1$ the metric $\textsl{g}_{-r^{2}}$ on $\mathbb{H}$ with

$\mathbb{H}\cong\mathbb{R}\oplus\mathbb{R}^{3}\cong\mathrm{span}_{\mathbb{R}}\left\\{\partial_{t}\right\\}\oplus\mathrm{Im}~{}\mathbb{H}$
$\vec{q}=x^{0}+x^{1}\mathbf{i}+x^{2}\mathbf{j}+x^{3}\mathbf{k}\leftrightarrow(\mathrm{Re}~{}\vec{q},\mathrm{Im}~{}\vec{q})$
(9)
$\textsl{g}_{-r^{2}}(t^{\prime}\partial_{t}+X,t^{\prime\prime}\partial_{t}+Y):=-t^{\prime}t^{\prime\prime}r^{-4}(r^{2}-1)+\langle
X,Y\rangle_{\mathbb{R}^{3}}$

is a semi-Riemannian metric for which the vector field $\partial_{t}$, which
is nothing but the identification $\frac{\partial}{\partial
x^{0}}\equiv\partial_{t}$, is globally defined and time-like.

If one denotes $c^{2}:=r^{-4}(r^{2}-1)$ to make analogy with physics we have
the Minkowski space $(\mathbb{R}^{4},\textsl{g}_{-c^{2}})$ with the following
description:

(10)
$\textsl{g}_{-c^{2}}(t^{\prime}\partial_{t}+X,t^{\prime\prime}\partial_{t}+Y):=-t^{\prime}t^{\prime\prime}c^{2}+\langle
X,Y\rangle_{\mathbb{R}^{3}}.$

###  3.2. $\mathrm{SU}(2)$ as a spacetime and obstructions

This section sees $\mathrm{SU}(2)$ as the subset of $\mathbb{H}$ consisting of
unit elements. It conveys to denote each element of $\mathrm{SU}(2)$ by
$q\in\mathbb{H}$ such that $|q|^{2}=1$.

Although $\mathrm{S}^{1}$ is naturally included in $\mathrm{SU}(2)$, there are
plenty of choices to explicitly write one of such to act on $\mathrm{SU}(2)$.
We shall proceed by picking
$\mathrm{S}^{1}=\\{e^{\mathbf{i}\theta}:\theta\in\mathbb{R}\\}.$ To construct
a $\mathrm{S}^{1}$-action on $\mathrm{SU}(2)$, instead of right or left
multiplication, we look to the restriction of a $\mathrm{SU}(2)$-action to a
$\mathrm{S}^{1}$-action on $\mathrm{S}^{7}$: such a construction comes out the
$\star$-diagram below, appearing first in the work of Gromoll and Meyer
[GM74], as well later in [Dur01, Spe16, CS18, CGS22].

Consider the compact Lie group

(11) $\mathrm{Sp}(2)=\left\\{\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\in\mathrm{S}^{7}\times\mathrm{S}^{7}~{}\Big{|}~{}a\overline{b}+c\overline{d}=0\right\\},$

where $a,b,c,d\in\mathbb{H}$ are quaternions with their standard conjugation,
multiplication, and norm. The projection $\pi:\mathrm{Sp}(2)\to\mathrm{S}^{7}$
of an element to its first row defines a principal $\mathrm{SU}(2)$-bundle
with principal action:

(12) $\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\overline{q}=\begin{pmatrix}a&c\overline{q}\\\
b&d\overline{q}\end{pmatrix}.$

As in [GM74], we consider the $\star$-action

(13) $q\begin{pmatrix}a&c\\\
b&d\end{pmatrix}=\begin{pmatrix}qa\overline{q}&qc\\\
qb\overline{q}&qd\end{pmatrix},$

whose quotient is an exotic 7-sphere, via the quotient map

$\pi^{\prime}\left(\begin{pmatrix}a&c\\\
b&d\end{pmatrix}\right):=\left(2\overline{c}d,|c|^{2}-|d|^{2}\right),$

placing everything in the diagram

(14)
$\textstyle{\mathrm{SU}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{\mathrm{SU}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{\mathrm{Sp}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\Sigma^{7}_{GM}}$$\textstyle{\mathrm{S}^{7}}$

(In [CGS22], such construction is exploited to produce stair of bundles with
curvature properties ).

We consider

$\mathrm{SU}(2)=\left\\{\begin{pmatrix}a\\\
0\end{pmatrix}\in\mathrm{S}^{7}\times\\{0\\}\subset\mathbb{H}\oplus\mathbb{H}\right\\}$

and define the following action of $\mathrm{S}^{1}$ on $\mathrm{SU}(2)$:

(15) $q\cdot a:=qa\overline{q}.$

It should then be clear that

$\mathcal{V}_{a}\cong\mathrm{span}_{\mathbb{R}}\left\\{\mathbf{i}a-a\mathbf{i}:a\in\mathrm{SU}(2)\right\\}.$

Hence, if we now consider the $\mathrm{S}^{1}$-twisted action in
$\mathrm{SU}(2)\times\mathrm{S}^{1}$:

(16)
$q\ast(a,q^{\prime}):=(qa\overline{q},qq^{\prime}),~{}q^{\prime}\in\mathrm{S}^{1},a\in\mathrm{SU}(2)$

one gets the following description of the vertical space for the action given
by equation (16):

$\mathcal{V}_{(a,q^{\prime})}\cong\mathrm{span}_{\mathbb{R}}\left\\{(\mathbf{i}a-a\mathbf{i},\mathbf{i}q^{\prime})\right\\}.$

Observe that the action (15) is nothing but the $\star$-action presented in
equation (3) used to perform a Cheeger deformation. However, to avoid
confusion with equation (13), we wrote it as $\ast$ instead of $\star$.

We regard $\mathrm{SU}(2)\times\mathrm{S}^{1}$ with the following
$\mathrm{S}^{1}$-invariant semi-Riemannian metric

$\widetilde{\textsl{g}}_{-r^{2}}(\cdot,\cdot):=\left\langle\cdot,\cdot\right\rangle_{\mathrm{SU}(2)}-r{{}^{-2}}d\theta^{2},$

where $d\theta^{2}$ stands for the standard metric on $\mathrm{S}^{1}$:
observe that to our description of $\mathrm{S}^{1}$ we have that
$d\theta^{2}(\mathbf{i}t,\mathbf{i}t^{\prime})$ coincides with
$\mathrm{Re}(\mathbf{i}t\overline{\mathbf{i}t^{\prime}})=tt^{\prime}$, for
$t,t^{\prime}\in\mathbb{R}$. The metric
$\langle\cdot,\cdot\rangle_{\mathrm{SU}(2)}$ is the isometric immersion metric
obtained from $\langle\cdot,\cdot\rangle_{\mathbb{H}}$.

We then proceed to compute the horizontal space, complementary to
$\mathcal{V}_{(a,q^{\prime})}$ for $\widetilde{\textsl{g}}_{-r^{2}}$. To do
so, let us search for a vector $(X\mathbf{i}a,-\mathbf{i}q^{\prime}t)$ with
$X\mathbf{i}a\in T_{a}\mathrm{SU}(2)$ which satisfies

$\widetilde{\textsl{g}}_{-r^{2}}((X\mathbf{i}a,-\mathbf{i}q^{\prime}t),(\mathbf{i}a-a\mathbf{i},\mathbf{i}q^{\prime}))=0$

for some $t\in\mathbb{R},~{}X\mathbf{i}a\in\mathbb{H}$. First observe that
$X\mathbf{i}a\in T_{a}\mathrm{SU}(2)$ if, and only if,
$\mathrm{Re}(X\mathbf{i}a\bar{a})=0$. That is, $\mathrm{Re}(X\mathbf{i})=0$.

We compute:

$\displaystyle 0$
$\displaystyle=\mathrm{Re}((\mathbf{i}a-a\mathbf{i})\overline{X\mathbf{i}a})-r^{-2}\mathrm{Re}(\mathbf{i}q^{\prime}\overline{(-\mathbf{i}q^{\prime}t)})$
$\displaystyle=\mathrm{Re}(\mathbf{i}a\overline{a}(-\mathbf{i})\overline{X}-a\mathbf{i}\overline{a}(-\mathbf{i})\overline{X})-r^{-2}\mathrm{Re}(\mathbf{i}q^{\prime}\overline{q^{\prime}}\mathbf{i}t)$
$\displaystyle=\mathrm{Re}(\mathbf{i}a\overline{a}(-\mathbf{i})\overline{X}-a\mathbf{i}\overline{a}(-\mathbf{i})\overline{X})+r^{-2}t$
$\displaystyle=\mathrm{Re}(\overline{X}+a\mathbf{i}\overline{a}\mathbf{i}\overline{X})+r^{-2}t$
$\displaystyle=\mathrm{Re}(\overline{X}-a\mathbf{i}\overline{\mathbf{i}a}\overline{X})+r^{-2}t.$

Therefore,

$\mathcal{H}_{(a,q^{\prime})}=\mathrm{span}_{\mathbb{R}}\left\\{(X\mathbf{i}a,-\mathbf{i}q^{\prime}t):\mathrm{Re}((1-a\mathbf{i}\overline{\mathbf{i}a})\overline{X})=-r^{-2}t,~{}\mathrm{Re}~{}X\mathbf{i}=0\right\\}.$

Hence, to promote a semi-Riemannian metric on $\mathrm{SU}(2)$ via this
procedure, one observes that since
$|(X\mathbf{i}a,-\mathbf{i}q^{\prime}t)|_{-r^{2}}^{2}=|X|^{2}_{\mathbb{H}}-r^{2}t^{2}$,
via the identification $\mathbf{i}\leftrightarrow\partial_{\theta}$, we have:

(17)
$T_{a}\mathrm{SU}(2)\cong\mathrm{span}_{\mathbb{R}}\left\\{t\partial_{\theta},X\mathbf{i}a:\mathrm{Re}((1-a\mathbf{i}\overline{\mathbf{i}a})\overline{X})=-r^{-2}t,~{}\mathrm{Re}(X\mathbf{i})=0\right\\}$

with the following semi-Riemannian metric

(18)
$\textsl{g}_{-r^{2}}(t\partial_{\theta}+X\mathbf{i}a,t^{\prime}\partial_{\theta}+Y\mathbf{i}a)=\langle
X,Y\rangle_{\mathbb{H}}-tt^{\prime}r^{-2}.$

However, observe that the defined $\mathrm{S}^{1}$-action on $\mathrm{SU}(2)$
has some fixed points. Therefore, for some points $a\in\mathrm{SU}(2)$, the
time-like vector $\partial_{\theta}$ vanishes identically, and the metric
degenerates to a Riemannian metric. Therefore, time does not exist for some
points in this compact model, and the metric presents a kind of “singularity”.
The following section discusses the nature of this singularity, aiming to
provide some physical insight into it.

###  3.3. The de Sitter spacetime and the universal covering in a
$\star$-diagram

The _de Sitter spacetime_ corresponds to a solution to the _Einstein field
equations_

$\mathrm{Ric}(\textsl{g})-\frac{1}{2}\mathrm{scal}_{\textsl{g}}\textsl{g}+\Lambda\textsl{g}=0$

that is topologically equivalent (in the homeomorphism sense) to
$\mathbb{R}\times\mathrm{SU}(2)$, being a mathematical model of the universe
that encompasses the observed accelerating expansion. We refer to [Cox43] for
a different geometric approach to the de Sitter spacetime.

Observe that the $(-r^{2})$-Cheeger deformation obtained in the previous
section allowed us to construct a semi-Riemannian submersion

(19)
$\bar{\pi}:\left(\mathrm{SU}(2)\times\mathrm{S}^{1},\widetilde{\textsl{g}}_{-r^{2}}\right)\rightarrow\left(\mathrm{SU}(2),\textsl{g}_{-r^{2}}\right)$

On the other hand, the universal cover of $\mathrm{SU}(2)\times\mathrm{S}^{1}$
is homeomorphic to $\mathrm{SU}(2)\times\mathbb{R}$. In this manner, consider
on $\mathrm{SU}(2)\times\mathbb{R}$ the covering semi-Riemannian metric
$h_{-r^{2}}$ induced via the covering map

$\mathrm{p}:\left(\mathrm{SU}(2)\times\mathbb{R},h_{-r^{2}}\right)\rightarrow\left(\mathrm{SU}(2)\times\mathrm{S}^{1},\widetilde{\textsl{g}}_{-r^{2}}\right).$

We can build the commutative diagram of semi-Riemannian submersions

(20)
---
$\textstyle{\left(\mathrm{SU}(2)\times\mathrm{S}^{1},\widetilde{\textsl{g}}_{-r^{2}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\pi}}$$\textstyle{\left(\mathrm{SU}(2),\textsl{g}_{-r^{2}}\right)}$$\textstyle{\left(\mathrm{SU}(2)\times\mathbb{R},h_{-r^{2}}\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{p}}$$\scriptstyle{\bar{\pi}\circ\mathrm{p}}$

which recovers the de Sitter spacetime
$\left(\mathrm{SU}(2)\times\mathbb{R},h_{-r^{2}}\right)$, i.e., we have
presented a (topologically) _global_ construction to the de Sitter spacetime
(that is, _coordinate free_) naturally associated with a Cheeger deformation.

Diagram (20) allow us to see, in some sense,
$\left(\mathrm{SU}(2),\textsl{g}_{-r^{2}}\right)$ as a _restrict world
perspective of the de Sitter spacetime_. Said in other worlds, the submersion
$\bar{\pi}\circ\mathrm{p}$ projects the vision of any observer in
$\left(\mathrm{SU}(2)\times\mathbb{R},\mathrm{h}_{-r^{2}}\right)$ to the low-
dimension compact model $\left(\mathrm{SU}(2),\mathrm{g}_{-r^{2}}\right)$.
Moreover, observe that the $\mathrm{S}^{1}$-action on
$\mathrm{SU}(2)\times\mathrm{S}^{1}$ is free, so there is no time-collapsing.
In this sense, a way of justifying the singularity presented on the metric
$\textsl{g}_{-r^{2}}$ relies on the $\mathrm{S}^{1}$-forced immersion on
$\mathrm{SU}(2)$ as its orbits. The singularity can be understood as a “lack
of perception of the full spacetime”.

We recall that a smooth group action $G$ on a smooth manifold $M$ is said to
be semi-free if every isotropy group is discrete. It is said to be almost
semi-free if some fixed points exist other than discrete isotropy subgroups.
Motivated by the former example, we define:

######  Definition 7.

A Riemannian manifold $(M,\textsl{g})$ with an effective isometric almost
semi-free action by $\mathrm{S}^{1}$ is said to be an _almost Lorentzian
manifold_. The (almost) Lorentzian metric defined on the principal stratum
$M^{reg}$ in $(M,\textsl{g})$ is given by a sufficiently negative
$-r^{2}$-Cheeger deformation of g.

In Section 5, we shall fix the metric on the fixed points so that we build a
global Lorentzian metric on $\mathrm{SU}(2)$ as well as in any (almost)
Lorentzian manifold.

In the next section, we take the previous description to obtain a higher
dimension analogous to

$\bar{\pi}\circ\mathrm{p}:\left(\mathrm{SU}(2)\times\mathbb{R},\mathrm{h}_{-r^{2}}\right)\rightarrow\left(\mathrm{SU}(2),\mathrm{g}_{-r^{2}}\right)$

obtaining (almost) Lorentzian metrics in the classical seven-dimensional
sphere $\mathrm{S}^{7}$ and in the Gromoll–Meyer exotic sphere: Both
constructions coming from the diagram (14). Before proceeding, however, we
take advantage of equation (4) to check that Ricci curvature of
$\left(\mathrm{SU}(2),g_{-r^{2}}\right)$ is positive but not constant. To
prove this, we exploit the equation

$\kappa_{-r^{2}}=K_{\mathrm{SU}(2)}+z_{-r^{2}},$

where $K_{\mathrm{SU}(2)}$ is the unreduced sectional curvature of the metric
$\langle\cdot,\cdot\rangle_{\mathrm{SU}(2)}$ induced by
$\langle\cdot,\cdot\rangle_{\mathbb{H}}:=\mathrm{Re}(\cdot\overline{\cdot})$,
which coincides with the negative of the Riemannian metric in $\mathrm{SU}(2)$
induced by a multiple of the Cartan–Killing form in $\mathfrak{su}(2)$. Aiming
generality on the computation, we will not use the explicit metric description
furnished earlier but provide a general perspective using the formulae in
Section 3.

######  Theorem 3.3.

For any $r>1$ we have that $(\mathrm{SU}(2),\textsl{g}_{-r^{2}})$ is an almost
Lorentzian manifold with time-like vectors given by

$\langle X,Y\rangle_{\mathbb{H}}-tt^{\prime}r^{-2}<0$

and light-like vectors are given by

$\langle X,Y\rangle=tt^{\prime}r^{-2}$

where

$\displaystyle\mathrm{Re}((1-a\mathbf{i}\overline{\mathbf{i}a}\bar{X})$
$\displaystyle=-r^{-2}t$
$\displaystyle\mathrm{Re}((1-a\mathbf{i}\overline{\mathbf{i}a}\bar{Y}))$
$\displaystyle=-r^{-2}t^{\prime}$
$\displaystyle\mathrm{Re}(X\mathbf{i})=\mathrm{Re}(Y\mathbf{i})$
$\displaystyle=0$

$\forall~{}a\in\mathrm{SU}(2),~{}\forall
X,Y\in\mathbb{H},~{}t,t^{\prime}\in\mathbb{R}$. Moreover,
$\mathrm{Ric}_{\textsl{g}_{-r^{-2}}}$ is positive.

###### Proof.

The first part of the statement was already obtained by our explicit
description given by equations (17) and (18). For the claim on the positivity
of Ricci curvature, here we only compute the Ricci curvature in the regular
part of the $\mathrm{S}^{1}$-action, that is, in the principal stratum. The
analysis containing the behavior of Cheeger deformations for the fixed points
shall be approached in Theorem 5.5. Assuming that $\mathrm{S}^{1}$ is totally
geodesic in $\mathrm{SU}(2)$ for the initial metric g, we can take $P=1$ for
the equation (4) in Theorem 3.2. Observe that such an assumption loses no
generality due to Remark 1. Hence, we have that

$z_{-r^{2}}(\overline{X},\overline{Y})=-3r^{2}(1-r^{2})^{-1}\left\|\nabla^{\mathbf{v}}_{\overline{X}}\overline{Y}\right\|_{\mathrm{S}^{1}}^{2}$

and so

$\mathrm{\kappa}_{-r^{2}}(\overline{X},\overline{Y})=\kappa_{0}(\overline{X},\overline{Y})-3r^{2}(1-r^{2})^{-1}\left\|\nabla^{\mathbf{v}}_{\overline{X}}\overline{Y}\right\|_{\mathrm{S}^{1}}^{2}.$

Take $\\{e_{1},e_{2}\\}\in\mathcal{H}_{a}$ a g-orthonormal basis to the
horizontal space at $a\in\mathrm{SU}(2)$ and
$\\{e_{3}\\}\in\mathbf{i}\mathbb{R}$ a g-unit vector. We shall add a
$\sqrt{-1}$-multiplying factor on this element since we are considering
$r^{2}-1>0$, so $(1-r^{2})^{1/2}$ is a complex number. Then
$\\{\sqrt{-1}(1-r^{2})^{1/2}e_{3}\\}$ is a $\textsl{g}_{-r^{2}}$-unit vector
in $\mathcal{V}_{a}$. Hence $\\{e_{1},e_{2},\sqrt{-1}(1-r^{2})^{1/2}e_{3}\\}$
is a $\textsl{g}_{-r^{2}}$-orthonormal basis in $T_{a}\mathrm{SU}(2)$.

Using the notation introduced in Section 3, we denote as
$\overline{X}=X+U^{*}$ any tangent vector to $(\mathrm{SU}(2),\textsl{g})$.
This manner

$\displaystyle\mathrm{Ric}_{\textsl{g}_{-r^{2}}}(\overline{X})=\sum_{i=1}^{2}\mathrm{\kappa}_{-r^{2}}((1-r^{2})^{-1}(\overline{X}),e_{i})+\mathrm{\kappa}_{-r^{2}}((1-r^{2})^{-1}(\overline{X}),\sqrt{-1}(1-r^{2})^{1/2}e_{3})$
$\displaystyle=\sum_{i=1}^{2}K_{\mathrm{SU}(2)}(e_{i},X+(1-r^{2})^{-1}U^{*})-(1-r^{2})^{-1}K_{\mathrm{SU}(2)}(e_{3},X+(1-r^{2})^{-1}U^{*})$
$\displaystyle-3r^{2}(1-r^{2})^{-1}\left\\{\sum_{i=1}^{2}\left\|\nabla^{\mathbf{v}}_{e_{i}}X+(1-r^{2})^{-1}U^{*}\right\|_{\mathrm{S}^{1}}^{2}+\left\|\nabla^{\mathbf{v}}_{e_{3}}X+(1-r^{2})^{-1}U^{*}\right\|_{\mathrm{S}^{1}}^{2}\right\\}.$

To the following formulae, we recall that the O’Neill tensor is given by
$A_{X}Y:=\frac{1}{2}[X,Y]^{\mathbf{v}}$ for any $X,Y\in\mathcal{H}$. The
superscript $\mathbf{v}$ denotes ’the vertical component of.’ We shall often
consider $A^{*}_{X}$, the g-dual of $A_{X}$. Moreover, the shape operator
$S_{X}$, i.e., the g-dual of each orbit’s second fundamental $\sigma$, is the
map $S_{X}:\mathcal{V}\rightarrow\mathcal{V}$. According to [GW09, Theorem
1.5.1, p. 28]

$\displaystyle K_{\mathrm{SU}(2)}(e_{i},X+(1-r^{2})^{-1}U^{*})$
$\displaystyle=K_{\mathrm{SU}(2)}(e_{i},X)+(1-r^{2})^{-2}K_{\mathrm{SU}(2)}(e_{i},U^{*})$
$\displaystyle+2(1+r^{2})^{-1}R_{\textsl{g}}(e_{i},X,U,e_{i})-3r^{2}(1-r^{2})^{-1}|A_{X}e_{i}|_{\textsl{g}}^{2}$
$\displaystyle K_{\mathrm{SU}(2)}(e_{3},X+(1-r^{2})^{-1}U^{*})$
$\displaystyle=K_{\mathrm{SU}(2)}(e_{3},X)+2(1+r^{2})^{-1}R_{\textsl{g}}(e_{3},X,U,e_{3})$
$\displaystyle\nabla^{\mathbf{v}}_{e_{i}}(X+(1-r^{2})^{-1}U^{*})$
$\displaystyle=A_{e_{i}}X-(1-r^{2})^{-1}S_{e_{i}}U^{*}$
$\displaystyle\nabla^{\mathbf{v}}_{e_{3}}(X+(1-r^{2})^{-1}U^{*})$
$\displaystyle=S_{X}(e_{3})$

so the totally geodesic fiber assumption ensures that

$\mathrm{Ric}_{\textsl{g}_{-r^{2}}}(\overline{X})=\mathrm{Ric}^{\mathbf{h}}(X)+(1-r^{2})^{-2}\sum_{i=1}^{2}|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}-(1-r^{2})^{-1}|A^{*}_{X}e_{3}|_{\textsl{g}}^{2}\\\
+2(1+r^{2})^{-2}R_{\textsl{g}}(e_{3},X,U,e_{3})+2\sum_{i=1}^{2}(1+r^{2})^{-1}R_{\textsl{g}}(e_{i},X,U^{*},e_{i})-3r^{2}(1-r^{2})^{-1}\sum_{i=1}^{2}|A_{X}e_{i}|_{\textsl{g}}^{2}.$

Moreover, it can also be checked from the same formulae that

$R_{\textsl{g}}(e_{i},X,U^{*},e_{i})=\textsl{g}\left((\nabla^{\mathbf{v}}_{i}A)_{i}X,U^{*}\right)$
$R_{\textsl{g}}(e_{3},X,U,e_{3})=0.$
$\textsl{g}\left((\nabla^{\mathbf{v}}_{i}A)_{i}X,U^{*}\right)=0.$

Hence,

$\mathrm{Ric}_{\textsl{g}_{-r^{2}}}(\overline{X})=\mathrm{Ric}^{\mathbf{h}}(X)+(1-r^{2})^{-2}\sum_{i=1}^{2}|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}-(1-r^{2})^{-1}|A^{*}_{X}e_{3}|_{\textsl{g}}^{2}-3r^{2}(1-r^{2})^{-1}\sum_{i=1}^{2}|A_{X}e_{i}|_{\textsl{g}}^{2}.\qed$

##  4\. A semi-Riemannian metric on the standard sphere vs. on the Gromoll and
Meyer exotic sphere

In complete analogy to the previous section, in this section, we deal once
more with the bundle diagram (14) to provide Lorentzian metrics on the
classical $\mathrm{S}^{7}$-sphere as well as in the Gromoll–Meyer sphere
$\Sigma_{GM}^{7}$. As we shall see, the metrics to be provided enjoy the
property that time collapse on the existent fixed points in $\mathrm{S}^{7}$
and $\Sigma_{GM}^{7}$. Suppose it is the case that compact $7$-dimensional
manifolds recover some physical significance (we need to find out if this is
the case; however, see [Wit85] for an account). However, the time-collapsing
occurs for the same points up to some identification; see Proposition 4.4. In
this manner, how to perceive a difference due to the choice of a smooth
structure needs to be clarified, as we explain next.

###  4.1. An induction on the construction of the Gromoll–Meyer exotic sphere

We review the work initiated by C. Durán in [Dur01] and further developed by
L. D. Sperança in his Ph.D. thesis [Spe12] on producing exotic manifolds in an
_equivariant manner_. As we shall see, this can be considered an induction of
the Gromoll–Meyer exotic construction described in (14). The reference [CS18]
already contains a significative description that we choose to recall once
more, bringing the parallels and possibly making more explicit how exoticity
emerges from such constructions. More importantly, the feasibility of
producing Lorentzian metrics is due to this machinery.

Consider a principal bundle $G\hookrightarrow P\to M$ with principal action
$\bullet$ whose total space is compact and connected. Assume that $G$ is also
compact and connected and that there is another action by $G$ on $P$, which we
denote by $\star$, such that $\bullet$ and $\star$ commute. This makes $P$ a
$G\times G$-manifold. If we assume that $\star$ is also free, we get the
following $\star$-diagram of bundles:

(21)
$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bullet}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\star}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{\pi^{\prime}}$$\textstyle{M^{\prime}}$$\textstyle{M}$

The manifold $M$ in the (21) diagram denotes the quotient of $P$ for the
action $\bullet$, while $M^{\prime}$ denotes the quotient of $P$ for action
$\star$.

######  Definition 8.

The (21) diagram is called $\star$-(star) diagram. In short, we will denote it
by $M^{\prime}\leftarrow P\rightarrow M$.

Following [CS18], since $\bullet$ and $\star$ commute, $\bullet$ induces an
action on quotient $M^{\prime}$, and $\star$ induces an action on $M$. We
denote orbit spaces according to such induced actions by $M^{\prime}/\bullet$
and $M/\star$, or, when the context is clear, simply by $M^{\prime}/G$ and
$M/G.$

Let $M$ be a $G$-manifold (on the left) and $\\{U_{i}\\}$ be a collection of
open sets of $M$ that is $G$-invariant. Reducing $U_{i}$ if necessary, we can
assume that $GU_{i}=U_{i}.$ Given two open $G$-invariant set $U_{i},~{}U_{j},$
denoted by $U_{ij}:=U_{i}\cap U_{j}.$

A collection $\phi_{ij}:U_{ij}\to G$ is said to be a collection $\star$ if it
satisfies the cocycle condition,

(22) $\phi_{ij}(x)\phi_{jk}(x)\phi_{ki}(x)=\phi_{ii}(x),~{}\forall x\in
U_{ij},$

and also,

(23) $\phi_{ij}(gx)=g\phi_{ij}(x)g^{-1},~{}\forall i,j,~{}\forall g\in
G,~{}\forall x\in U_{ij}.$

The condition (23) guarantees that the adjoint map

$\widehat{\phi}_{ij}:U_{ij}\to U_{ij}$ (24)
$\widehat{\phi}_{ij}:x\mapsto\phi_{ij}(x)x.$

is equivariant. In this way, $\widehat{\phi}_{ij}$ defines an equivalent
diffeomorphism over $U_{ij}$.

Let $\\{\phi_{ij}:U_{ij}\to G\\}$ be a collection $\star.$ We define

$\bigcup_{\widehat{\phi}_{ij}}U_{i}$

as the quotient space associated with the following equivalence relation:

$x\in U_{ij}\sim\widehat{\phi}_{ij}x\in U_{ij}.$

I.e., given $x,y\in M,$ we say that $x$ is equivalent to $y$ if $\exists i,j$
for which $x,y\in U_{ij}$ and yet, $\phi_{ij}(x)x=y.$

The foundation of star diagram construction lies in the next theorem. We will
assume the existence of a collection $\star$ on the $G$-manifold $M$ based on
a principal bundle $\pi:P\to M.$ In this case, we say that $\pi$ is linked to
the collection $\star$ defined via $M$.

######  Theorem 4.1.

Let $\pi:P\to M$ be the principal bundle associated with a collection $\star$
given by $\\{\phi_{ij}:U_{ij}\to G\\}.$ Then $P$ admits a new action, denoted
by $\star$, such that

  1. (1)

The action $\star$ is free in $P$,

  2. (2)

The quotient $P/\star$ is a $G$-manifold equivalently diffeomorphic to

$M^{\prime}:=\bigcup_{\widehat{\phi}_{ij}}U_{i},$

  3. (3)

The star diagram (21) obtained from this construction is such that if
$(G\times G)_{p}$ denotes the isotropy group on $p$ with respect to the action
by juxtaposition

$(r,s)p:=rps^{-1},$

where $G\times\\{e\\}$ represents the action $\star$, $\\{e\\}\times G$
represents the principal action, and “$e$” denotes the neutral element of $G$
, then there exists $g\in G$ such that

(25) $(G\times G)_{p}=\\{(h,ghg^{-1}):g\in G_{\pi(p)}\\}.$

Let $M^{\prime}\leftarrow P\rightarrow M$ be a $\star$-bundle. Denote
$\bullet$ the principal action in $P.$ Remembering that since $\bullet$ and
$\star$ commute, $\star$ induces an action by $G$ in the quotient $M$ and
$\bullet$ induces an action by $G$ on $M^{\prime}$. Thus, we will see $M$ and
$M^{\prime}$ as $G$-manifolds associated with such actions.

An efficient method to compare the geometries of $M$ and $M^{\prime}$ consists
in constructing a metric $G\times G$-invariant on $P$ so that the projections
$\pi$ and $\pi^{\prime}$ induce Riemannian submersion metrics in $M$ and
$M^{\prime}$, respectively. In particular, if we start with a metric $G\times
G$-invariant on $P$, if $g_{M}$ denotes the Riemannian submersion metric
induced by $\pi$ on $M$, we prove the following theorem:

######  Theorem 4.2 (Corollary 5.2 in [CS18]).

Let $M^{\prime}\leftarrow P\rightarrow M$ be a $\star$-bundle. Then, a metric
$\bullet$-invariant exists on $M^{\prime}$ such that $M/G$ and $M^{\prime}/G$
are isometric as metric spaces.

Therefore, let

$\mathcal{H}^{{}^{\prime\prime}}:=\\{X\in\Gamma(TP):X\perp T(G\times G)\\}$

that is, $\mathcal{H}^{{}^{\prime\prime}}$ denotes the horizontal space
according to the action $(r,s)p\mapsto rps^{-1}.$ If $M$ and $M^{\prime}$ are
provided with Riemannian submersion metrics, then the constraints of $d\pi$
and $d\pi^{\prime}$ to $\mathcal{H}^{{}^{\prime\prime}}$ induce isometries
between $\mathcal{H}^{{}^{\prime\prime}}$ and $\mathcal{H}\subset TM$,
$\mathcal{H}^{\prime}\subset TM^{\prime},$ respectively.

Remember that the action of $G$ on $P$ induces actions on $TP$ (by the
differential of the representation). We abuse notation so that the action
induced on $TP$ is written as $gX,~{}g\in G,~{}X\in\Gamma(TP).$

######  Definition 9.

A $1$-connection form $\omega_{0}:\Gamma(TP)\to\mathfrak{g}$ in a principal
bundle $\pi:G\hookrightarrow P\to M$ consists of a smooth mapping such that

  1. (1)

$\omega_{0}(U^{*})=U,~{}\forall U\in\mathfrak{g},$ where $U^{*}$ denotes the
field of action associated with $U$,

  2. (2)

$(\omega_{0})_{pg}(Xg)=\mathrm{Ad}_{g^{-1}}(\omega_{0})_{p}(X).$

Given a $1$-form of connection $\omega:\Gamma(TP)\to\mathfrak{g}$ and an
$G$-invariant metric $g_{M}$ on $M$, we induce in $P$ a Kaluza-Klein metric
defined by

(26) $\langle X,Y\rangle:=g_{M}(d\pi X,d\pi
Y)+Q\left(\omega(X),\omega(Y)\right),$

where $Q$ is a bi-invariant metric on $G$. If $g_{M}$ is $G$-invariant and
$\omega$ satisfies $\omega_{gp}(gX)=\omega_{p}(X),~{}\forall g\in
G,X\in\Gamma(TP),$ so $\langle\cdot,\cdot\rangle$ is $G\times G$-invariant.

######  Proposition 4.3.

There is a $1$-connection form $\omega:\Gamma(TP)\to\mathfrak{g}$ such that
for all $X\in\Gamma(TP)$ and all $r\in G$

$\omega_{rp}(rX)=\omega_{p}(X).$

Furthermore, $\langle\cdot,\cdot\rangle$ defined by (26) is $G\times
G$-invariant.

###### Proof.

After taking the average of $\omega_{0},$

$\omega_{p}(X):=\int_{G}(\omega_{0})_{pr}(rX)dr$

we immediately conclude the result since $\bullet$ and $\star$ commute. ∎

From now on, we fix a Kaluza-Klein metric $G\times G$-invariant in $P$. As the
action of $G\times G$ on $P$ is by isometries, there is only one Riemannian
submersion metric on $M^{\prime}$. Furthermore, this is $G$-invariant. Observe
that if $x=\pi(p)$ and $x^{\prime}=\pi^{\prime}(p),$ then
$\pi^{-1}(Gx)=(G\times G)p={\pi^{\prime}}^{-1}(x)$, Thus, a bijection is
defined

$i:M/G\to M^{\prime}/G.$

In particular, $i$ is an isometry. Indeed, if $\gamma:[0,1]\to P$ is a
horizontal geodesic $\mathcal{H}^{\prime\prime}$, then $\pi\circ\gamma$ and
$\pi^{\prime}\circ\gamma$ are horizontal geodesics to the orbits of $G$ in $M$
and $M^{\prime}$, respectively. Therefore, given $p_{1},p_{2}\in P$, if
$\gamma$ denotes the smallest geodesic segment connecting the orbits $(G\times
G)p_{1}$ and $(G\times G)p_{2},$ then $\pi\circ\gamma$ and
$\pi^{\prime}\circ\gamma$ have the same length and are the minimizing
geodesics between $G\pi(p_{1})$ and $G\pi(p_{2})$; $G\pi^{\prime}(p_{1})$ and
$G\pi^{\prime}(p_{2});$ respectively. This verifies Theorem 4.2. ∎

###  4.2. Lorentzian metric via $\star$-diagrams

We now use the appropriate Riemannian metrics in $\star$-diagrams to easily
promote Lorentzian metrics out of these. Later, we give explicit constructions
to Lorentzian metrics on classical and exotic $7$-dimensional spheres.

Take $\textsl{g}^{\prime}$ to be the $\bullet$-invariant metric on
$M^{\prime}$ obtained from Theorem 4.2, and let $\textsl{g}^{\prime}_{-r^{2}}$
its $-r^{2}$-Cheeger deformation for sufficiently large $r>0$. Since we do not
have any information on whether the orbits of the $\bullet$-action on
$M^{\prime}$ are totally geodesic, the intrinsic geometry of the fibers is
decoded by a metric tensor $P^{\prime}$. So any $r>0$ such that
$1-r^{2}P^{\prime}<0$ makes $\textsl{g}_{-r^{2}}$ a semi-Riemannian metric on
the principal stratum ${M^{\prime}}^{reg}$ of $M^{\prime}$: Recall that
outside the principal stratum the orbit tensor may lose rank, so we cannot
control the signature of the metric there.

Observe, in addition, that there is a straightforward way of promoting a semi-
Riemannian metric on $M$ for which the orbit spaces $M/G$ and $M^{\prime}/G$
are still isometric. We claim that it suffices to consider the (same
parameter) $-r^{2}$-Cheeger deformation of g, a $\star$-invariant metric from
which $\textsl{g}^{\prime}$ descends in Theorem 4.2. This shall be guaranteed
by the following proposition, which we only state and appears in [CS18]. It is
also essential that it shows how $\star$-diagrams encode exoticity through the
bundle transitions since, locally, each _local orbit type_ is the same for
both $M$ and $M^{\prime}$.

More precisely, for each $x,y\in M$, Theorem 3.57 in [AB15] shows the
existence of well-defined $G$-invariant tubular neighborhoods centered on $Gx$
and $Gy$ so that: Denoting by $M^{\sim}_{x}$ the connected components of the
partition on $M$ by orbits with the same type (that is, with conjugate
isotropy subgroups) as $Gx,$ we say that any $y,z\in M^{\sim}_{x}$ has the
same local orbit type if there is a $G$-equivariant map between some
$G$-invariant tubular neighborhood $\mathrm{Tub}(Gx)$ and $\mathrm{Tub}(Gy)$.

######  Proposition 4.4 (Proposition 5.3 in [CS18]).

Given $x\in M$, there is $x^{\prime}\in\pi^{\prime}(\pi^{-1}(x))$ and an
isomorphism $\Phi\colon\mathrm{Tub}(Gx)\to\mathrm{Tub}(Gx^{\prime})$
satisfying: given $\mathcal{O}\subseteq\mathrm{Tub}(Gx)$, if
$\exp|_{\mathcal{O}}\colon\mathcal{O}\to M$ is a diffeomorphism onto its
image, then $\exp|_{\Phi(\mathcal{O})}\colon\Phi(\mathcal{O})\to M^{\prime}$
is a diffeomorphism onto its image.

Since Cheeger deformations preserve horizontal distributions, we have:

######  Theorem 4.5.

Let $M^{\prime}\leftarrow P\rightarrow M$ be a star diagram with
$\mathrm{S}^{1}$ as its structure group. Then there is a Lorentzian metric
$\textsl{g}_{-r^{2}}$ in the principal stratum of $M^{reg}$ of the
$\star$-action $M$, and a $\textsl{g}^{\prime}_{-r^{2}}$ Lorentzian metric in
the principal stratum ${M^{\prime}}^{reg}$ of the $\bullet$-action on
$M^{\prime}$ such that the orbit spaces $M/G$ and $M^{\prime}/G$ are
isometric.

We finally proceed with explicit constructions.

###  4.3. Explicit (almost) Lorentzian metrics on homotopy spheres

####  4.3.1. The $(\mathrm{S}^{1})-\star$-action on $\mathrm{S}^{7}$

Consider the $\star$ action on $\mathrm{S}^{7}$. It is defined by the
$q$-conjugation on each factor of $(a,b)\in\mathbb{H}\oplus\mathbb{H}$
satisfying $|a|^{2}+|b|^{2}=1$. It is straightforward to check that the
vertical space at a point
$(a,b)\in\mathrm{S}^{7}\subset\mathbb{H}\oplus\mathbb{H}$ is given by

(27) $\mathcal{V}_{(a,b)}=\left\\{\begin{pmatrix}[V,a]\\\
[V,b]\end{pmatrix},~{}V\in\mathfrak{\mathbf{i}\mathbb{R}}\right\\},$ (28)
$[V,a]:=Va-aV.$

We take $\begin{pmatrix}Xa\\\ Yb\end{pmatrix}$ to be orthogonal to the
$\star$-orbit through $(a,b).$ Throughout this section we assume that
$\langle\cdot,\cdot\rangle_{\mathrm{S}^{7}}$ is the restriction of the metric
of $\mathbb{H}\oplus\mathbb{H}$ to $\mathrm{S}^{7}$. Therefore,
$(Xa,Yb)^{T}\in T_{(a,b)}\mathrm{S}^{7}$ if, and only if,
$\mathrm{Re}(|a|^{2}X+|b|^{2}Y)=0$. We prove:

######  Lemma 1.

The vector $\begin{pmatrix}Xa\\\ Yb\end{pmatrix}\in
T_{(a,b)}\mathrm{\mathrm{S}^{7}}$ satisfy the equation

(29)
$\left(|a|^{2}-C_{\overline{a}}\right)X+\left(|b|^{2}-C_{\overline{b}}\right)Y=0,$

where $C_{\overline{a}}X:=\overline{a}Xa$ and $C_{\overline{b}}$ is defined
similarly.

###### Proof.

Recall that the Riemannian metric at $(a,b)$ is the sum of quaternionic inner
products on $\mathbb{H}\oplus\mathbb{H}$. Therefore, one has:

$\displaystyle 0=\left\langle\begin{pmatrix}Xa\\\
Yb\end{pmatrix},\begin{pmatrix}[V,a]\\\ [V,b]\end{pmatrix}\right\rangle$
$\displaystyle=\mathrm{Re}\left(C_{\overline{a}}(\overline{V}X)-\overline{V}C_{\overline{a}}(X)+C_{\overline{b}}(\overline{V}Y)-\overline{V}C_{\overline{b}}(Y)\right)$
$\displaystyle=\mathrm{Re}\left(|a|^{2}\overline{V}X-C_{a}(\overline{V})X+|b|^{2}\overline{V}Y-C_{b}(\overline{V})b\right)$
$\displaystyle=\mathrm{Re}\left(\left(\left(|a|^{2}-C_{a}\right)\overline{V}\right)X+\left(\left(|b|^{2}-C_{b}\right)\overline{V}\right)Y\right)$
$\displaystyle=\left\langle\left(|a|^{2}-C_{\overline{a}}\right)X+\left(|b|^{2}-C_{\overline{b}}\right)Y,V\right\rangle.$

One concludes the proof since the last equality must hold for every $V$. ∎

We now consider the $\mathrm{S}^{1}$-twisted action on
$\mathrm{S}^{7}\times\mathrm{S}^{1}$ defined as

(30)
$q\ast((a,b)^{T},q^{\prime}):=((qa\overline{q},qb\overline{q})^{T},qq^{\prime}),$

where $(a,b)^{T}\equiv\begin{pmatrix}a\\\ b\end{pmatrix}.$

In complete analogy to Lemma 1 one has that the horizontal space at
$((a,b),q^{\prime})$ with respect to the action (30), computed in the metric

$\widetilde{\textsl{g}}_{-r^{-2}}:=\langle\cdot,\cdot\rangle_{\mathrm{\mathrm{S}^{7}}}-r^{-2}d\theta^{2},$

is given by

$\mathcal{H}_{((a,b),q^{\prime})}\cong\mathrm{span}_{\mathbb{R}}\left\\{((Xa,Yb)^{T},iq^{\prime}t)\in
T_{(a,b)}\mathrm{S}^{7}\oplus
T_{q^{\prime}}\mathrm{S}^{1}:(|a|^{2}-C_{\overline{a}})X+(|b|^{2}-C_{\overline{b}})Y=tr^{-2}\right\\}.$

Therefore, once more identifying $\partial_{\theta}\leftrightarrow\mathbf{i}$
one gets the following description of the tangent space of $\mathrm{S}^{7}$ at
$(a,b)$

$T_{(a,b)}\mathrm{S}^{7}\cong\mathrm{span}_{\mathbb{R}}\left\\{t\partial_{\theta},(Xa,Yb)^{T}:(Xa,Yb)\in
T_{(a,b)}\mathrm{S}^{7},(|a|^{2}-C_{\overline{a}})X+(|b|^{2}-C_{\overline{b}})Y=tr^{-2}\right\\},$

with the following semi-Riemannian metric:

$\textsl{g}_{-r^{2}}(t\partial_{\theta}+(Xa,Yb)^{T},t^{\prime}\partial_{\theta}+(X^{\prime}a,Y^{\prime}b)^{T})=\langle
X,X^{\prime}\rangle_{\mathbb{H}}+\langle
Y,Y^{\prime}\rangle_{\mathbb{H}}-r^{-2}tt^{\prime}.$

####  4.3.2. The $(\mathrm{S}^{1})-\bullet$-action on $\Sigma^{7}_{GM}$

Consider the $\bullet$ action on $\Sigma_{GM}^{7}$. It is given by the
conjugation by $q\in\mathrm{S}^{1}$ only in the first factor
$(2\bar{c}d,|c|^{2}-|d|^{2})$. Hence, the vertical space at a point
$(2\overline{c}d,|c|^{2}-|d|^{2})\in\Sigma^{7}\subset\mathbb{H}\oplus\mathbb{H}$
is given by

(31)
$\mathcal{V}_{(2\overline{c}d,|c|^{2}-|d|^{2})}=\mathrm{span}_{\mathbb{R}}\left\\{\begin{pmatrix}2\mathbf{i}\overline{c}d-2\overline{c}d\mathbf{i}\\\
0\end{pmatrix}\right\\}.$

Take $\begin{pmatrix}\frac{1}{2}X\overline{c}d\\\ Y\end{pmatrix}\in
T_{(2\overline{c}d,|c|^{2}-|d|^{2})}\Sigma_{GM}^{7}$ to be
$\widetilde{\textsl{g}}_{-r^{2}}$-orthogonal to the $\star$-orbit through
$(2\overline{c}d,|c|^{2}-|d|^{2})$, where

$\widetilde{\textsl{g}}_{-r^{2}}=\langle\cdot,\cdot\rangle_{\mathbb{H}\oplus\mathbb{H}}-r^{2}d\theta^{2}.$

Observe that $(\frac{1}{2}X\overline{c}d,Y)\in
T_{(2\overline{c}d,|c|^{2}-|d|^{2})}$ if, and only if,

$|\bar{c}d|^{2}\mathrm{Re}(X)=-\mathrm{Re}(Y)(|c|^{2}-|d|^{2}).$

Moreover, copying the proof of Lemma 1 we can check that

######  Lemma 2.

The vector $\begin{pmatrix}\frac{1}{2}X\overline{c}d\\\ Y\end{pmatrix}$
satisfy the equation

$|\bar{c}d|^{2}X-C_{\bar{c}d}X=0,$

where $C_{\bar{c}d}$ is the conjugation by $\bar{c}d$.

We now consider the $\mathrm{S}^{1}$-twisted action on
$\Sigma^{7}_{GM}\times\mathrm{S}^{1}$ defined as

(32)
$q\ast((2\overline{c}d,|c|^{2}-|d|^{2})^{T},q^{\prime}):=((q\overline{c}d\overline{q},|c|^{2}-|d|^{2})^{T},qq^{\prime}),$

where
$(2\overline{c}d,|c|^{2}-|d|^{2})^{T}\equiv\begin{pmatrix}2\overline{c}d\\\
|c|^{2}-|d|^{2}\end{pmatrix}.$

Therefore, it is straightforward to check that the horizontal for $\ast$ is
given by

$\mathcal{H}_{((\bar{c}d,|c|^{2}-|d|^{2}),q^{\prime})}\cong\left\\{\left(\left(-\frac{1}{2}X\overline{c}di,Y\right),itq^{\prime}\right):|\bar{c}d|^{2}X+C_{\bar{c}d}X=r^{-2}t,~{}|\bar{c}d|^{2}\mathrm{Re}(X)=-\mathrm{Re}(Y)(|c|^{2}-|d|^{2})\right\\}.$

Hence, any horizontal vector has the norm

$\left|\left(\left(-\frac{1}{2}X\overline{c}d\mathbf{i},Y\right),\mathbf{i}tq^{\prime}\right)\right|^{2}_{\widetilde{\textsl{g}}_{-r^{2}}}=\frac{1}{4}|X|^{2}|c|^{2}|d|^{2}+|Y|^{2}-t^{2}r^{-2}$

and so the Lorentzian metric induced on $\Sigma_{GM}^{7}$ can be defined as

(33)
$\textsl{g}_{-r^{2}}(-\frac{1}{2}X\overline{c}d\mathbf{i}+Y,-\frac{1}{2}X^{\prime}\overline{c}d\mathbf{i}+Y^{\prime}):=\frac{1}{4}|c|^{2}|d|^{2}\langle
X,Y\rangle_{\mathbb{H}}+\langle
X^{\prime},Y^{\prime}\rangle_{\mathbb{H}}-t^{2}r^{-2}.$

Observe that the $\mathrm{S}^{1}$-action constructed in $\mathrm{S}^{7}$ has
some fixed points, the same holding for the $\mathrm{S}^{1}$-action built on
$\Sigma_{GM}^{7}$. In this manner, just as in the case of the
$\mathrm{S}^{1}$-action constructed in $\mathrm{SU}(2)$, these metrics present
time singularities, which we emphasize once more, could be interpreted as
restricted world perspectives of observers living in
$\mathrm{S}^{7}\times\mathbb{R}$ or $\Sigma_{GM}^{7}\times\mathbb{R}$, the
corresponding universal coverings of $\mathrm{S}^{7}\times\mathrm{S}^{1}$ and
$\Sigma_{GM}^{7}\times\mathrm{S}^{1}$. In the next section, we provide a
better study of such singularities besides furnishing a classification.

##  5\. Fixed points singularities and Ricci curvature

######  Lemma 3.

Let $\mathrm{S}^{7}$ with the $\mathrm{S}^{1}$-action defined by equation
(13). Then, the only fixed points $(a,b)\in\mathrm{S}^{7}$ by the
$\mathrm{S}^{1}$-action are

$(a,b)\in\mathrm{S}^{7}\cap\left(\mathbb{R}\times\mathbb{R}\right).$

Moreover, for each $(a,b)\in\mathrm{S}^{7}$ such that $a\in\mathrm{S}^{1},b=0$
or $b\in\mathrm{S}^{1},a=0$, we have that
$\mathrm{S}^{1}_{(a,b)}=\\{e,(a,b)\\}\cong\mathbb{Z}_{2}$ and out of these
points the isotropy subgroups are trivial.

###### Proof.

Since we are considering $(a,b)\in\mathrm{S}^{7}$ where both $a,b$ are
quaternions, let $q\in\mathrm{S}^{1}$ be such that $q\neq 1$ but
$qa\bar{q}=a,qb\bar{q}=b$. It follows that $[a,q]=0=[b,q]$, where
$[\cdot,\cdot]$ denotes the quaternionic bracket. Assuming that
$\mathrm{Im}a,\mathrm{Im}b\neq 0$ write
$a=|a|\mathrm{e}^{\mathrm{Im}a\theta_{a}/|\mathrm{Im}a|}$,
$b=|b|\mathrm{e}^{\mathrm{Im}b\theta_{b}/|\mathrm{Im}b|}$ and
$q=\mathrm{e}^{\mathbf{i}\theta_{q}}$. Here,
$\theta_{a},\theta_{b},\theta_{q}\in[0,2\pi[$ are such that

$\cos(\theta_{a})=\mathrm{Re}a/|a|,~{}\sin(\theta_{a})=|\mathrm{Im}a|/|a|,$

making the corresponding analogous definitions for both $b,q$.

Hence,

$[a,q]=0\Leftrightarrow\mathrm{e}^{2\left(\mathrm{Ima}\theta_{a}/|\mathrm{Im}a|-i\theta_{q}\right)}=1.$

From the definition of exponential to quaternions, one gets that
$\mathrm{Im~{}a}\theta_{a}=|\mathrm{Im}~{}a|\mathbf{i}\theta_{q}$. Hence,
$a=a_{0}+|\mathrm{Im}a|\theta_{q}/\theta_{a}\mathbf{i}+0\mathbf{j}+0\mathbf{k}$.
The same argumentation implies that
$b=b_{0}+|\mathrm{Imb}|\theta_{q}/\theta_{b}\mathbf{i}+0\mathbf{j}+0\mathbf{k}$.
Hence, $\theta_{q}=\theta_{a}=\theta_{b}$ and so
$q=a_{0}/|a|+\mathbf{i}a_{1}/|a|=b_{0}/|b|+\mathbf{i}b_{1}/|b|$. However,
using that $|a|^{2}+|b|^{2}=1$ we have that $|q|^{2}=2|a|^{2}+2|b|^{2}=2$, a
contradiction. Therefore, either $\mathrm{Im}a=0$ or $\mathrm{Im}b=0$. In the
first case, $a\in\mathbb{R}$ and $b=b_{0}+|\mathrm{Im}b|\mathbf{i}$ since
$\theta_{q}=\theta_{b}$. Moreover, $q=b_{0}/|b|+b_{1}\mathbf{i}/|b|$ and we
once more derive the same contradiction unless $\mathrm{Re}a=0$. Hence, either
$\mathrm{Im}b=0$ or $a=0$. That considered, we have that
$(a,b)\in\mathrm{S}^{7}$ such that $a,b\in\mathbb{R}$ are fixed points by the
action defined in equation(13).

Assume now that $b=0$ but $\mathrm{Im}a\neq 0$. Then
$\mathrm{Ima}\theta_{a}=|\mathrm{Im}a|\mathbf{i}\theta_{q}$ and
$\theta_{a}=\theta_{q}$, that is, in this case $a=q$. So
$\mathrm{S}^{1}$-itself is fixed by this action. The same analysis holds for
$a=0$ but $b\neq 0$, finishing the proof.∎

At these points, we now study the behavior of what is called _fake horizontal
vectors_ , first introduced in [CeSS23]. Namely, since the vertical space
collapses at fixed points, say that $x\in M$ is one such point. Then, fixed a
horizontal geodesic $c$ emanating from $x$, we proceed to characterize the
horizontal vectors at $x$, which behave like time-like vectors along $c$ for
$s\neq 0$.

######  Lemma 4 (Lemma 3.2 in [CeSS23]).

Let $(M,\textsl{g})$ be a path-connected Riemannian manifold with an effective
isometric action by a compact connected Lie group $G$. Let $x$ be a $G$-fixed
point and $X\in\mathcal{H}_{x}$. Assume that the horizontal geodesic
$\gamma:[0,\epsilon]\to M$ defined as $\gamma(s)=\exp_{x}(sX)$ intersects the
principal stratum for any $s>0.$ Define

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

Then,

  1. $1.$

the image $\tilde{S}_{X}({\mathfrak{g}_{x}})$ is contained in
$\mathcal{H}_{x}$. Moreover, for $\epsilon>0$ sufficiently small, the
following defines a smooth bundle on $\gamma([0,\epsilon))$

$\tilde{\mathcal{H}}_{s}=\begin{cases}\mathcal{H}_{\gamma(s)}~{}~{}\text{if}~{}s>0\\\
(\tilde{S}_{X}(\mathfrak{g}_{x}))^{\bot}~{}~{}\text{if}~{}s=0.\end{cases}$

  2. $2.$

the kernel of the restriction $\tilde{S}_{X}|_{\mathfrak{g}_{X}}$ coincides
with $\mathfrak{g}_{X}$, the Lie algebra of $G_{X}=\\{r\in
G_{x}~{}|~{}\rho(r)X=X\\}$, where $\rho(r):T_{x}M\rightarrow T_{x}M$ is the
isotropy representation at $r\in G_{x}$.

In our case, in a fixed point $x$ we have that $\mathfrak{g}_{x}=\mathfrak{g}$
but also that
$\dim_{\mathbb{R}}\mathfrak{g}_{x}=\dim_{\mathbb{R}}\mathbf{i}\mathbb{R}=1$.
Therefore, for each $X\in\mathcal{H}_{x}$ either $\mathfrak{g}_{X}=0$ or
$\mathfrak{g}_{X}\cong\mathfrak{g}\cong\mathbf{i}\mathbb{R}$. If
$\mathfrak{g}_{X}=\mathbf{i}\mathbb{R}$, then $\widetilde{S}_{X}\equiv 0$ and
hence, the parallel transport of each horizontal vector along the geodesic
generated by $X$ leads to a horizontal vector at $x$. Hence, $X$ is precisely
a _fake-horizontal vector_. The geodesic $s\mapsto\gamma(s)$ is time-like for
$s>0$. We conclude:

######  Theorem 5.1.

Let $(M,\textsl{g}_{-r^{2}})$ be a path-connected semi-Riemannian manifold
with an effective isometric action by $\mathrm{S}^{1}$ and $x$ be a $G$-fixed
point. Assuming that $\textsl{g}_{-r^{2}}$ is Lorentzian in the principal
stratum, given $X\in\mathcal{H}_{x}\cong T_{x}M$, the geodesic generated by
$s\mapsto\gamma(s)$ is time-like for any $s>0$ if, and only if,
$\mathfrak{g}_{X}=\mathfrak{s}^{1}\cong\mathbf{i}\mathbb{R}$.

We can fix the metric $\textsl{g}_{-r^{2}}$ with the former theorem to produce
a global Lorentzian metric on any almost Lorentzian manifold.

######  Theorem 5.2.

Let $(M,\textsl{g})$ be an almost Lorentzian manifold with the Riemannian
metric $\textsl{g}_{-r^{2}}$. The there exists a Lorentzian metric
$\widetilde{\textsl{g}}_{-r^{2}}$ on $M$ which coincides with
$\textsl{g}_{-r^{2}}$ in the principal stratum in $M$.

###### Proof.

According to Theorem 5.1, let $x\in M$ be a fixed point for the
$\mathrm{S}^{1}$ almost semi-free action. Take $X\in T_{x}M$ such that
$\mathfrak{g}_{X}=\mathfrak{s}^{1}$. We define

$\widetilde{\textsl{g}}_{-r^{2}}=\textsl{g}|_{\left(\mathrm{span}_{\mathbb{R}}\left\\{X\right\\}\right)^{\perp_{\textsl{g}}}}+(1-r^{2})^{-1}\textsl{g}|_{\mathrm{span}_{\mathbb{R}}\\{X\\}}.$

Moreover, we let $\widetilde{\textsl{g}}_{-r^{2}}:=\textsl{g}_{-r^{2}}$ be the
Lorentzian metric in the principal stratum $M^{reg}$. The well-definition of
this metric is straightforward. ∎

To exemplify the desired fake horizontal vector at a fixed point given by
Lemma 3, let $(a,b)\in\mathbb{R}\oplus\mathbb{R}$ such that
$|a|^{2}+|b|^{2}=1$. Then, the $\mathrm{S}^{1}$ action by conjugation in each
factor fixes $(a,b)$. Let $s\mapsto c(t)=(a(s),b(s))$ a smooth curve in
$\mathrm{SU}(2)$ such that $(a(0),b(0))=(a,b)$ and assume that
$(a^{\prime}(0),b^{\prime}(0))=(Xa,bY)$ where
$\mathrm{Re}(X|a|^{2}+Y|b|^{2})=0$. Then,

$\displaystyle q\ast(a(s),b(s))$ $\displaystyle:=(qa(s)\bar{q},qb(s)\bar{q})$
$\displaystyle\rho(q)(Xa,bY)$ $\displaystyle=(qaX\bar{q},qbY\bar{q})$
$\displaystyle\rho(q)(Xa,bY)$ $\displaystyle=(Xa,bY)\Leftrightarrow
qaX=Xaq,~{}qbY=byq$ $\displaystyle\Leftrightarrow
aX,bY\in\mathbb{C}\cong\mathrm{span}_{\mathbb{R}}\left\\{1,\mathbf{i}\right\\}.$

Hence, the geodesic generated by $(aX,bY)\in
T_{(a,b)}\mathrm{SU}(2)\cap\mathbb{C}\oplus\mathbb{C}$ for
$(a,b)\in\mathrm{SU}(2)\cap(\mathbb{R}\times\mathbb{R})$ is time-like.

###  5.1. $\mathrm{S}^{1}$-Fat bundles and a dual leaf type result

We recall that for a general (non-necessarily metric) submersion, the
_fatness_ condition is an intrinsic property of its horizontal distribution
$\mathcal{H}$. Namely, a submersion $\pi:M\rightarrow B$ is said to be fat if,
for every non-zero $X\in\mathcal{H}$,
$\mathcal{V}=[\widetilde{X},\mathcal{H}]^{\mathbf{v}}$ for some horizontal
extension of $X$, where we denote by $\mathcal{V}$ the vertical distribution
associated to $\pi$. We observe that often in the literature, the nomenclature
for fatness is widely employed, assuming the preexistence of a Riemannian
metric with totally geodesic leaves.

Treating the fatness condition for $\mathrm{S}^{1}$-principal bundles is quite
different from other structure groups simply due to the fact
$\mathfrak{s}^{1}$ is the (1$\mathbb{R}$-dimensional) Abelian Lie algebra
generated by $\mathbf{i}\in\mathbb{C}$. In this manner, any connection form
$\omega$ on the bundle and its corresponding curvature form $\Omega$ are
nothing but ordinary $1$, $2$-forms, respectively, where $d\omega=\Omega$.

It can be inferred from Chern-Weil theory ([Zil00, Corollary 2.11, p. 10])
that a principal $\mathrm{S}^{1}$-bundle is fat if, and only if the curvature
form $\Omega$ is a pullback of some symplectic form on the base $B$: Fat
$\mathrm{S}^{1}$-principal bundles are (bi)univocally associated with
symplectic manifolds.

We recall the following examples of $\mathrm{S}^{1}$-fat Riemannian
submersions (see [Zil00])

  1. (a)

the Hopf bundles
$\mathrm{S}^{1}\hookrightarrow\mathrm{S}^{2n+1}\rightarrow\mathbb{CP}^{n}$

  2. (b)

any $\mathrm{S}^{1}$-principal Riemannian bundle
$(M,\textsl{g})\rightarrow(B,\textsl{g}_{B})$ with fibers of length $2\pi$
such that $\textsl{g}_{B}$ is either

    1. (i)

almost Kähler and so g is $K$-contact

    2. (ii)

Kähler and so g is Sasakian

    3. (iii)

Kähler Einstein and so g is Sasakian Einstein

A metric on a smooth manifold $M$ is named $K$-contact if it admits a unit
Killing field $V$ with $\mathrm{sec}_{\textsl{g}}(X,V)=1$ for every non-
degenerated vertizontal plane $X\wedge V$. We say further that the metric is
regular if the integral curves of $V$ define a free-circle action, and it is
quasi-regular if they define an almost free-circle action. According to
[Wad75], the latter is equivalent to requiring that all integral curves of $V$
are closed. If it is regular, $M\rightarrow B$ is a fat Riemannian submersion
with totally geodesic fibers.

A metric is called Sasakian if it admits a unit Killing vector field $X$ such
that $R_{\textsl{g}}(V,X)Y=\textsl{g}(V,Y)X-\textsl{g}(X,Y)V$ for every
$X,Y\perp V$. As before, the Sasakian structure is called regular if the
integral curves of $V$ define a free circle action and quasi-regular if they
define an almost free circle action.

####  5.1.1. Holonomy and Dual Holonomy fields

Let $(M,\textsl{g})$ be a Riemannian manifold with a Riemannian foliation
$\mathcal{F}$. Considering the natural decomposition
$TM=\mathcal{V}\oplus\mathcal{H}$ where $\mathcal{V}$ stands for the sub-
bundle of $TM$ containing the vectors tangent to the leaves of $\mathcal{F}$,
if $c:I\rightarrow M$ is a geodesic such that
$\dot{c}(s)\in\mathcal{H}_{c(s)}$ for any $s\in I\subset\mathbb{R}$ a vector
field $\nu$ such that $\nu(s)\in\mathcal{V}_{c(s)}$ for every $s\in I$ is
named a dual-holonomy field provided if

(34) $\nabla_{\dot{c}}\nu=S_{\dot{c}}\nu-A^{*}_{\dot{c}}\nu.$

Analogously, basic horizontal fields along a vertical curve $\gamma$ are
horizontal solutions of

(35) $\nabla_{\dot{\gamma}}X=-A^{*}_{X}\dot{\gamma}-S_{X}(\dot{\gamma}).$

In the above equations, $S_{X}$ always stands for the second fundamental form
of a leaf. The tensor $A^{*}_{X}$ is g-dual of the O’Neill tensor
$A_{X}Y=\frac{1}{2}[X,Y]^{\mathbf{v}}$, $X,Y\in\mathcal{H}$. Under the totally
geodesic fibers assumption, both dual-holonomy and holonomy fields along
horizontal geodesics and basic horizontal fields along vertical geodesics are
the Jacobi fields induced by, respectively, local horizontal lifting and
holonomy transport.

Moreover, $A^{*}_{X}\dot{\gamma}$ is basic along the geodesic $\exp(sV)$,
where $V\in\mathcal{V}$ and $X\in\mathcal{H}$ is basic. If g has non-negative
sectional curvature, then for each leaf $L_{p}\in\mathcal{H}$,

$T_{p}L_{p}^{\\#}\cap\mathcal{V}_{p}=\mathrm{span}_{\mathbb{R}}\\{A_{X}Y(p):X,Y\in\mathcal{H}_{p}\\},$

where $L_{p}^{\\#}$ is a dual leaf through $p$:

$L^{\\#}_{p}:=\\{q\in M:\text{there exists a piecewise smooth horizontal curve
from $p$ to $q$}\\},$

as described in [Wil07].

Suppose that the foliation $\mathcal{F}$ is given by the connected components
of the fibers of a Riemannian submersion $\pi:M\rightarrow B$. Recall that the
_holonomy group_ at some $b\in B$ is defined considering: Fixed $b\in B$,
consider all simple closed curves in $B$ based at $b$. We can define a
diffeomorphism of the fiber $F=\pi^{-1}(b)$ over $b$ by lifting the curve
horizontally to points of $F$. The collection of all of these diffeomorphisms
forms an appropriate composition rule. Such a group is trivial in the case of
a Riemannian product $F\times B\rightarrow B$. Generally, the holonomy group
is not a Lie group, but it certainly is for _Homogeneous bundles_. This is
precisely the case for fat Riemannian submersions.

For many foliations, such as the Riemannian foliation induced by the isometric
action of a compact connected Lie group, it is known that the dual-foliation
is a _Singular Riemannian Foliation_ (see [Wil07]). A crucial point to the
verification of this is the usage of completeness of each dual leaf.

On the other hand, it is known (for instance, via the now classic example of
J. Wolf ([Wol64, Corollary 11.1])) that even under very high symmetrical
conditions, a Lorentzian manifold needs not to be geodesically complete. We
can show, however, that the family of compact Riemannian manifolds with almost
semi-free $\mathrm{S}^{1}$-actions, when regarded with the metric
$\widetilde{\textsl{g}}_{-r^{2}}$ given by Theorem 5.1, is time-like and
space-like complete. In particular, we show that the dual leaf foliation
induced by a fixed horizontal distribution has complete leaves for
$\mathrm{S}^{1}$-fat foliations. Indeed, the dual foliation has a single leaf,
which coincides with $M$. This was already observed for Riemannian metrics in
[CG23], so we show it works as well for Lorentzian metrics induced by circle
actions.

######  Theorem 5.3.

Let $\mathrm{S}^{1}\rightarrow M\rightarrow B$ be a principal
$\mathrm{S}^{1}$-fat bundle with connection form $\omega$. If g is the unique
(up to scaling) connection metric g on $M$, any (semi) Riemannian dual
foliation for the Lorentzian metric $\widetilde{\textsl{g}}_{-r^{2}}$ (for
$r>1),$ has only one leaf, which coincides with $M$. Moreover, if the action
is almost semi-free, the conclusion holds in the principal stratum $M^{reg}$.

###### Proof.

For the proof, note that if g is the unique (up to scaling) connection metric
on $\mathrm{S}^{1}\rightarrow M\rightarrow B$ and
$A:\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{V}$ is the O’Neill tensor,
denoting by $A^{*}_{X}$ the g-dual to $A_{X}$ for any fixed $X\in\mathcal{H}$
one has that ${A^{*}_{-r^{2}}}_{X}V=(1-r^{2})^{-1}A^{*}_{X}V$, where
${A^{*}_{-r^{2}}}_{X}$ is the $\textsl{g}_{-r^{2}}$-dual tensor to $A_{X}$. In
particular, since the orbits of $\mathrm{S},^{1}$ are totally geodesic for g,
these are totally geodesic for $\textsl{g}_{-r^{2}}$ as well. So
$\mathrm{sec}_{\textsl{g}_{-r^{2}}}(X,V)=-(1-r^{2})^{-1}|A^{*}_{X}V|_{\textsl{g}}^{2}$
for any $\textsl{g}_{-r^{2}}$-orthonormal vertizontal plane. In this manner,
the fatness condition ensures that
$\mathrm{sec}_{\textsl{g}_{-r^{2}}}(X,V)<0$.

We claim that each dual leaf is open. Indeed, since the dual foliation only
depends on the choice of a horizontal distribution, we take $\mathcal{H}$ as
the horizontal distribution given by the g-orthogonal complement of the
Riemannian foliation induced by the orbits of $\mathrm{S}^{1}$. Now take a
point $x\in L_{x}^{\\#}$, where $L_{x}^{\\#}$ is the dual leaf through $x$.
The fatness condition ensures that any given $V\in\mathcal{V}_{x}$ has a
constant norm along any horizontal geodesic emanating from $x$. This follows
easily from equation (11) in [Spe18]: Since
$\mathrm{sec}_{\textsl{g}_{-r^{2}}}(X,V)<0$ then any dual holonomy $\nu$ field
along the geodesic $c$ generated by $X$ has constant norm, since
$\frac{1}{2}\frac{d^{2}}{dt^{2}}|\nu|^{2}<\mathrm{sec}_{\textsl{g}}(X,V)$
implies that $|\nu|^{2}=e^{t\lambda}$ for $\lambda\in(-\sqrt{2},\sqrt{2})$. If
$\lambda\neq 0$ then $|\nu|^{2}\rightarrow\infty$ for $t\nearrow\infty$ (if
$\lambda>0$) or $-t\nearrow\infty$ (if $\lambda<0$). Both cases contradict the
compactness of $\mathrm{S}^{1}$, since every compact structure group enjoys
the property $|\nu(t)|\leq L|\nu(0)|$ for some $L>0$, see Definition 1.3 in
[Spe18]. The claim holds since then the dimension of
$T_{c(t)}L_{c(t)}^{\\#}\cap\mathcal{V}_{c(t)}$ is constant and so $\dim M=\dim
L_{x}^{\\#}$. ∎

Recall that if $\mathrm{S}^{1}$ acts almost freely on a smooth compact
connected manifold $M$, then the quotient space $M/G$ has the structure of an
orbifold, see [CJ22, Proposition 1.5.1, p. 17]. In the following corollary, we
use the terminology _orbibundle_ to refer to a smooth manifold with an
effective isometric, almost free action, so the quotient is seen as an
orbifold.

######  Corollary 5.4.

The following manifolds admit Lorentzian metrics, which are time-like and
space-like geodesically complete:

  1. (a)

the Hopf orbibundle
$\mathrm{S}^{1}\hookrightarrow\mathrm{S}^{3}\rightarrow\mathbb{CP}^{1}$,

  2. (b)

any $\mathrm{S}^{1}$-principal Riemannian orbibundle
$(M,\textsl{g})\rightarrow(B,\textsl{g}_{B})$ with (non-trivial) fibers of
length $2\pi$ such that $\textsl{g}_{B}$ is either

    1. (i)

almost Kähler and so g is $K$-contact

    2. (ii)

Kähler and so g is Sasakian

    3. (iii)

Kähler Einstein and so g is Sasakian Einstein

###  5.2. A general theorem for Lorentzian manifolds with positive Ricci
curvature metrics

We finish this section proving that our manifolds in this paper have positive
Ricci curvature—the same holding for the product space that furnishes the
considered Lorentzian metrics via Cheeger deformations.

######  Theorem 5.5.

Let $(M,\textsl{g})$ be a closed Riemannian manifold of dimension $n\geq 3$
with non-negative sectional curvature. Assume a free isometric
$\mathrm{S}^{1}$-action on $M$ exists. Then for any $r>1$, the Lorentzian
metric $\mathrm{g}_{-r^{2}}$ has positive Ricci curvature, the same holding
for the Lorentzian manifold $(M\times\mathrm{S}^{1},\textsl{g}-r^{-2}Q)$,
where $Q$ is the standard bi-invariant metric on $\mathfrak{s}^{1}$.

If the $\mathrm{S}^{1}$-action is almost semi-free, there exists $r^{2}$
sufficiently large such that the metric $\textsl{g}_{-r^{2}}$ has positive
Ricci curvature in $M$. In particular, the metric
$\widetilde{\textsl{g}}_{-r^{2}}$ given by Theorem 5.1 has positive Ricci
curvature.

###### Proof.

The proof is a generalization of the proof of Theorem 3.3. We take
$\mathrm{S}^{1}$ to be totally geodesic for g, and so the orbit tensor
$P:\mathfrak{m}_{x}\rightarrow\mathfrak{m}_{x}$, for every $x\in M$, can be
taken equal to the identity, where we decompose
$\mathfrak{g}=\mathfrak{m}_{x}\oplus\mathfrak{g}_{x}$ orthogonaly with respect
to any fixed bi-invariant metric induced from $\mathrm{Iso}(\textsl{g})$. If
the $G$-action is free, then $\mathfrak{m}_{x}\cong\mathfrak{g}$ for every
$x\in M$. Otherwise, in the presence of fixed points
$\mathfrak{g}_{x}=\mathfrak{g}$ for every fixed point $x$ and $P$ collapses to
the $0$-operator along any geodesic coming from the principal stratum: see
Lemma 4.

We compute the Ricci curvature of the metric $\textsl{g}_{-r^{2}}$. With this
aim, we write any vector $\overline{X}\in T_{x}M$ as $\overline{X}=X+U^{*}$
for $U\in\mathbf{i}\mathbb{R}$ where the superscript $\ast$ reinforces that
$U^{*}$ is an action vector at $x\in M$. Taking $\\{e_{1},\ldots,e_{n-1}\\}$
as a g-orthonormal basis to $\mathcal{H}_{x}$ and
$\\{e_{n}\\}\in\mathbf{i}\mathbb{R}$ is a unit vector, one gets that
$\\{e_{1},\ldots,e_{n-1}\\}\cup\left\\{\sqrt{-1}(1-r^{2})^{1/2}e_{n}\right\\}$
is a $\textsl{g}_{-r^{2}}$-orthogonal basis to $T_{x}M$ such that
$\textsl{g}_{-r^{2}}(\sqrt{-1}(1-r^{2})^{1/2}e_{n},\sqrt{-1}(1-r^{2})^{1/2}e_{n})=-1$.
We have added the $\sqrt{-1}$ multiplying factor since we are considering
$r^{2}-1>0$. So $(1-r^{2})^{1/2}$ is a complex number. Therefore, just as in
the proof of Theorem 3.3, this choice of basis works as a simple way to write
the needed basis explicitly.

Now the needed unreduced sectional curvature can be easily computed for any
fixed $\overline{X}\in T_{x}M$ as for $i\in\\{1,\ldots,n-1\\}$

$\displaystyle K_{\textsl{g}_{-r^{2}}}(\overline{X},e_{i})$
$\displaystyle=\kappa_{-r^{2}}((1-r^{2})^{-1}\overline{X},e_{i})$
$\displaystyle=K_{\textsl{g}}((1-r^{2})^{-1}\overline{X},e_{i})-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}e_{i}|_{\textsl{g}}^{2}$
$\displaystyle=K_{\textsl{g}}(X,e_{i})+(1-r^{2})^{-2}K_{\textsl{g}}(U^{*},e_{i})+2(1-r^{2})^{-1}R_{\textsl{g}}(e_{1},X,U^{*},e_{i})-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}e_{i}|_{\textsl{g}}^{2}$
$\displaystyle=K_{\textsl{g}}(X,e_{i})+(1-r^{2})^{-2}|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}e_{i}|_{\textsl{g}}^{2}$

where we have used the ’vertizontal’ curvature under totally geodesic fibers
is given by $|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}$ and that the mixed terms
curvature component $R_{\textsl{g}}(e_{1},X,U^{*},e_{i})$ vanishes.

Moreover,

$\displaystyle
K_{\textsl{g}_{-r^{2}}}(\overline{X},\sqrt{-1}(1-r^{2})^{1/2}e_{n})$
$\displaystyle=\kappa_{-r^{2}}((1-r^{2})^{-1}\overline{X},\sqrt{-1}(1-r^{2})^{-1/2}e_{n})$
$\displaystyle=K_{\textsl{g}}((1-r^{2})^{-1}\overline{X},\sqrt{-1}(1-r^{2})^{-1/2}e_{n})-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}\sqrt{-1}(1-r^{2})^{1/2}e_{n}|_{\textsl{g}}^{2}$
$\displaystyle=-(1-r^{2})^{-1}K_{\textsl{g}}(X,e_{n})-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}\sqrt{-1}(1-r^{2})^{1/2}e_{n}|_{\textsl{g}}^{2}$
$\displaystyle=-(1-r^{2})^{-1}|A^{*}_{X}e_{n}|_{\textsl{g}}^{2}-3r^{2}(1-r^{2})^{-1}|\nabla_{(1-r^{2})^{-1}\overline{X}}\sqrt{-1}(1-r^{2})^{1/2}e_{n}|_{\textsl{g}}^{2}.$

The equation on the last line is obtained again due to the vanishing of mixed
curvature terms and the fact that the fiber is one-dimensional. This also
implies that

$\nabla_{(1-r^{2})^{-1}\overline{X}}\sqrt{-1}(1-r^{2})^{1/2}e_{n}=0.$

Just observe that the possible non-vanishing term is
$\sqrt{-1}(1-r^{2})^{-1/2}\nabla_{X}e_{n}$ which coincides with (a multiple
of) the shape operator of the fibers, that is $0$ due to the totally geodesic
assumption. Hence,

(36)
$K_{\textsl{g}_{-r^{2}}}(\overline{X},\sqrt{-1}(1-r^{2})^{1/2}e_{n})=-(1-r^{2})^{-1}|A^{*}_{X}e_{n}|_{\textsl{g}}^{2}.$

The same argument to deal with $\nabla_{(1-r^{2})^{-1}\overline{X}}e_{i}$ can
be employed ensuring that

$\nabla_{X+(1-r^{2})^{-1}U^{*}}e_{i}=\nabla_{X}e_{i}=A_{X}e_{i}.$

Therefore,

$\mathrm{Ric}_{\textsl{g}_{-r^{2}}}(\overline{X})=\mathrm{Ric}^{\mathbf{h}}(X)+(1-r^{2})^{-2}\sum_{i=1}^{n-1}|A^{*}_{e_{i}}U^{*}|^{2}_{\textsl{g}}-3r^{2}(1-r^{2})^{-1}\sum_{i=1}^{n-1}|A_{X}e_{i}|_{\textsl{g}}^{2}\\\
-(1-r^{2})^{-1}|A^{*}_{X}e_{n}|_{\textsl{g}}^{2}.$

Since
$\mathrm{Ric}^{\mathbf{h}}(X)=\mathrm{Ric}_{M/G}-3\sum_{i=1}^{n-1}|A_{X}e_{i}|_{\textsl{g}}^{2}$
we have

$\mathrm{Ric}_{\textsl{g}_{-r^{2}}}(\overline{X})=\mathrm{Ric}_{M/G}(X)+(1-r^{2})^{-2}\sum_{i=1}^{n-1}|A^{*}_{e_{i}}U^{*}|^{2}_{\textsl{g}}-3(1-r^{2})^{-1}\sum_{i=1}^{n-1}|A_{X}e_{i}|_{\textsl{g}}^{2}\\\
-(1-r^{2})^{-1}|A^{*}_{X}e_{n}|_{\textsl{g}}^{2}\geq 0.$

We prove that $\textsl{g}-r^{-2}Q$ has positive Ricci curvature. With this
aim, we observe that the vertical space associated with the $\star$-acion
defined by equation (3) on $M\times\mathrm{S}^{1}$ has as vertical space at
$(x,q)\in M\times\mathrm{S}^{1}$ the space

$\mathcal{V}_{(x,q)}=\left\\{(U^{*},U):U\in\mathbf{i}\mathbb{R}\right\\}.$

Moreover, its $\textsl{g}-r^{-2}Q$-orthogonal complement
$\mathcal{H}^{\star}_{(x,q)}$ can be described as

$\mathcal{H}^{\star}_{(x,q)}=\left\\{(X+r^{-2}W^{*},W):X\in\mathcal{H}_{x},~{}W\in\mathbf{i}\mathbb{R}\right\\}.$

Observe that if one takes $\\{e_{1},\ldots,e_{n-1}\\}$ to be a g-orthonormal
set spanning $\mathcal{H}_{x}$ and a unit vector
$e_{n}\in\mathbf{i}\mathbb{R}$ (for
$\textsl{g}|_{\mathfrak{\mathrm{missing}}{S}^{1}}=Q$) then the following
constitute a desired basis for computing the Ricci curvature of
$\textsl{g}-r^{-1}Q$:

(37)
$\left\\{\left(e_{i}+r^{-2}e_{n}^{*},e_{n}\right):i\in\\{1,\ldots,n-1\\}\right\\}\cup\left\\{\left((1-r^{-2})^{-\frac{1}{2}}e_{n}^{*},(1-r^{-2})^{-\frac{1}{2}}e_{n})\right)\right\\}.$

Using that any tangent vector to $T_{(x,q)}M$ can be written as
$\widetilde{X}:=\left(X+r^{-2}W^{*}+U^{*},W+U\right)$ we compute

$\displaystyle
K_{\textsl{g}-r^{-2}}\left(\widetilde{X},\left(e_{i}+r^{-2}e_{n}^{*},e_{n}\right)\right)$
$\displaystyle=K_{\textsl{g}}\left(X+r^{-2}W^{*}+U^{*},e_{i}+r^{-2}e_{n}^{*}\right),$
$\displaystyle i\in\\{1,\ldots,n-1\\}$ $\displaystyle
K_{\textsl{g}-r^{-2}}\left(\widetilde{X},((1-r^{-2})^{-\frac{1}{2}}e_{n}^{*},(1-r^{-2})^{-\frac{1}{2}}e_{n}))\right)$
$\displaystyle=K_{\textsl{g}}\left(X+r^{-2}W^{*}+U^{*},(1-r^{-2})^{-\frac{1}{2}}e_{n}^{*}\right)$
$\displaystyle
K_{\textsl{g}}\left(X+r^{-2}W^{*}+U^{*},e_{i}+r^{-2}e_{n}^{*}\right)$
$\displaystyle=K_{\textsl{g}}(X,e_{i})+r^{-4}K_{\textsl{g}}(X,e^{*}_{n})+r^{-2}K_{\textsl{g}}(W^{*},e_{i})$
$\displaystyle+r^{-2}K_{\textsl{g}}(U^{*},e_{i})+2R_{\textsl{g}}(X,e_{i},e^{*}_{n},X)+4r^{-2}R_{\textsl{g}}(X,e_{i},e^{*}_{n},W^{*})$
$\displaystyle+4R_{\textsl{g}}(X,e_{i},e^{*}_{n},U^{*})+2r^{-2}R_{\textsl{g}}(X,e_{i},e_{i},W^{*})+2R_{\textsl{g}}(X,e_{i},e_{i},U^{*}).$
$\displaystyle
K_{\textsl{g}}\left(X+r^{-2}W^{*}+U^{*},(1-r^{-2})^{-\frac{1}{2}}e_{n}^{*}\right)$
$\displaystyle=(1-r^{-2})^{-1}K_{\textsl{g}}(X+r^{-2}W^{*}+U^{*},e_{n}^{*})$
$\displaystyle=(1-r^{-2})^{-1}K_{\textsl{g}}(X,e_{n}^{*})$
$\displaystyle+2(1-r^{-2})^{-1}R_{\textsl{g}}(X,r^{-2}W^{*}+U^{*},r^{-2}W^{*}+U^{*},e_{n}^{*}).$

However,

$R_{\textsl{g}}(X,r^{-2}W^{*}+U^{*},r^{-2}W^{*}+U^{*},e_{n}^{*})=0$
$R_{\textsl{g}}(X,e_{i},e_{i},U^{*})=R_{\textsl{g}}(X,e_{i},e_{i},W^{*})=R_{\textsl{g}}(X,e_{i},e_{n}^{*},X)=0$

Summing up

$\mathrm{Ric}(\textsl{g}-r^{-2})(\widetilde{X})=\left\\{\dfrac{(r^{2}-1)(n-1)+r^{6}}{r^{4}(r^{2}-1)}\right\\}K_{\textsl{g}}(X,e_{n}^{*})+\mathrm{Ric}^{\mathbf{h}}(X)\\\
+r^{-2}\left\\{\sum_{i=1}^{n-1}K_{\textsl{g}}(W^{*},e_{i})+K_{\textsl{g}}(U^{*},e_{i})\right\\}+4r^{-2}\sum_{i=1}^{n-1}\left\\{R_{\textsl{g}}(X,e_{i},e^{*}_{n},W^{*})+r^{2}R_{\textsl{g}}(X,e_{i},e_{n}^{*},U^{*})\right\\}.$

Since
$K_{\textsl{g}}(X,e_{n}^{*})=|A^{*}_{X}e_{n}^{*}|_{\textsl{g}}^{2},~{}K_{\textsl{g}}(W^{*},e_{i})=|A^{*}_{e_{i}}W^{*}|_{\textsl{g}}^{2},~{}K_{\textsl{g}}(U^{*},e_{i})=|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}$
and

$R_{\textsl{g}}(X,e_{i},e_{n}^{*},W^{*})+r^{2}R_{\textsl{g}}(X,e_{i},e_{n}^{*},U^{*})=\textsl{g}((\nabla^{\mathbf{v}}_{e^{*}_{n}}A)_{X}e_{i}+A_{e_{i}}(A^{*}_{X}e_{n}),W+r^{2}U^{*})$

we conclude

(38)
$\mathrm{Ric}(\textsl{g}-r^{-2})(\widetilde{X})=\left\\{\dfrac{(r^{2}-1)(n-1)+r^{6}}{r^{4}(r^{2}-1)}\right\\}|A_{X}e_{n}^{*}|_{\textsl{g}}^{2}+\mathrm{Ric}^{\mathbf{h}}(X)\\\
+r^{-2}\left\\{\sum_{i=1}^{n-1}|A^{*}_{e_{i}}W^{*}|_{\textsl{g}}^{2}+|A^{*}_{e_{i}}U^{*}|_{\textsl{g}}^{2}\right\\}+4r^{-2}\sum_{i=1}^{n-1}\textsl{g}((\nabla^{\mathbf{v}}_{e^{*}_{n}}A)_{X}e_{i}+A_{e_{i}}(A^{*}_{X}e_{n}),W+r^{2}U^{*}).$

Finally, we claim that

$\textsl{g}((\nabla^{\mathbf{v}}_{e^{*}_{n}}A)_{X}e_{i}+A_{e_{i}}(A^{*}_{X}e_{n}),W+r^{2}U^{*})=\textsl{g}\left(A^{*}_{e_{i}}e^{*}_{n},A^{*}_{X}(W^{*}+r^{-2}U^{*})\right)$

what shall conclude the proof since

$\textsl{g}\left(A^{*}_{e_{i}}e^{*}_{n},A^{*}_{X}(W^{*}+r^{-2}U^{*})\right)=\textsl{g}\left(e_{n}^{*},A_{e_{i}}A^{*}_{X}(W^{*}+r^{-2}U^{*}\right)=|A_{e_{i}}A^{*}_{X}(W^{*}+r^{-2}U^{*})|_{\textsl{g}}\geq
0.$

To verify the claim, it suffices to observe that (for the computation, we
extend either $X$ or $V$ to geodesics and use equations (34), (35)).

$\displaystyle(\nabla^{\mathbf{v}}_{e^{*}_{n}}A)_{X}e_{i}$
$\displaystyle=\nabla_{e^{*}_{n}}(A_{X}e_{i})-A_{\nabla^{\mathbf{h}}_{e^{*}_{n}}X}e_{i}-A_{X}(\nabla^{\mathbf{h}}_{e^{*}_{n}}e_{i})$
$\displaystyle=\nabla_{e^{*}_{n}}(A_{X}e_{i})+A_{A^{*}_{X}e^{*}_{n}}e_{i}+A_{X}(A^{*}_{e_{i}}e^{*}_{n})$
$\displaystyle=\nabla_{e^{*}_{n}}(A_{X}e_{i})-A_{e_{i}}(A^{*}_{X}e^{*}_{n})+A_{X}(A^{*}_{e_{i}}e^{*}_{n})$
$\displaystyle=-A_{e_{i}}(A^{*}_{X}e^{*}_{n})+A_{X}(A^{*}_{e_{i}}e^{*}_{n}).$

So

$\displaystyle\textsl{g}((\nabla^{\mathbf{v}}_{e^{*}_{n}}A)_{X}e_{i}+A_{e_{i}}(A^{*}_{X}e_{n}),W+r^{2}U^{*})$
$\displaystyle=\textsl{g}(A_{X}(A^{*}_{e_{i}}e^{*}_{n}),r^{-2}W^{*}+U^{*}).$

The almost semi-free case is a simple application of Corollary C item (a) in
[CeSS23] since the orbits in the complement of the principal stratum are
isolated. ∎

## Acknowledgements

The authors gladly acknowledge the anonymous referees for the careful read,
leading to useful suggestions that improved the quality of the paper. The São
Paulo Research Foundation (FAPESP) supports L. F. C. grants 2022/09603-9,
2023/14316-1 and partially supports L. G. grants 2023/13131-8, 2021/04065-6.

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See pages 1-last of paper.pdf