# Spacelike hypersurfaces in twisted product spacetimes with complete fiber
and Calabi-Bernstein-type problems

Alberto Soria
Higher Technical School of Telecommunications Engineering,
Universidad Politécnica de Madrid
Av. Complutense 30, 28040 Madrid, Spain
<EMAIL_ADDRESS>
(August 27, 2024)

###### Abstract

In this article spacelike hypersurfaces immersed in twisted product spacetimes
$I\times_{f}F$ with complete fiber are studied. Several conditions ensuring
global hyperbolicity are presented, as well as a relation that needs to hold
on each spacelike hypersurface in $I\times_{f}F$ for it to be a simple warped
product. When the fiber is assumed to be closed (compact and without boundary)
and the ambient spacetime has a suitable expanding behaviour, non-existence
results for constant mean curvature hypersurfaces are obtained. Under the same
hypothesis, a characterization of compact maximal hypersurfaces and other for
totally umbilic ones with a suitable restriction on their mean curvature are
presented. The description of maximal hypersurfaces in twisted product
spacetimes of the form $I\,{}_{f}\\!\\!\times F$ with a one-dimensional
Lorentzian fiber is also included. Finally, the mean curvature equation for a
spacelike graph on the fiber is computed and as an application, some Calabi-
Bernstein-type results are proven. We also include in an Appendix some known
conformal geometry results describing the transformation of relevant tensors
and operators under the action of a conformal map in a pseudo-Riemannian
background.

##  1 Introduction

Twisted and warped products are particular cases of pseudo-Riemannian
manifolds which have interesting geometric properties and often have an
important role in General Relativity (GR). Indeed, warped products constitute
a significant group of the exact solutions of the Einstein field equations.
Among these is the Schwarzschild solution, which describes the outer geometry
of the spacetime around massive bodies such planets, stars, black holes, etc.
and the Robertson-Walker solutions, determining the geometry of a simply-
connected expanding or contracting universe satisfying the cosmological
principle of homogeneity and isotropy. In the area of geometry, twisted
products have played an important role in the study of complex space forms
[34], in Lagrangian submanifolds [23] and in curvature netted hypersurfaces
[32] among others. Let us first give the definition of pseudo-Riemannian
twisted product manifold:

######  Definition 1.

Let $(B,g_{B})$ and $(F,g_{F})$ be pseudo-Riemannian manifolds. Let $f:B\times
F\rightarrow(0,+\infty)$ be a positive smooth function. Consider the canonical
projections $\pi_{B}:B\times F\rightarrow B$ and $\pi_{F}:B\times F\rightarrow
F$. Then the (simple) twisted product $B\times_{f}F$ of $(B,g_{B})$ and
$(F,g_{F})$ is the differentiable manifold $B\times F$ endowed with the
pseudo-Riemannian metric $\overline{g}$ defined as

$\overline{g}(X,Y)=g_{B}(d\pi_{B}(X),d\pi_{B}(Y))+f^{2}g_{F}(d\pi_{F}(X),d\pi_{F}(Y))$

for all vector fields $X,Y\in\mathfrak{X}(B\times F)$. In this setting,
$(B,g_{B})$ is the base manifold and $(F,g_{F})$ the fiber manifold.

Twisted products were first introduced in [12] by Bishop as umbilic products,
being so named because the leaves of the canonical foliation made up with
copies of the fiber are totally umbilical in the ambient space. Later on, Chen
[18, 19] would refer to them as twisted products, and were generalized to
double twisted products of two pseudo-Riemannian manifolds by Ponge and
Reckziegel in [41]. In case that $f$ is exclusively defined on the base
$(B,g_{B})$, twisted products reduce to the so-called warped products. Warped
spacetimes with a Lorentzian open interval $(I,-dt^{2})$ as base manifold and
with isotropic and homogeneous fiber are called Robertson-Walker. A natural
extension for such spacetimes was proposed for the first time in [7], and
constitutes the family of Generalized Robertson-Walker (GRW) spacetimes, where
the fiber is not required to have constant sectional curvature. For a survey
on GRW spacetimes, we refer the reader to [35]. In this work we focus our
interest on twisted products spacetimes with $(I,-dt^{2})$ as base manifold
and a complete Riemannian fiber $(F,g_{F})$ of arbitrary dimension. However,
some results will be obtained in twisted product spacetimes where the
Lorentzian interval plays the role of the fiber.

It is possible to find in the literature several characterizations of twisted
product pseudo-Riemannian manifolds. As Ponge and Reckziegel show in [41], a
pseudo-Riemannian product manifold $(M_{1}\times M_{2},g)$ where the leaves of
the associated canonical foliations $L_{1}=\\{M_{1}\times\\{q\\},\,q\in
M_{2}\\}$ and $L_{2}=\\{\\{p\\}\times M_{2},\,p\in M_{1}\\}$ intersect
perpendiculary everywhere has the structure of a twisted product of the form
$M_{1}\times_{f}M_{2}$ if and only if the leaves of $L_{1}$ are totally
geodesic and the leaves of $L_{2}$ totally umbilic in $M_{1}\times_{f}M_{2}$.
A local characterization for Riemannian and Lorentzian manifolds to be twisted
products of the form $I\times_{f}F$ was also put forward by Chen in [22],
where he proved that the existence of a so-called torqued vector field was
equivalent to such local structure. This result was later on exploited by
Mantica and Molinari in [36], where any twisted spacetime is proven to be
purely electric according to the Bel-Debever criterion and a condition for it
to be a GRW spacetime is put forward. It is possible to find in the literature
some other characterizations for twisted product manifolds to be a warped,
like the one shown in [30], where the authors find a necessary and sufficient
condition on the ambient Ricci tensor to be warped, and which they refer to as
mixed Ricci-flatness. In Proposition 3 of Section 3 we also find a condition
that needs to hold on immersed spacelike surfaces in a twisted product
spacetime of the form $I\times_{f}F$ to be a truly GRW spacetime.

Spacelike hypersurfaces constitute a remarkable class of manifolds in the area
of Riemannian and Lorentzian geometry. Since Lichnerowicz’s work [33] the
study of maximal and constant mean curvature (CMC) hypersurfaces has been a
relevant problem in the context of differential geometry, partial differential
equations, and has also played a fundamental role with regard to the dynamic
aspects of GR. In addition, when a spacetime admits a foliation by CMC
hypersurfaces there is a preferred choice for the time coordinate (read[42]
for more details). It is well known that the whole nature of a spacetime can
be determined by the geometry of spacelike hypersurfaces, as Choquet-Bruhat
and Geroch’s result [25] about the Cauchy problem for the Einstein equations
reveals. In particular, CMC and maximal hypersurfaces have been of great
utility as initial data in the problem since the associated PDE system is
reduced to a linear first order differential one and a nonlinear second order
elliptic differential equation [33]. Globally hyperbolic spacetimes do not
always admit CMC Cauchy surfaces and the problem of determining them in a
generic sense is still open. Several existence results for compact Cauchy
surfaces can be found in the literature, like the fundamental one put forward
by Bartnik [9], which in turn has motivated others in the same line, like the
one put forward by Galloway and Ling in [31], where the existence of the CMC
Cauchy surface is consequence of an ambient curvature hypothesis related to
the strong energy condition. Naturally, the study of CMC and maximal
hypersurfaces in twisted product ambients is also an issue of remarkable
interest. The validity of many results in twisted product ambients depends on
the monotonic properties of the associated twisted function. In this work
several non-existence results of CMC hypersurfaces are obtained in
$I\times_{f}F$ when the appropriate expanding conditions take place and
$(F,g_{F})$ is assumed to be a closed (compact and without boundary)
Riemannian manifold. We also present a characterization of maximal spacelike
hypersurfaces in twisted products of the form $I\times_{f}F$ and in those of
type $I\,{}_{f}\\!\\!\times F$ (i.e. with Lorentzian interval as fiber), and
another one for totally umbilical hypersurfaces with a suitable restriction on
their mean curvature in $I\times_{f}F$.

Maximal surfaces occupy a prominent place among CMC ones. As Brill and
Flaherty remark in [14], maximal hypersurfaces can be used to describe the
transition between expansive and contractive phases in some significant models
of the universe. An important result involving maximal hypersurfaces is the
so-called Calabi-Bernstein theorem in the Minkowski spacetime, which
establishes that the only entire hypersurfaces satisfying the maximal
hypersurface differential equation are spacelike affine hyperplanes (see [46]
for more details). This result was first obtained by Calabi [16] in the
$(n+1)$-dimensional Minkowski spacetime $\mathcal{M}^{1,n}$ for $n\leq 3$. The
proof was based on the original Bernstein result for minimal surfaces in the
three-dimensional Euclidean space. In [24] Cheng and Yau proved the theorem
for arbitrary dimension, which shows a different nature for the problem with
respect to the Euclidean context, where the analogous result just holds up to
ambient dimension $8$ (see [13]). Since then many other related Calabi-
Bernstein type results have been obtained in different ambient spaces. We
refer the reader to [1, 2, 4, 5, 6, 10, 15, 17, 38, 39, 40, 43] for more
information. In the last section we obtain results in $I\times_{f}F$ in the
same line than the Calabi-Bernstein main one, where $(F,g_{F})$ is assumed to
be closed.

The paper is organized as follows: in Section 2 we first introduce our
terminology and present general properties of twisted product spacetimes of
the form $I\times_{f}F$. Section 3 begins with the description of the causal
character of spacelike hypersurfaces in $I\times_{f}F$ in terms of the fiber
geometry, the twisted function and the so-called time-height function $\tau$
(see for instance [37]), which defines the time-distance at which each point
on the surface is located. In addition, a relation that needs to hold on each
spacelike hypersurface immersed in $I\times_{f}F$ for it to be a GRW spacetime
is presented. We also address some global hyperbolicity questions in
$I\times_{f}F$. Specifically, we prove that $I\times_{f}F$ turns out to be
globally hyperbolic whenever it admits a complete spacelike hypersurface where
the restriction of $f$ is bounded. Section 4 is devoted to present the main
results of the work. Several non-existence results of CMC spacelike
hypersurfaces in expanding or contracting twisted product spacetimes are put
forward. Exploiting well-known properties of conformal maps, totally umbilical
hypersurfaces with a suitable restriction on their mean curvature are
characterized in $I\times_{f}F$, as well as all compact maximal hypersurfaces.
Likewise, the analogous result for maximal hypersurfaces in twisted products
of the form $I\,{}_{\alpha}\\!\\!\times M$ is also presented, where the
Lorentzian interval $(I,-dt^{2})$ plays the role of the fiber manifold. In
addition to assuming the compactness of $(F,g_{F})$, we will also require
$I\times_{f}F$ to be either a expanding or contracting background, or to admit
a so-called transition hypersurface which delimits two different regions, an
expanding one to its past and a contracting one to its future. Finally, in
Section 5 the expression for the mean curvature of a spacelike graph over the
fiber in $I\times_{f}F$ is derived, and some existence and uniqueness results
for the associated maximal Calabi-Bernstein PDE equation are presented. At the
end of the work we add an Appendix with elementary conformal geometry results
which illustrate how the Levi-Civita connection, the shape operator, the mean
curvature and the Laplacian on any non-degenerate submanifold transform under
the action of a global conformal map in a pseudo-Riemannian background.

##  2 Preliminaries

Consider $(F,g_{F})$ and $(B,g_{B})$ two pseudo-Riemannian manifolds of
arbitrary dimension. Let $f:B\times F\rightarrow\mathbb{R}$ be a positive
smooth function. The manifold $B\times F$ endowed with the metric

$\overline{g}=\pi^{*}_{B}(g_{B})+f^{2}\pi^{*}_{F}(g_{F}),$

where $\pi_{B}$ and $\pi_{F}$ are the canonical projections onto the _base_
$(B,g_{B})$ and the _fiber_ $(F,g_{F})$ respectively, is known as twisted
product manifold and it is denoted as $B\times_{f}F$ (see [41], for instance).
As it is usual, vector fields $X\in\mathfrak{X}(B)$ and $V\in\mathfrak{X}(F)$
can be lifted to vector fields $\overline{X},\overline{V}$ in
$\mathfrak{X}(B\times F)$. We will denote then as $L(B)$ and $L(F)$ the sets
of lifted vector fields from $\mathfrak{X}(B)$ and $\mathfrak{X}(F)$
respectively. For simplicity, we will generally not distinguish between
$f\circ\pi_{B}$ and $f$ or between $X$ and $\overline{X}$, as it will be clear
from context. There exist useful formulae for evaluating the Levi-Civita
connection on these fields (see [41]). Indeed, given $X,Y\in L(B)$ and $V,W\in
L(F)$ and denoting by $\overline{\nabla}$, $\nabla^{B}$ and $\nabla^{F}$ the
Levi-Civita connections on $B\times_{f}F$, $(B,g_{B})$, and $(F,g_{F})$
respectively, it is not difficult to see that

$\displaystyle\overline{\nabla}_{X}Y$ $\displaystyle=$
$\displaystyle\nabla^{B}_{X}Y$ (1) $\displaystyle\overline{\nabla}_{X}V$
$\displaystyle=$ $\displaystyle\overline{\nabla}_{V}X=X(\log f)V$ (2)
$\displaystyle\overline{\nabla}_{V}W$ $\displaystyle=$
$\displaystyle\nabla^{F}_{V}W+V(\log f)W+W(\log f)V-\overline{\nabla}(\log
f)\,\overline{g}(V,W).$ (3)

In this work we will consider a twisted product Lorentzian $(n+1)$-manifold
constructed with a $n$-Riemannian manifold $(F,g_{F})$ and an interval
$I\subset\mathbb{R}$ endowed with the negative metric, i.e., $(I,-dt^{2})$,
where $(F,g_{F})$ and $(I,-dt^{2})$ can be fiber or base alternatively
depending on the case. In case that the fiber is $(F,g_{F})$, the twisted
product spacetime will be denoted by $I\times_{f}F$, and its metric will be of
the form $\overline{g}=-dt^{2}+f^{2}\pi^{*}_{F}(g_{F})$. Likewise, when the
fiber is $(I,-dt^{2})$ we will refer to it as $I\,{}_{f}\\!\\!\times F$ and
the metric is of type $\overline{g}=-f^{2}dt^{2}+\pi^{*}_{F}(g_{F})$. For
convenience and for the sake of clarity, we will also use the convention
$\overline{g}=-dt^{2}+f^{2}g_{F}$ and $\overline{g}=-f^{2}dt^{2}+g_{F}$. Note
that when function $f$ is just defined on $(I,-dt^{2})$, we restrict in the
first case to the so-called Generalized Robertson-Walker (GRW) spacetimes (see
[7] for more details) and when $f$ is defined on $(F,g_{F})$ to the standard
static spacetimes (see [40]).

The study of twisted products is relevant because they constitute a much wider
family of manifolds with respect to the warped product one. Even though
twisted products satisfying suitable curvature conditions are actually warped
products as mentioned before, there are many of them which do not meet this
property. Given a pseudo-Riemannian manifold $(M,g)$, a vector field
$X\in\mathfrak{X}(M)$ is said to be a conformal Killing if there exists a
smooth function $\mu:M\rightarrow\mathbb{R}$ so that $\mathcal{L}_{X}g=\mu g$.
The following result illustrates the existence of twisted products with one-
dimensional base which are not isometric to GRW spacetimes. To show it, we
will exploit the existence of conformal Killing vector fields in warped
products of the form $I\times_{f}F$.

######  Proposition 1.

There exist twisted product spacetimes which are not isometric to warped
product ones.

###### Proof.

Let us consider $f:\mathbb{R}\rightarrow\mathbb{R}$ a non-constant periodic
function of period $T$, differentiable and strictly positive. If $(t,x)$ stand
for the usual coordinates in $\mathbb{R}^{2}$, $(\mathbb{R}^{2},g_{f})$ with
$g_{f}=-dt^{2}+f^{2}(t+x)dx^{2}$ is a twisted product Lorentzian manifold. If
we consider $\mathbb{S}^{1}=\mathbb{R}/T\mathbb{Z}$ where $T$ is the period of
$f$, then $\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$ is naturally
endowed with the Lorentzian twisted metric
$\tilde{g}_{f}=-dt^{2}+\tilde{f}dx^{2}$, where $\tilde{f}([t],[x]):=f(t+x)$.
We know that for $f=1$, $g_{f}$ is incomplete [47] and hence
$(\mathbb{T}^{2},\tilde{g}_{f})$ is incomplete too. In this case, there is no
timelike conformal vector field on $(\mathbb{T}^{2},\tilde{g}_{f})$. In fact,
if such vector field did exist, $(\mathbb{T}^{2},\tilde{g}_{f})$ would be
geodesically complete since every compact Lorentzian manifold aditting a
conformal Killing vector is geodesically complete [44]. In particular,
$(\mathbb{T}^{2},\tilde{g}_{f})$ can not be isometric to a warped product. The
torus $(\mathbb{T}^{2},\tilde{g}_{f})$ can also be used to construct new
examples of twisted products which do not admit a warped product structure.
Consider a compact Riemannian manifold $(F,g_{F})$ and let us construct the
warped product $\mathbb{T}^{2}\times_{\tilde{f}}F$. Since
$(\mathbb{T}^{2},\tilde{g}_{f})$ is the base space of this warped product, it
is totally geodesic in $\mathbb{T}^{2}\times_{\tilde{f}}F$ when it is seen as
$\mathbb{T}^{2}\times\\{p_{0}\\}$ with $p_{0}\in F$. Therefore, the fact that
$(\mathbb{T}^{2},\tilde{g}_{f})$ is incomplete tells us that
$\mathbb{T}^{2}\times_{\tilde{f}}F$ is incomplete too. Note that
$\mathbb{T}^{2}\times_{\tilde{f}}F$ is identical to the twisted product
$\mathbb{S}^{1}\times_{\tilde{f}}(\mathbb{S}^{1}\times F)$ whose metric is of
the type $-dt^{2}+f^{2}(t+x)(dx^{2}+g_{\hat{F}})$ on
$\mathbb{R}\times(\mathbb{R}\times\hat{F})$, with $\hat{F}$ the universal
cover of $F$. Using again the main result of [44], there is no timelike
conformal vector field on
$\mathbb{S}^{1}\times_{\tilde{f}}(\mathbb{S}^{1}\times F)$; in particular
$\mathbb{S}^{1}\times_{\tilde{f}}(\mathbb{S}^{1}\times F)$ can not be
isometric to a GRW spacetime. ∎

As we have just seen, the analysis of spacelike hypersurfaces in twisted
product spacetimes can not be reduced to the existing one in GRW. Let us first
show general properties of interest of twisted product spacetimes of the form
$I\times_{f}F$. The vector field $\partial_{t}:=\frac{\partial}{\partial t}$
is defined globally on $I\times_{f}F$. Function $t$ is a smooth global time
function and as consequence $I\times_{f}F$ is stably causal (see [8], pg. 64)
Furthermore, it is globally hyperbolic if and only if its fiber is complete
(Thm. 3.66 in [8] also applies to twisted product spacetimes). Besides, it is
clearly time-orientable and consequently it constitutes a relativistic
spacetime. In [20] Chen characterizes GRW spacetimes by the presence of the
so-called concircular vectors, which were intruduced by Fialkow for the first
time in [29]. Specifically, a Lorentzian manifold $(M,g)$ of dimension greater
or equal than three is a GRW spacetime if and only if there exists a timelike
concircular vector $K$, i.e. a timelike vector satisfying

$\overline{\nabla}_{X}K=\mu X$

for all $X\in\mathfrak{X}(M)$, and for some smooth function
$\mu:M\rightarrow\mathbb{R}$. If we denote by $\overline{\nabla}$ the Levi-
Civita connection of $I\times_{f}F$ and make use of the covariant derivative
formulas (1), (2) and (3), we can see that for any vector field
$X\in\mathfrak{X}(I\times_{f}F)$ and for $K:=f\partial_{t}$ we have

$\overline{\nabla}_{X}K=df(X^{F})\partial_{t}+(\partial_{t}f)X=(\partial_{t}f)X+\mbox{\boldmath$\omega$}(X)K,$
(4)

where $X^{F}$ is the fiber-parallel component of $X$, and $\omega$ is the one-
form defined as

$\mbox{\boldmath$\omega$}(X)=\left(d(\log f)\circ\pi_{F}\right)(X),$ (5)

where $\pi_{{}_{F}}:I\times F\rightarrow F$ is the canonical projection onto
$(F,g_{F})$. In addition $\mbox{\boldmath$\omega$}(K)=0$, so $K$ is a torqued
vector field in the sense of [22], defined globally on $I\times_{f}F$, and
$\boldsymbol{\omega}$ is defined to be the torqued one-form. From (4) it is
easy to prove

$\mathcal{L}_{K}\overline{g}=2(\partial_{t}f)\overline{g}+\mbox{\boldmath$\omega$}\otimes\mbox{\boldmath$K$}+\mbox{\boldmath$K$}\otimes\mbox{\boldmath$\omega$},$

where $K$ is the $1$-form metrically equivalent to $K$. In particular, when
$U,V\in\mathfrak{X}(F)$,

$(\mathcal{L}_{K}\overline{g})(U,V)=f(\mathcal{L}_{\partial_{t}}\overline{g})(U,V)=2(\partial_{t}f)\overline{g}(U,V).$
(6)

Thus, $\partial_{t}$ is a spatially conformally stationary reference frame on
$I\times_{f}F$ in the sense of [45]. The case where function $\partial_{t}f$
is constant signed deserves special interest, since the volume of any immersed
spacelike submanifold which is Lie-dragged along the vector $\partial_{t}$ is
monotonically altered. Indeed, the observers in $\partial_{t}$ detect
expansion if $\partial_{t}f\geq 0$ or compression when $\partial_{t}f\leq 0$
along their proper times.

##  3 Spacelike hypersurfaces in $I\times_{f}F$

Given an $n$-dimensional connected manifold $\Sigma$, an immersion
$\psi:\Sigma\longrightarrow I\times_{f}F$ is said to be spacelike if the
ambient Lorentzian metric $\overline{g}$ induces, via $\psi$, a Riemannian
metric $g_{{}_{\Sigma}}$ on $\Sigma$. In such case $(\Sigma,g_{\Sigma})$ is
called a spacelike hypersurface. In case that $(\Sigma,g_{{}_{\Sigma}})$ is
orientable, there is a unique choice of a unit timelike vector field $N$
normal to $\Sigma$ with the same time-orientation than the vector field
$\partial_{t}$ (note that in this case $\overline{g}(\partial_{t},N)<0$). The
second fundamental form vector of $\Sigma$ is defined as
$\vec{\mathrm{II}}(X,Y)=(\overline{\nabla}_{X}Y)^{\perp}$, where
$X,Y\in\mathfrak{X}(\Sigma)$ and $Z^{\top}$ and $Z^{\perp}$ denote the tangent
and orthogonal component of any vector field $Z$ along $\Sigma$ respectively.
If we consider the Weingarten operator $A(X)=-\overline{\nabla}_{X}N$, we can
express the second fundamental form vector as

$\vec{\mathrm{II}}(X,Y)=-\overline{g}(A(X),Y)N,$

and consequently its covariant form reads
$\mathrm{II}(X,Y)=\overline{g}(\vec{\mathrm{II}}(X,Y),N)=\overline{g}(A(X),Y)$.
The mean curvature vector is defined by
$\vec{H}=\frac{1}{n}\mathrm{trace}\,(\vec{\mathrm{II}})$ and the mean
curvature function $H=-\frac{1}{n}\mathrm{trace}\,(A)$. The Gauss equation
relates the ambient connection $\overline{\nabla}$ with the induced connection
$\nabla$ on $\Sigma$ and in this setting becomes

$\overline{\nabla}_{X}Y=\nabla_{X}Y-\overline{g}(A(X),Y)N.$

On the other hand, we define on $\Sigma$ the hyperbolic angle
$\theta:\Sigma\rightarrow\mathbb{R}$ as the function given by

$\cosh\theta=-\overline{g}\left(N,\partial_{t}\right).$ (7)

The so-called “time-height” function
$\tau=\pi_{I}\circ\psi:\Sigma\rightarrow\mathbb{R}$ (see for instance [37])
measures the time-distance at which each point of $\Sigma$ lies, where
$\pi_{I}:I\times_{f}F\rightarrow I$ is the canonical projection onto the base
$(I,-dt^{2})$. Function $\tau$ is the restriction of $t$ to the hypersurface
$\Sigma$, so the notation $\tau=t|_{\Sigma}$ will be sometimes considered. The
expression for the induced metric $g_{\Sigma}$ on $\Sigma$ is given by

$g_{\Sigma}=-d\tau\otimes d\tau+f^{2}\pi_{F}^{*}(g_{F}).$ (8)

With the following result we describe the causal character of spacelike
hypersurfaces in twisted product spacetimes of the form $I\times_{f}F$ in
terms $\tau$, $f$ and the fiber geometry.

######  Proposition 2.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be an oriented spacelike
hypersurface and let $\tau=t|_{\Sigma}$ its associated time-height function.
Then the following relations on the hypersurface $\Sigma$ are equivalent:

  * (i)

$g_{\Sigma}$ is a Riemannian metric,

  * (ii)

$|d\tau(U)|\leq f|U^{F}|_{g_{F}}$ for all $U\in\mathfrak{X}(\Sigma)$,

  * (iii)

$|\nabla\tau|^{2}\leq f|\nabla\tau^{F}|_{g_{F}}$,

  * (iv)

$\overline{g}(\partial_{t},N)<0$,

where $|\,.\,|_{g_{F}}$ and $|\,.\,|$ are the norms associated to $g_{F}$ and
$g_{\Sigma}$ respectively, and $N$ is the future unit normal to $\Sigma$.

###### Proof.

Let us first prove that $(i)$ and $(ii)$ are equivalent. Metric $g_{\Sigma}$
is Riemannian if and only if
$g_{\Sigma}(U,U)=-d\tau(U)^{2}+f^{2}g_{F}(U^{F},U^{F})\geq 0$ for every
$U\in\mathfrak{X}(\Sigma)$, which is equivalent to $(ii)$. To prove that
$(ii)$ implies $(iii)$, it suffices to consider $U=\nabla\tau$. For the
equivalence of $(i)$ and $(iii)$, if $g_{\Sigma}$ is Riemannian, then
$g_{\Sigma}(\nabla\tau,\nabla\tau)\geq 0$, which implies $(iii)$.
Reciprocally, given any $U\in\mathfrak{X}(\Sigma)$ orthogonal to $\nabla\tau$,
$g_{\Sigma}(U,U)=f^{2}g_{F}(U^{F},U^{F})\geq 0$ since $d\tau(U)=0$, so $g_{F}$
is a non-degenerate positive definite metric, i.e. Riemannian. Finally we
prove the equivalence of $(i)$ and $(iv)$. The metric $g_{\Sigma}$ is
Riemannian when each tangent plane $T_{p}\Sigma$ is endowed with a positive
definite metric for all $p\in\Sigma$. This happens if and only if the normal
subspace $(T_{p}\Sigma)^{\bot}=\mathrm{span}\\{N\\}$ is timelike, i.e.
$\overline{g}(\partial_{t},N)<0$ for both vectors lie in the same future
timelike cone. ∎

Any vector field $Z$ along a spacelike hypersurface $\psi:\Sigma\rightarrow
I\times_{f}F$ decomposes into its tangent and normal components to $\Sigma$ as
$Z=Z^{\top}+Z^{\bot}=Z^{\top}-\overline{g}(Z,N)N$. On the other hand, its
decomposition into its horizontal and vertical components reads
$Z=Z^{0}\partial_{t}+Z^{F}$, where $Z^{0}=-\overline{g}(\partial_{t},Z)$. In
the particular case that $Z\in\mathfrak{X}(\Sigma)$, $Z^{0}=d\tau(Z)$. One can
easily check that $\nabla\tau=-(\partial_{t})^{\top}$. Squaring the
decompositions $\partial_{t}=-\nabla\tau-\overline{g}(\partial_{t},N)N$ and
$N=-\overline{g}(N,\partial_{t})\partial_{t}+N^{F}$ we get

$|\nabla\tau|^{2}=|N^{F}|^{2}=-1+\overline{g}(\partial_{t},N)^{2}=\sinh^{2}\theta,$
(9)

where relation (7) has been considered. When $\psi:\Sigma\rightarrow
I\times_{f}F$ is spacelike, (9) implies that
$\overline{g}(N,\partial_{t})\leq-1$ along $\Sigma$. In the twisted product
$I\times_{f}F$ the level hypersurfaces of the function $\pi_{I}$ constitute a
distinguished family of spacelike hypersurfaces: the so-called spacelike
slices. This family provides a foliation of the spacetime, whose leaves are
the restspaces of the reference frame given by the vector field
$\partial_{t}$, i.e. $\partial_{t}$ is the future unit normal vector field to
each slice. Every spacelike slice is totally umbilical [41]. In fact, the
slice $\tau=t_{0}$ has $A=-\partial_{t}(\log f)(t_{0},p)\textrm{Id}$ as shape
operator, where we denote by “Id” the identity endomorphism. As a consequence
its mean curvature function is $H=\partial_{t}(\log f)(t_{0},p)$. In contrast
to the warped product case, the mean curvature of the slices is not
necessarily constant along them. In case that $I\times_{f}F$ is globally
hyperbolic, every spacelike slice is a Cauchy hypersurface.

As already mentioned, formula (6) shows that when $\partial_{t}f\geq 0$ (resp.
$\partial_{t}f\leq 0$) the observers in $\partial_{t}$ (comoving observers)
measure a universe in expansion (resp. in contraction). It may also happen
that $I\times_{f}F$ admits a maximal slice $S_{t_{0}}\equiv\\{t=t_{0}\\}$
separating an expanding phase to its past from a contracting one to its future
(read [14] for more details). All these cases can be put together in the
following definition:

######  Definition 2.

Let $I\times_{f}F$ be twisted product spacetime. Then

  * (i)

if $\partial_{t}f>0$, we say that it is an expanding twisted product
spacetime.

  * (ii)

if $\partial_{t}f<0$, we say that it is a contracting twisted product
spacetime.

  * (iii)

if the spacetime is expanding for $t\leq t_{0}$ and it is contracting for
$t\geq t_{0}$, where $t_{0}\in\mathbb{R}$ is constant, then the maximal
hypersurface $S_{t_{0}}\equiv\\{t=t_{0}\\}$ is said be a transition spacelike
slice.

Twisted product manifolds of the form $I\times_{f}F$ generalize GRW
spacetimes. As Chen explains in [21], the existence of a torqued vector field
on a pseudo-Riemannian manifold implies a local twisted product structure.
Relation (4) shows that a torqued vector $K$ is concircular when the
associated torqued one-form $\mbox{\boldmath$\omega$}=d(\log f)\circ\pi_{F}$
vanishes identically. By formula (5), this happens if and only if $X^{F}(f)=0$
for any $X^{F}\in\mathfrak{X}(F)$, or equivalently

$\nabla^{F}f=0.$ (10)

When this occurs, $f$ becomes constant on each slice and the ambient spacetime
is a GRW spacetime. In the following result we obtain a condition that is
fulfilled on every spacelike hypersurface when $I\times_{f}F$ is actually a
warped product.

######  Proposition 3.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be an immersed spacelike
hypersurface in the twisted product spacetime $I\times_{f}F$. In case that
$I\times_{f}F$ is isometric to a warped product, the vector field

$\mathfrak{T}=-(\partial_{t}f)\nabla\tau+\nabla f$ (11)

vanishes on $\Sigma$, where $\tau$ is the associated time-height function of
$\Sigma$, and $\nabla$ its induced connection.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a spacelike hypersurface and
$U\in\mathfrak{X}(\Sigma)$. The expression for the restriction to $\Sigma$ of
the torqued one-form $\mbox{\boldmath$\omega$}=d(\log f)\circ\pi_{F}$
associated to the vector $K=f\partial_{t}$ reads

$\mbox{\boldmath$\omega$}(U)=\frac{df}{f}\left(U^{F}\right)=\frac{df}{f}\left(-d\tau(U)\partial_{t}+U\right)=\frac{1}{f}\overline{g}(-(\partial_{t}f)\nabla\tau+\nabla
f,U),$ (12)

where the decomposition $U=d\tau(U)\partial_{t}+U^{F}$ into its horizontal and
vertical components has been considered. In case that $I\times_{f}F$ is a
warped product the torqued one-form $\omega$ is zero, which occurs if and only
if $\mathfrak{T}=-(\partial_{t}f)\nabla\tau+\nabla f=0$ along $\Sigma$. ∎

######  Remark 1.

The vanishing of vector (11) on $\Sigma$ reduces to $\nabla f=0$ if $\Sigma$
is a slice, so relation (10) is recovered since $\nabla f=f^{-2}\nabla^{F}f$
in this case.

In general, the completeness of a twisted product of the form $B\times_{f}F$
can not be ensured even in case that the base manifold $(B,g_{B})$ and the
fiber $(F,g_{F})$ are complete pseudo-Riemannian manifolds. In this work we
are interested in the study of twisted products $I\times_{f}F$ with complete
Riemannian fiber $(F,g_{F})$. As it happens in the warped case, $I\times_{f}F$
is globally hyperbolic under this hypothesis [8]. We next prove that global
hyperbolicity is also derived from the fact that $I\times_{f}F$ admits a
complete spacelike hypersurface where the restriction of the twisted function
$f$ is bounded. Some of the forthcoming results are an adaptation from [7] to
the twisted product spacetime context. To prove them we first need the
following lemma, which establishes sufficient conditions for a map between
Riemannian manifolds to be a covering one, and generalizes Lemma 3.3, Chapter
7 in [28].

######  Lemma 1.

Let $(M,g_{M})$ be a complete Riemannian manifold and let
$h:(M,g_{M})\rightarrow(N,g_{N})$ be a local diffeomorphism onto a Riemannian
manifold $(N,g_{N})$ where for all $p\in M$ and for all $U\in T_{p}M$,

$|U|_{g_{M}}\leq f|dh_{p}(U)|_{g_{N}}$ (13)

for some positive bounded function $f:M\rightarrow\mathbb{R}$, where
$|\,.\,|_{g_{M}}$ and $|\,.\ |_{g_{N}}$ are the norms with respect to the
metrics $g_{M}$ and $g_{N}$ respectively. Then $h$ is a covering map.

###### Proof.

As it is shown in [27] (pp. 383), this result would follow from the fact that
$h$ has the path lifting property for curves in $(N,g_{N})$, i.e. given a
smooth curve $c:[0,1]\rightarrow(N,g_{N})$ and a point $q\in M$ with
$h(q)=c(0)$, there is another curve $\overline{c}:[0,1]\rightarrow(M,g_{M})$
starting at $q$ so that $h\circ\overline{c}=c$. Let us define the set

$A=\\{a\in[0,1]\,|\,c:[0,a]\rightarrow(N,g_{N})\,\,\text{can be lifted
to}\,\,(M,g_{M})\\}.$

Function $h$ is a local diffeomorphism at $q\in M$, which means that there are
open neighbourhoods of $U(q)$ and $V(c(0))$ so that $h:U(q)\rightarrow
V(c(0))$ is a diffeomorphism. Then, there exists some $\epsilon>0$ so that
$\overline{c}:[0,\epsilon]\rightarrow M$ satisfies $h\circ\overline{c}=c$.
Hence $A\neq\emptyset$. Let us define $t_{0}:=\mathrm{sup}A$. We will first
prove that any $c:[0,t_{0}]\rightarrow(N,g_{N})$ with $t_{0}<1$ can not be
lifted to $\overline{c}$. Otherwise, using again that $h$ is a local
diffeomorphism, there would be open neighbourhoods $U(\overline{c}(t_{0}))$
and $V(c(t_{0}))$ such that $h:U(\overline{c}(t_{0}))\rightarrow V(c(t_{0}))$
is a diffeomorphism. This would imply the existence of some $t^{\prime}>t_{0}$
such that $c:[0,t^{\prime}]\rightarrow(N,g_{N})$ could be lifted to
$\overline{c}$, which is an absurd by the definition of $t_{0}$. Therefore,
the largest interval $I_{t_{0}}$ that allows the lift must be open in $[0,1]$,
i.e. of the form $[0,t_{0})$. Let us next prove that $I_{t_{0}}$ is also
closed by showing that $t_{0}\in I_{t_{0}}$ This way, $I_{t_{0}}=[0,1]$ for it
is open and closed at the same time, and the lemma will be proven. To show
that $t_{0}\in I_{t_{0}}$, consider an increasing sequence
$\\{t_{n}\\}_{n\in\mathbb{N}}\subset[0,t_{0})$ so that
$\lim\limits_{n\rightarrow\infty}t_{n}=t_{0}$. The sequence
$\\{\overline{c}(t_{n})\\}_{n\in\mathbb{N}}$ must be contained in a compact
set $K\subset M$. Otherwise, the sequence of distances
$\\{d(\overline{c}(t_{n}),\overline{c}(0))\\}_{n\in\mathbb{N}}$ would not be
bounded and the length $L_{0}^{t_{n}}(c)$ of $c$ between $t=0$ and $t=t_{n}$
would satisfy

$L_{0}^{t_{n}}(c)=\int_{0}^{t_{n}}|c^{\prime}(t)|_{g_{N}}dt=\int_{0}^{t_{n}}|dh_{\overline{c}(t)}(\overline{c}^{\prime}(t))|_{g_{N}}\geq\frac{1}{C}\int_{0}^{t_{n}}|\overline{c}^{\prime}(t)|_{g_{M}}dt\geq\frac{1}{C}d(\overline{c}(t_{n}),\overline{c}(0)),$
(14)

where in the two last inequalities we have applied that $|f|\leq C$ for some
$C>0$ in combination with hypothesis (13). Note that relation (14) implies the
unboundedness of $L_{0}^{t_{0}}(c)$ since
$\\{d(\overline{c}(t_{n}),\overline{c}(0))\\}_{n\in\mathbb{N}}$ is not
bounded, which is an absurd. As a consequence,
$\\{\overline{c}(t_{n})\\}_{n\in\mathbb{N}}\subset K$, so it has an
accumulation point $r\in M$. Consider a neighbourhood $U(r)$ of $r$ so that
$h|_{U(r)}$ is a diffeomorfism. Then $c(t_{0})\in h(U(r))$, and by the
continuity of $c$, there exists an interval $J=(t_{0}-\delta,t_{0}+\delta)$
for some $\delta>0$ such that $c(J)\subset h(U(r))$. Choose some
$n_{0}\in\mathbb{N}$ so that for all $n\geq n_{0}$, $\overline{c}(t_{n})\in
U(r)$ and consider the lift $\overline{g}$ of $c:J\rightarrow(N,g_{N})$, which
contains $r$. The lifts $\overline{g}$ and $\overline{c}$ coincide on
$[0,t_{n})\cap J$ for $n\geq n_{0}$ since $h|_{U(r)}$ is bijective, and
consequently on $[0,t_{0})\cap J$. Hence, $\overline{g}$ is an extension of
$\overline{c}$ to $J$, and $\overline{c}$ is defined at $t_{0}$, which implies
that $t_{0}\in I_{t_{0}}$. ∎

######  Lemma 2.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a spacelike hypersurface and
$\pi_{F}|_{\Sigma}:\Sigma\rightarrow F$ its projection onto the fiber
$(F,g_{F})$. Then $\pi_{F}|_{\Sigma}$ is a local diffeomorphism. In case that
$\Sigma$ is complete and $f$ is bounded on $\Sigma$,
$\pi_{F}|_{\Sigma}:\Sigma\rightarrow F$ is a covering map.

###### Proof.

The squared norm of any $U\in\mathfrak{X}(\Sigma)$ with respect to the
hypersurface metric (8) satisfies

$g_{\Sigma}(U,U)=-(d\tau(U))^{2}+f^{2}g_{F}(U^{F},U^{F})\leq
f^{2}g_{F}(d\pi_{F}(U),d\pi_{F}(U))$ (15)

as a consequence of the decomposition $U=d\tau(U)\partial_{t}+U^{F}$. This
relation implies that $\pi_{F}$ is locally injective. Otherwise there would be
some non-zero $U\in\mathfrak{X}(\Sigma)$ such that $d\pi_{F}(U)=0$ at some
point of it, which would mean that $g_{\Sigma}(U,U)=0$, contradicting the fact
that $g_{\Sigma}$ is non-degenerate. Besides,
$\mathrm{dim}\,\Sigma=\mathrm{dim}\,F$, so $d\pi_{F}$ is locally bijective and
hence a local diffeomorphism. On the other hand, the validity of (15) makes
the hypothesis (13) hold true for the map
$h:=\pi_{F}|_{\Sigma}:\Sigma\rightarrow F$. Since $\Sigma$ is complete and $f$
bounded, $\pi_{F}$ is a covering map as a consequence of Lemma 1. ∎

A usual way to prove that a Riemannian manifold is complete is by considering
divergent curves, i.e. those ones that can not be enclosed by any compact set.
The Hopf-Rinow theorem (see [28] for instance) entails that a Riemannian
manifold containing a divergent curve of finite length can not be complete. We
exploit this fact to prove the following completeness results for spacelike
hypersurfaces in $I\times_{f}F$:

######  Proposition 4.

Let $I\times_{f}F$ be a twisted product spacetime and assume that it admits a
complete immersed spacelike hypersurface $\psi:\Sigma\rightarrow I\times_{f}F$
where $f$ is bounded. Then the fiber $(F,g_{F})$ is complete.

###### Proof.

Given any $U\in\mathfrak{X}(\Sigma)$, the inequality

$|U|_{g_{\Sigma}}\leq C|d\pi_{F}(U)|_{g_{F}}$ (16)

follows from (15), where $C$ is any upper bound for $f$. The hypothesis of
Lemma 2 are satisfied, so the map $\pi_{F}|_{\Sigma}:\Sigma\rightarrow F$ is a
covering map. In particular this means that the path lifting property for
curves in $(F,g_{F})$ is satisfied in this context. Hence, any divergent curve
$c:[0,+\infty)\rightarrow(F,g_{F})$ can be lifted to a divergent
$\overline{c}:[0,\infty)\rightarrow\Sigma$ so that
$c=\pi_{F}\circ\overline{c}$. By (16), the lengths of $c$ and $\overline{c}$
satisfy

$L_{0}^{\infty}(c)=\int_{0}^{\infty}|c^{\prime}(t)|_{g_{F}}dt=\int_{0}^{\infty}|d\pi_{F}(\overline{c}^{\prime}(t))|_{g_{F}}dt\geq\frac{1}{C}\int_{0}^{\infty}|\overline{c}^{\prime}(t)|_{g_{\Sigma}}dt=\frac{1}{C}L_{0}^{\infty}(\overline{c}).$
(17)

The right-hand side of (17) is infinite since $\overline{c}$ is a divergent
curve in a complete manifold, which in particular implies that $c$ also has
infinite length in $(F,g_{F})$. Since this happens to any arbitrary divergent
curve $c$ in the fiber, $(F,g_{F})$ is complete too. ∎

######  Proposition 5.

Let $I\times_{f}F$ be a twisted product spacetime. The following assertions
hold true:

  * (i)

If $I\times_{f}F$ admits a compact spacelike hypersurface
$\psi:\Sigma\rightarrow I\times_{f}F$, then the fiber $(F,g_{F})$ is compact.

  * (ii)

If the universal cover of the fiber is compact and $\psi:\Sigma\rightarrow
I\times_{f}F$ is any complete spacelike hypersurface on which $f$ is bounded,
then $\psi:\Sigma\rightarrow I\times_{f}F$ is compact.

###### Proof.

Compact Riemannian manifolds are complete. Since $f$ is bounded on $\Sigma$,
we apply Lemma 2 and $\pi_{F}|_{\Sigma}$ is a covering map. Hence
$F=\pi_{F}(\Sigma)$ is compact and that proves $\it{(i)}$. On the other hand,
the assumptions of $\it{(ii)}$ also imply that $\pi_{F}|_{\Sigma}$ is a
covering map. Since the universal cover of the fiber is compact by hypothesis,
both $F$ and $\Sigma=(\pi_{F}|_{\Sigma})^{-1}(F)$ are compact too. ∎

##  4 Main results

We have already mentioned the big interest that $CMC$ and maximal
hypersurfaces have in several areas of mathematics and physics. In the current
section results involving CMC, maximal, totally geodesic and totally umbilical
spacelike hypersurfaces in twisted products of the form $I\,{}_{f}\\!\\!\times
M$ and $I\times_{f}M$ are presented. For the validity of many of them an
appropriate expanding behaviour and the compactness of the fiber become
essential hypothesis. The geometry of the surface is encoded in the time-
height function $\tau$, which measures the time-distance of any point lying on
it. This motivates the computation of the Laplacian of $\tau$ on such
hypersurfaces as a fundamental tool to characterize these families of
manifolds. In the next result we obtain the expression for $\triangle\tau$,
where $\triangle$ denotes the Laplacian on the hypersurface with respect to
its induced metric.

######  Proposition 6.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be an immersed spacelike
hypersurface in $I\times_{f}F$, and $\tau$ its corresponding time-height
function. The expression for the Laplacian of $\tau$ with respect to the
induced metric on $\Sigma$ is

$\triangle\tau=-(n+|\nabla\tau|^{2})\partial_{t}\log
f-nH\overline{g}(\partial_{t},N),$ (18)

where $H$ is the mean curvature of $\Sigma$ along its future unit normal $N$.

###### Proof.

The Laplacian of $\tau$ can be obtained considering the decomposition of the
torqued vector $K=f\partial_{t}$ into its tangential and normal components to
the hypersurface $\psi:\Sigma\rightarrow I\times_{f}F$,

$K=K^{\top}-\overline{g}(K,N)N=-f\nabla\tau-\overline{g}(K,N)N.$ (19)

Let $U\in\mathfrak{X}(\Sigma)$. The ambient covariant derivative of (19) along
$U$ reads

$\overline{\nabla}_{U}K=\overline{\nabla}_{U}\left(-f\nabla\tau\right)-\overline{\nabla}_{U}\left(\overline{g}(K,N)N\right).$

Using formula (4) in the left-hand side of the above expression and applying
the Leibniz rule on the right-hand side, it becomes

$(\partial_{t}f)U+\omega(U)K=-U(f)\nabla\tau-f\overline{\nabla}_{U}\left(\nabla\tau\right)-U(\overline{g}(K,N))N-\overline{g}(K,N)\overline{\nabla}_{U}N.$

Inserting relation (12) for $\omega(U)$ and contracting the whole expression
with any other $V\in\mathfrak{X}(\Sigma)$ gives

$\left(\overline{g}(U,V)+\left(\nabla_{U}\tau\right)\left(\nabla_{V}\tau\right)\right)(\partial_{t}f)=-f\overline{g}\left(\nabla_{U}\left(\nabla\tau\right),V\right)+\overline{g}(K,N)\mathrm{II}(U,V),$

where $II$ is the second fundamental form tensor of $\Sigma$. The trace of the
above expression gives the stated relation for the Laplacian, after solving
for $\triangle\tau$.

∎

Formula (18) can be applied to obtain rigidity results for slices in
$I\times_{f}F$. In the following proposition, we prove that a spacelike
hypersurface $\psi:\Sigma\rightarrow I\times_{f}F$ is necessarily a slice
provided its mean curvature satisfies a suitable bound involving
$\partial_{t}f$, where such derivative is required to be constant signed along
$\Sigma$.

######  Proposition 7.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a compact spacelike hypersurface
in a twisted product spacetime $I\times_{f}F$. If

  * (i)

$\partial_{t}f\geq 0$ on $\Sigma$ and $H\geq\frac{\partial_{t}(\log
f)}{\overline{g}(\partial_{t},N)}$,

  * (ii)

or $\partial_{t}f\leq 0$ on $\Sigma$ and $H\leq\frac{\partial_{t}(\log
f)}{\overline{g}(\partial_{t},N)}$,

then $\Sigma$ is a slice.

###### Proof.

Note that formula (18) can also be written as

$\triangle\tau=-n\left(\partial_{t}(\log
f)+H\overline{g}(\partial_{t},N)\right)-\partial_{t}(\log f)|\nabla\tau|^{2}.$

Condition $H\geq\frac{\partial_{t}(\log f)}{\overline{g}(\partial_{t},N)}$
(resp. $H\leq\frac{\partial_{t}(\log f)}{\overline{g}(\partial_{t},N)}$) is
equivalent to $\partial_{t}(\log f)+H\overline{g}(\partial_{t},N)\geq 0$
(resp. $\partial_{t}(\log f)+H\overline{g}(\partial_{t},N)\leq 0$). Hence, by
(18) the inequality $\triangle\tau\leq 0$ (resp. $\triangle\tau\geq 0$) holds.
Therefore $\tau$ is constant in both cases $(i)$ and $(ii)$ as a consequence
of the compactness hypothesis on $\Sigma$. ∎

The following non-existence results of CMC hypersurfaces in $I\times_{f}F$ are
also derived from relation (18), where an appropriate expanding behaviour of
the ambient space is required:

######  Theorem 1.

The following assertions hold true in expanding and contracting twisted
product spacetimes:

  * (i)

There are no compact spacelike hypersurfaces with negative CMC in expanding
$I\times_{f}F$.

  * (ii)

There are no compact spacelike hypersurfaces with positive CMC in contracting
$I\times_{f}F$.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a compact CMC spacelike
hypersurface. Hence $\tau$ reaches a global maximum at $(t^{0},x^{0})$ and
minimum at $(t_{0},x_{0})$ on $\Sigma$. As a consequence,
$\triangle\tau|_{(t_{0},x_{0})}\geq 0$ and $\triangle\tau|_{(t^{0},x^{0})}\leq
0$. Using the Laplacian expression (18), we obtain that the following
inequality holds at $(t_{0},x_{0})$:

$0\leq\triangle\tau|_{(t_{0},x_{0})}=-n\partial_{t}(\log
f)|_{(t_{0},x_{0})}+nH,$

which means

$H\geq\partial_{t}(\log f)|_{(t_{0},x_{0})}.$ (20)

This lower bound for $H$ implies that in an expanding twisted product
spacetime case, there are no compact spacelike hypersurfaces with negative
constant mean curvature. An analogous reasoning for the maximum
$(t^{0},x^{0})$ leads to

$H\leq\partial_{t}(\log f)|_{(t^{0},x^{0})},$ (21)

which also proves that neither are there any positive CMC spacelike
hypersurfaces in a contracting background. ∎

######  Remark.

Note that for the validity of inequalities (20) and (21) it is sufficient for
the ambient spacetime to have a twisted product structure in a neighbourhood
of $(t_{0},x_{0})$ and $(t^{0},x^{0})$. Such inequalities are optimal, since
slices $\Sigma_{t_{0}}\equiv\\{t=t_{0}\\}$ in GRW spacetimes are CMC
hypersurfaces with $H=\partial_{t}(\log f)(t_{0})$.

Finding conditions for the Laplacian of $\tau$ to be constant signed is an
effective approach to obtain rigidity results for slices in twisted product
spacetimes. In some occasions, the sign of $\triangle\tau$ is not clearly
determined under generic geometric conditions. Conformal transformations are
useful maps which allow to modify the geometry of manifolds in a very precise
manner and can help to discern under which hypothesis this sign is well-
defined. In twisted product spacetimes, slices belong to the family of totally
umbilical hypersurfaces, which are those whose shape operator is proportional
to the identity endomorphism. As it is shown in Proposition 10 in the
Appendix, totally umbilic submanifolds are preserved under the action of
conformal transformations. If the twisted product spacetime
$\overline{M}=I\times_{f}F$ undergoes the conformal transformation defined by
$\tilde{g}=e^{2\phi}\overline{g}$ with $\phi=-\log f$, then $\tilde{g}$
becomes $\tilde{g}=-\alpha^{2}dt^{2}+g_{F}$, where $\alpha:=f^{-1}$, and
$\overline{M}$ acquires a twisted structure of the form
$I\,{}_{\alpha}\\!\\!\times F$, where this time $(I,-dt^{2})$ is the fiber.
When the metric is of the form $\tilde{g}=-\alpha^{2}dt^{2}+g_{F}$, slices in
$(\overline{M},\tilde{g})$ are totally geodesic [41]. Indeed, the future
timelike unit normal of the slice $\Sigma_{t_{0}}\equiv\\{t=t_{0}\\}$ is
$\alpha^{-1}\partial_{t}$, so given any $X,Y\in\mathfrak{X}(\Sigma_{t_{0}})$,
the second fundamental form $II$ of $\Sigma_{t_{0}}$ is

$\displaystyle II(X,Y)$ $\displaystyle=$
$\displaystyle-\tilde{g}(\tilde{\nabla}_{X}\left(\alpha^{-1}\partial_{t}\right),Y)=-\tilde{g}(X(\alpha^{-1})\partial_{t}+\alpha^{-1}\tilde{\nabla}_{X}\partial_{t},Y)=-\alpha^{-1}\tilde{g}(\tilde{\nabla}_{X}\partial_{t},Y)=0$

since $\tilde{\nabla}_{X}\partial_{t}=X(\log\alpha)\partial_{t}$.

In this work we exploit some useful properties of conformal maps to obtain
characterizations of slices in both sorts of twisted product spacetimes
$I\times_{f}F$ and $I\,{}_{\alpha}\\!\\!\times F$ provided $f$ and $\alpha$
enjoy suitable monotonic behaviour with respect to $t$. We next compute the
Laplacian of the time-height function of any spacelike surface in
$I\,{}_{\alpha}\\!\\!\times F$.

######  Lemma 3.

Let $\psi:\Sigma\rightarrow(\overline{M}=I\,{}_{\alpha}\\!\\!\times
F,\tilde{g})$ an immersed spacelike hypersurface in
$I\,{}_{\alpha}\\!\\!\times F$. The Laplacian $\tilde{\triangle}\tau$ with
respect to the induced metric on $\Sigma$ reads

$\tilde{\triangle}\tau=\alpha^{-2}\left(\left(1+\cosh^{2}\theta\right)\partial_{t}\log\alpha+n\tilde{H}\alpha\cosh\theta-2\alpha\cosh\theta\tilde{N}(\log\alpha)\right),$
(22)

where $\tilde{H}$ is the mean curvature of $\Sigma$ along the $\tilde{g}$-unit
normal $\tilde{N}$ and
$\cosh\theta=-\tilde{g}\left(\tilde{N},{\alpha}^{-1}\partial_{t}\right)$ is
the hyperbolic cosine of $\theta$.

###### Proof.

The metric $\tilde{g}=-\alpha^{2}dt^{2}+g_{F}$ is obtained from the twisted
metric $\overline{g}=-dt^{2}+f^{2}g_{F}$ after applying the conformal
transformation $\tilde{g}=\alpha^{2}\overline{g}$, where $\alpha:=f^{-1}$. In
this case expression (44) in the Appendix establishes that the respective
Laplacians $\triangle$ and $\tilde{\triangle}$ associated to $\overline{g}$
and $\tilde{g}$ satisfy

$\tilde{\triangle}\tau=\alpha^{-2}(\triangle\tau+(n-2)\overline{g}(\nabla\log\alpha,\nabla\tau)).$
(23)

Since $|\triangle\tau|^{2}=\cosh^{2}{\theta}-1$ and the relation for the mean
curvatures $H$ and $\tilde{H}$ associated to the metrics $\overline{g}$ and
$\tilde{g}$ is $H=(\tilde{H}-\tilde{N}(\log\alpha))$ by formula (43) in the
Appendix, expression (18) for $\triangle\tau$ on $\psi:\Sigma\rightarrow
I\times_{f}F$ can be rewritten as

$\displaystyle\triangle\tau=(n-1+\cosh^{2}{\theta})\partial_{t}\log\alpha+n\alpha(\tilde{H}-\tilde{N}(\log\alpha))\cosh\theta.$
(24)

On the other hand, it is immediate to see that
$\tilde{\nabla}\tau=-\alpha^{-2}(\partial_{t})^{\top}$. The decomposition of
the $\tilde{g}$-unit vector $\alpha^{-1}\partial_{t}$ into its tangent and
normal component to $\Sigma$ imply

$\tilde{\nabla}\tau=\alpha^{-1}\left(-\alpha^{-1}\partial_{t}-\tilde{g}(\alpha^{-1}\partial_{t},\tilde{N})\tilde{N}\right).$
(25)

Likewise, the decomposition of the ambient gradient
$\tilde{\nabla}^{\tilde{g}}\log\alpha$ in $I\,{}_{\alpha}\\!\\!\times M$ gives

$\tilde{\nabla}\log\alpha=\tilde{\nabla}^{\tilde{g}}\log\alpha+\tilde{N}(\log\alpha)\tilde{N}.$
(26)

Considering (25) and (26), we obtain

$\overline{g}(\nabla\log\alpha,\nabla\tau)=\alpha^{2}\tilde{g}(\tilde{\nabla}\log\alpha,\tilde{\nabla}\tau)=-\partial_{t}\log\alpha+\alpha\cosh\theta\tilde{N}(\log\alpha).$
(27)

Finally, (22) follows after plugging (24) and (27) into (23).

∎

A natural question that arises is whether slices are the only maximal
hypersurfaces in $I\,{}_{\alpha}\\!\\!\times F$ when appropriate expanding
conditions are considered. With the help of conformal maps, we prove they are
indeed. To this purpose, we will first need the following lemma:

######  Lemma 4.

Let $I\,{}_{\alpha}\\!\\!\times F$ be a twisted product spacetime, and
$\psi:\Sigma\rightarrow I\,{}_{\alpha}\\!\\!\times F$ an immersed maximal
compact spacelike hypersurface, where $\mathrm{dim}\,F>2$. Assume that
$\partial_{t}\alpha$ is a constant signed function. Then there exists a
conformal transformation under which the Laplacian of the time-height function
$\tau$ of $\Sigma$ is constant signed.

###### Proof.

For any $p\in\mathbb{R}$, the Laplacians $\tilde{\triangle}$ and
$\hat{\triangle}$ of the conformally related metrics
$\tilde{g}=-\alpha^{2}dt^{2}+g_{F}$ of $I\,{}_{\alpha}\\!\\!\times F$ and
$\hat{g}=\alpha^{2p}\tilde{g}$ satisfy

$\hat{\triangle}\tau=\alpha^{-2p}(\tilde{\triangle}\tau+(n-2)\tilde{g}(\tilde{\nabla}\log(\alpha^{p}),\tilde{\nabla}\tau))=\alpha^{-2p}(\tilde{\triangle}\tau-\alpha^{-2}p(n-2)\tilde{g}(\tilde{\nabla}\log\alpha,\tilde{\nabla}\tau))$

by virtue of formula (44) in the Appendix. Particularizing expression (22) for
a maximal hypersurface in $(I\,{}_{\alpha}\\!\\!\times F,\tilde{g})$ and
substituting in the above relation gives

$\hat{\triangle}\tau=\alpha^{-2p-2}\left((1-p(n-2)+\cosh^{2}\theta)\partial_{t}(\log\alpha)+(-2+p(n-2))\alpha(\cosh\theta)\tilde{N}(\log\alpha)\right).$

Note that the second term in the right-hand side of this expression vanishes
for $p=\frac{2}{n-2}$ (we assume $n>2$). For this value of $p$ the relation
between the Laplacians turns out to be

$\hat{\triangle}\tau=\alpha^{\frac{-2n}{n-2}}(\sinh^{2}\theta)\partial_{t}\log\alpha=\left(\alpha^{\frac{-n}{n-2}}\right)^{2}(\sinh^{2}\theta)\frac{\partial_{t}\alpha}{\alpha},$
(28)

which clearly establishes the statement of the lemma provided $n>2$. ∎

######  Remark.

Note that the above conformal transformation is not well defined for $n=2$.
However, this problem can be addressed in the same way as in [40] (Remark 1).
Specifically, given $I\,{}_{\alpha}\\!\\!\times F$ we can consider the higher-
dimensional ambient space $I\,{}_{\alpha}\\!\\!\times\mathbb{S}^{2}\times F$,
and $\mathbb{S}^{2}\times\Sigma$ as immersed spacelike hypersurface, whose
mean curvature vector is the same than the one of $\Sigma$ lifted to the
extended spacetime. The conformal transformation can be performed in this new
background.

The characterization of spacelike maximal hypersurfaces in
$I\,{}_{\alpha}\\!\\!\times F$ follows as a consequence of the above lemma.

######  Theorem 2.

Let $(\overline{M}=I\,{}_{\alpha}\\!\\!\times F,\tilde{g})$ be a twisted
product spacetime, where $(F,g_{F})$ is compact. Suppose that either

  * (i)

$\partial_{t}\alpha$ is constant signed,

  * (ii)

or a slice $\Sigma_{t_{0}}\equiv\\{t=t_{0}\\}$ divides
$I\,{}_{\alpha}\\!\\!\times F$ in two different regions, with
$\partial_{t}\alpha<0$ if $t\leq t_{0}$ (past region), and
$\partial_{t}\alpha>0$ if $t\geq t_{0}$ (future region).

Then, an immersed compact hypersurface $\psi:\Sigma\rightarrow
I\,{}_{\alpha}\\!\\!\times F$ is maximal if and only if it is a slice.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a maximal hypersurface in
$I\,{}_{\alpha}\\!\\!\times F$. Lemma 4 can be applied under the assumptions
of $(i)$, so $\Sigma$ becomes a compact hypersurface in
$(\overline{M},\hat{g}=\alpha^{\frac{4}{n-2}}\tilde{g})$ where
$\hat{\triangle}\tau$ is constant signed, and hence $\psi:\Sigma\rightarrow
I\,{}_{\alpha}\\!\\!\times F$ is a slice in $I\,{}_{\alpha}\\!\\!\times F$. To
prove $(ii)$, let us consider the Laplacian of the function $(\tau-t_{0})^{2}$
in $(\overline{M},\hat{g}=\alpha^{\frac{4}{n-2}}\tilde{g})$

$\hat{\triangle}(\tau-t_{0})^{2}=2\left(|\hat{\nabla}\tau|^{2}+(\tau-
t_{0})\hat{\triangle}\tau\right).$ (29)

Given any $p\in\Sigma$ in the future region where $\partial_{t}\alpha>0$ and
$\tau-t_{0}\geq 0$, $\hat{\triangle}\tau\geq 0$ by virtue of (28), which makes
$\hat{\triangle}(\tau-t_{0})^{2}$ non-negative. An analogous argument for any
$p\in\Sigma$ in the past region where $\partial_{t}\alpha<0$ and $\tau-
t_{0}\leq 0$ entails that $\hat{\triangle}(\tau-t_{0})^{2}\geq 0$ too. In any
case, $\hat{\triangle}(\tau-t_{0})^{2}\geq 0$, so $\tau$ must be constant as a
consequence of the compactness of $\Sigma$. ∎

The results obtained so far allow us to find a characterization of the totally
umbilical hypersurfaces in $I\times_{f}F$ with a suitable restriction on their
mean curvature.

######  Theorem 3.

Let $I\times_{f}F$ be a twisted product spacetime with compact fiber, that is
either

  * (i)

expanding (or contracting),

  * (ii)

or it admits a transition slice $S_{t_{0}}\equiv\\{t=t_{0}\\}$.

Then the only totally umbilical compact hypersurfaces whose mean curvature
satisfies the equation $H=\overline{g}(N,\overline{\nabla}\log f)$ in
$I\times_{f}F$ are the slices.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a totally umbilical compact
hypersurface whose mean curvature satisfies
$H=\overline{g}(N,\overline{\nabla}\log f)$. Case $(i)$ can be addressed
applying Corollary 1 in the Appendix, which gives the maximality of $\Sigma$
in $I\,{}_{f^{-1}}\\!\\!\times F$. Since the first time derivative of
$\alpha:={f^{-1}}$ is constant signed, $\Sigma$ has to be necessarily a slice
in $I\,{}_{f^{-1}}\\!\\!\times F$ by Theorem 2, and consequently also in
$I\times_{f}F$. The proof of $(ii)$ follows the same argument as Theorem 2,
$(ii)$. Specifically, $\hat{\triangle}(\tau-t_{0})^{2}$ is proven to be non-
negative regardless of the region where any point of $\Sigma$ lies, and by the
compactness assumption on the hypersurface, $\tau$ is constant. ∎

######  Remark.

The above theorem is optimal in the sense that it does not hold provided the
signs of $\partial_{t}f$ in the future and past region of the transition slice
$S_{t_{0}}$ are opposite to the ones stated, as it occurs in the De Sitter
spacetime.

We finally conclude the section with the characterization of compact maximal
spacelike hypersurfaces in $I\times_{f}F$. To obtain this result we first need
to prove the following lemma, which establishes that any compact spacelike
hypersurface must be a totally geodesic slice provided its mean curvature and
the restriction of the time-derivative of $f$ to it have opposite signs.

######  Lemma 5.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a compact spacelike hypersurface
in the twisted product spacetime $I\times_{f}F$ and $H$ its mean curvature. If
$(\partial_{t}f)\cdot H\leq 0$ on $\Sigma$, then it is a totally geodesic
spacelike slice.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a compact spacelike hypersurface
in $I\times_{f}F$. The expression for $\triangle\tau$ in (18) becomes constant
signed provided $\partial_{t}f$ and $H$ have opposite signs, so $\tau$ is
constant as a consequence of the compactness of $\Sigma$. Since the terms on
the right hand-side of (18) have the same sign both must be zero, which in
particular implies the maximality of $\Sigma$. Any slice in $I\times_{f}M$ is
totally umbilical, so $\Sigma$ turns out to be totally geodesic too. ∎

######  Theorem 4.

Let $I\times_{f}F$ be a twisted product spacetime with compact fiber, that is
either

  * (i)

expanding (resp. contracting),

  * (ii)

or it admits a transition spacelike slice $S_{t_{0}}\equiv\\{t=t_{0}\\}$.

Then every compact maximal hypersurface $\psi:\Sigma\rightarrow I\times_{f}F$
is also a totally geodesic spacelike slice. If $(ii)$ happens, $S_{t_{0}}$ is
the only compact maximal hypersurface in $I\times_{f}F$.

###### Proof.

Let $\psi:\Sigma\rightarrow I\times_{f}F$ be a compact maximal hypersurface in
$I\times_{f}F$. Under the assumptions of $(i)$, $(\partial_{t}f)\cdot H\leq 0$
holds on $\Sigma$, so $\Sigma$ is totally geodesic as a consequence of Lemma
5. To prove $(ii)$, it is enough to see again that $\triangle(\tau-t_{0})^{2}$
is non-negative, which implies that $\tau=t_{1}$ for some
$t_{1}\in\mathbb{R}$. If $t_{1}>t_{0}$, the mean curvature of $\Sigma$ would
be strictly negative, whereas if $t_{1}<t_{0}$ it would be strictly positive,
which contradicts the maximality hypothesis in both cases. The only
possibility for the slice $\Sigma_{t_{1}}\equiv\\{t=t_{1}\\}$ to be maximal is
that $t_{1}=t_{0}$, i.e when it is the transition slice. ∎

######  Remark 2.

It would also be natural to consider twisted product spacetimes admitting a
maximal spacelike slice $S_{t_{0}}\equiv\\{t=t_{0}\\}$ such that they are
contracting for $t\leq t_{0}$ and expanding for $t\geq t_{0}$. Nevertheless,
in this case it is not possible to obtain a similar uniqueness result as in
the previous theorem. Indeed, it is enough to consider the well-known warped
model of the De Sitter spacetime (see for instance [3]).

##  5 Calabi-Bernstein-type results

A characterization of maximal compact hypersurfaces has been obtained in
Theorem 4 for expanding and contracting twisted product spacetimes
$I\times_{f}F$, and also for those which admit a transition slice. In this
section we focus on immersed spacelike hypersurfaces $\psi:\Sigma\rightarrow
I\times_{f}F$ which are graphs over the fiber $(F,g_{F})$. The calculation of
the graph mean curvature $H$ becomes essential to reformulate the maximality
results of Theorem 4 in a proper Calabi-Bernstein style by providing existence
and uniqueness results to the associated $H=0$ elliptic PDE under suitable
conditions. To this purpose, let $\Omega\subset F$ be an open set in
$I\times_{f}F$. Any $u\in C^{\infty}(\Omega)$ defines a local graph
$\Sigma_{u}=\\{(u(p),p):p\in\Omega\\}$ on the fiber, where the map
$x:\Omega\subset F\rightarrow I\times_{f}F$ given by $x(p)=(u(p),p)$ is an
immersion. Let us refer to the induced metric on the graph as $g_{u}$, which
is of the form (8), where the time-height function $\tau$ of the graph is
related to $u$ by $u=\tau\circ x$. The pull-back of $g_{u}$ by $x$ to the
fiber $(F,g_{F})$ reads

$x^{*}(g_{u})=-du^{2}+(f\circ x)^{2}g_{F}.$ (30)

The hypersurface $(\Sigma_{u},g_{u})$ is said to be an entire graph when
$\pi_{F}|_{\Sigma_{u}}=x^{-1}:\Sigma_{u}\rightarrow F$ is a global
diffeomorphism. Lemma 2 shows that $\pi_{F}|_{\Sigma_{u}}$ is a cover map
provided $(\Sigma_{u},g_{u})$ is complete and the restriction of $f$ to
$(\Sigma_{u},g_{u})$ is bounded. In particular, this occurs when
$I\times_{f}F$ admits a compact spacelike hypersurface $\psi:\Sigma\rightarrow
I\times_{f}F$, which also implies the compactness of $(F,g_{F})$.
Additionally, if the fiber is connected, $\pi_{F}|_{\Sigma}$ is a global
diffeomorphism and hence $(\Sigma_{u},g_{u})$ is a global graph over the
fiber. In the following proposition the causal character of spacelike graphs
is characterized in terms of the gradient of the graph function $u$.

######  Proposition 8.

Let $\Omega\subset F$ be an open set and $(\Sigma_{u},g_{u})$ a graph over
$(\Omega,g_{F})$ determined by the immersion $x:\Omega\subset F\rightarrow
I\times_{f}F$, where $x(p)=(u(p),p)\in I\times_{f}F$ for some function $u\in
C^{\infty}(\Omega)$. Then the validity of the inequality

$|\nabla^{F}u|<f\circ x$ (31)

at every $p\in\Omega$ is necessary and sufficient for the graph metric $g_{u}$
to be Riemannian, where $\nabla^{F}$ is the Levi-Civita connection of
$(F,g_{F})$.

###### Proof.

For every $p\in\Omega$, let us consider the push-forward of the gradient
$\nabla^{F}u\in\mathfrak{X}(F)$ by $dx$ to the tangent bundle of
$(\Sigma_{u},g_{u})$

$dx(\nabla^{F}u)=(|\nabla^{F}u|^{2}_{g_{F}}\circ
x^{-1})\partial_{t}+\nabla^{F}u.$ (32)

Squaring the above relation gives

$|dx(\nabla^{F}u)|^{2}=(x^{-1})^{*}\left(|\nabla^{F}u|^{2}_{g_{F}}((f\circ
x)^{2}-|\nabla^{F}u|^{2}_{g_{F}})\right),$ (33)

where $(x^{-1})^{*}$ stands for the pull-back by
$x^{-1}:(\Sigma_{u},g_{u})\rightarrow(F,g_{F})$. Since $g_{u}$ is a Riemannian
metric, the right-hand side of (33) becomes strictly positive when
$\nabla^{F}u\neq 0$, which can only happen if (31) holds. On the other hand,
(31) is also valid when $\nabla^{F}u=0$, so in any case (31) holds true. ∎

The mean curvature of the graph $(\Sigma_{u},g_{u})$ satisfies a PDE which
involves the function $u:\Omega\subset F\rightarrow\mathbb{R}$ and its first
and second order derivatives. This expression is well known when the ambient
is a warped product (see [5] for instance). In this section the corresponding
maximal equation for a graph in $I\times_{f}F$ is derived from (18). To this
purpose, we first obtain in the following lemma the expression for the
Laplacian of $\tau$ in terms of the fiber geometry:

######  Lemma 6.

Let $\Omega\subset F$ be an open set and $(\Sigma_{u},g_{u})$ a graph over
$(\Omega,g_{F})$ determined by the immersion $x:\Omega\subset F\rightarrow
I\times_{f}F$, where $x(p)=(u(p),p)\in I\times_{f}F$ for some function $u\in
C^{\infty}(\Omega)$. Assume that $|\nabla^{F}u|<f\circ x$. The expression for
$\triangle\tau$ in terms of the fiber geometry is

$x^{*}(\triangle\tau)=\rho(f\circ x)^{-n+2}g_{F}(\nabla^{F}(\rho(f\circ
x)^{n}),\nabla^{F}u)+\rho^{2}(f\circ x)^{2}\triangle_{F}u\,,$ (34)

where $\rho:\Omega\subset F\rightarrow\mathbb{R}$ is defined as
$\rho=\frac{1}{(f\circ x)\sqrt{(f\circ x)^{2}-|\nabla^{F}u|^{2}}}$ and
$\triangle_{F}$ is the Laplacian on $(F,g_{F})$.

###### Proof.

Let $\\{E_{i}\\}_{i=1}^{n}$ be a local basis tangent to the graph
$(\Sigma_{u},g_{u})$. The decomposition of $E_{i}$ into its horizontal and
vertical parts is $E_{i}=(\partial_{i}u)\partial_{t}+E_{i}^{F}$. The Laplacian
of $\tau$ can be computed using formula (45) in the Appendix, which in this
context reads

$\triangle\tau=\frac{1}{\sqrt{\det g_{u}}}\sum_{i=1}^{n}E_{i}\left(\sqrt{\det
g_{u}}\,(\nabla\tau)^{i}\right),$ (35)

where $(\nabla\tau)^{i}$ stands for the i-component of $\nabla\tau$ in the
basis $\\{E_{i}\\}_{i=1}^{n}$. A straightforward computation shows that the
components of $\nabla\tau$ and $\nabla^{F}u$ in the respective bases
$\\{E_{i}\\}_{i=1}^{n}$ and $\\{E_{i}^{F}\\}_{i=1}^{n}$ are related by

$x^{*}((\nabla\tau)^{i})=f^{2}\rho^{2}(\nabla^{F}u)^{i},$ (36)

where $\rho$ is defined as in the statement of this lemma. On the other hand,
the metrics $g_{u}$ and $g_{F}$ are related by (30), which makes their
determinants satisfy

$x^{*}(\det g_{u})=\rho^{-2}(f\circ x)^{2n-4}\det g_{F}.$ (37)

Plugging (36) and (37) into (35) gives

$x^{*}(\triangle\tau)=\frac{1}{(f\circ x)^{n-2}\rho^{-1}\sqrt{\det
g_{F}}}\sum_{i=1}^{n}E_{i}^{F}\left(\rho(f\circ x)^{n}\sqrt{\det
g_{F}}\,(\nabla^{F}u)^{i}\right),$

where we have used that $X(h)=X^{F}(h\circ x)$ for any function
$h:\Sigma_{u}\rightarrow\mathbb{R}$ and any $X=X^{F}(u)\partial_{t}+X^{F}$
tangent to $(\Sigma_{u},g_{u})$. Hence

$\displaystyle x^{*}(\triangle\tau)$ $\displaystyle=$ $\displaystyle(f\circ
x)^{-n+2}\rho\left(\left(\sum_{i=1}^{n}E_{i}^{F}\left(\rho(f\circ
x)^{n}\right)(\nabla^{F}u)^{i}\right)+\frac{\rho(f\circ x)^{n}}{\sqrt{\det
g_{F}}}\sum_{i=1}^{n}E_{i}^{F}\left(\sqrt{\det
g_{F}}(\nabla^{F}u)^{i}\right)\right)$ $\displaystyle=$
$\displaystyle\rho(f\circ x)^{-n+2}g_{F}(\nabla^{F}(\rho(f\circ
x)^{n}),\nabla^{F}u)+\rho^{2}(f\circ x)^{2}\triangle_{F}u,$

as stated above. ∎

The expression for the mean curvature $H$ of a graph over the fiber in
$I\times_{f}F$ follows directly from Lemma 6:

######  Proposition 9.

Let $\Omega\subset F$ be an open set and $(\Sigma_{u},g_{u})$ a graph over
$(\Omega,g_{F})$ determined by the immersion $x:\Omega\subset F\rightarrow
I\times_{f}F$, where $x(p)=(u(p),p)\in I\times_{f}F$ for some function $u\in
C^{\infty}(\Omega)$. Assume that $|\nabla^{F}u|<f\circ x$. Its mean curvature
$H$ satisfies the following equation:

$\displaystyle n\cdot x^{*}H=\mathrm{div}\,_{F}(\rho\nabla^{F}u)+(f\circ
x)^{2}\rho\left(n+\frac{|\nabla^{F}u|^{2}_{g_{F}}}{(f\circ
x)^{2}}\right)(\partial_{t}\log f)\circ x+n\rho g_{F}(\nabla^{F}\log(f\circ
x),\nabla^{F}u),$ (38)

where $\rho:\Omega\subset F\rightarrow\mathbb{R}$ is defined as
$\rho=\frac{1}{(f\circ x)\sqrt{(f\circ x)^{2}-|\nabla^{F}u|^{2}}}$ and
$\mathrm{div}\,_{F}$ is the divergence operator with respect to $g_{F}$.

###### Proof.

Relation (38) for $H$ can be derived by rewritting the whole equality (18) in
terms of the geometry of $(F,g_{F})$ and solving for $H$. The expression for
$\triangle\tau$ in terms of the fiber geometry has already been obtained in
Lemma 6, which together with relation $x^{*}(|\nabla\tau|^{2})=(f\circ
x)^{2}\rho^{2}|\nabla^{F}u|^{2}_{g_{F}}$ derived from (36) can be plugged into
(18) to obtain

$(f\circ x)^{-n}g_{F}(\nabla^{F}(\rho(f\circ
x)^{n}),\nabla^{F}u)+\rho\triangle_{F}u=\frac{-1}{\rho(f\circ
x)^{2}}\left(n+\rho^{2}(f\circ
x)^{2}|\nabla^{F}u|^{2}_{g_{F}}\right)(\partial_{t}\log f)\circ x+n\cdot
x^{*}H.$ (39)

Notice that

$\nabla^{F}(\rho(f\circ x)^{n})=(\nabla^{F}\rho)(f\circ x)^{n}+n\rho(f\circ
x)^{n-1}(\overline{\nabla}f\circ dx),$

so the first term of the left-hand side of (39) becomes

$(f\circ x)^{-n}g_{F}(\nabla^{F}(\rho(f\circ
x)^{n}),\nabla^{F}u)=g_{F}(\nabla^{F}\rho,\nabla^{F}u)+n\rho(f\circ
x)^{-1}(\overline{\nabla}f\circ dx)(\nabla^{F}u).$

Using (32) in combination with formula
$\textrm{div}_{F}(\rho\nabla^{F}u)=g_{F}(\nabla^{F}\rho,\nabla^{F}u)+\rho\triangle_{F}u$
finally gives (38) after solving for $H$ in (39),

$\displaystyle n\cdot x^{*}H$ $\displaystyle=$ $\displaystyle=$
$\displaystyle\mathrm{div}_{F}(\rho\nabla^{F}u)+\left(\left(\frac{1}{\rho(f\circ
x)^{2}}+\rho|\nabla^{F}u|^{2}_{g_{F}}\right)n+\rho|\nabla^{F}u|^{2}_{g_{F}}\right)(\partial_{t}\log
f)\circ x$ $\displaystyle+n\rho g_{F}(\nabla^{F}(\log f\circ x),\nabla^{F}u)$
$\displaystyle=$ $\displaystyle\mathrm{div}\,_{F}(\rho\nabla^{F}u)+(f\circ
x)^{2}\rho\left(n+\frac{|\nabla^{F}u|^{2}_{g_{F}}}{(f\circ
x)^{2}}\right)(\partial_{t}\log f)\circ x+n\rho g_{F}(\nabla^{F}\log(f\circ
x),\nabla^{F}u).$

∎

######  Remark 3.

It is immediate to check that relation (38) for the mean curvature of graphs
over the fiber in a twisted product context reduces to the one given in [5]
for GRW spacetimes when the twisted function is assumed to depend just on the
time variable.

The results obtained in Theorem 4 for hypersurfaces in $I\times_{f}F$ whose
mean curvature satisfies $H=0$ with $H$ as in (38) are Calabi-Bernstein-type,
since the only maximal ones under the stated expanding conditions are the
slices. The precise formulation for these results in a proper Calabi-Bernstein
style reads as follows:

######  Theorem 5.

Let $(M,g)$ be a connected compact Riemannian $n$-manifold where $D$ is the
Levi-Civita connection of $g$. Let $I\subset\mathbb{R}$ be an open interval
and $f:I\times M\longrightarrow\mathbb{R}$ a positive smooth function
satisfying some of the following possibilities:

i) $\partial_{t}f\leq 0$, for all $t\in I$, or

ii) $\partial_{t}f\geq 0$, for all $t\in I$, or

iii) there exists $t_{1}\in I$ such that, $\partial_{t}f\geq 0$, for all
$t\leq t_{1}$ and $\partial_{t}f\leq 0$, for all $t\geq t_{1}$.

Then, the only solution to the elliptic non-linear problem

$\displaystyle\mathrm{div}\,(\rho Du)+(f\circ
x)^{2}\rho\left(n+\frac{|Du|^{2}}{(f\circ x)^{2}}\right)(\partial_{t}\log
f)\circ x+n\rho g_{F}(D\log(f\circ x),Du)=0,$

where $x:M\rightarrow I\times M$, $x(p)=(u(p),p)$, $\rho=\frac{1}{(f\circ
x)\sqrt{(f\circ x)^{2}-|Du|^{2}}}$ and $|Du|<f\circ x$ are the constant
functions $u=t_{0}$, with $\partial_{t}f(t_{0},p)=0$, for all $p\in M$.

##  6 Appendix

We devote this Appendix to compile elementary results within the area of
conformal geometry which are useful for this work. Specifically, given a non-
degenerate submanifold in a pseudo-Riemannian background, we show the relation
between the connections, the Weingarten operators, the mean curvatures and the
Laplacians associated to two conformally related metrics. Let us first
introduce the concept of conformal transformation.

######  Definition 3.

Let $(M,g_{1})$ be a pseudo-Riemannian manifold of dimension
$n=\mathrm{dim}M\geq 2$. A conformal transformation is defined as a
transformation of the metric tensor of the form

$g_{1}\rightarrow g_{2}=e^{2\phi}g_{1},$

where $\phi\in C^{\infty}(M)$.

In the particular case where $g_{1}$ has Lorentzian signature, these
transformations preserve the angles between geometrical objects, as well as
null geodesics and light cones. The following result establishes the
preservation of non-degenerate totally umbilical submanifolds in pseudo-
Riemannian contexts under the action of a global conformal transformation.

######  Proposition 10.

Let $\psi:\Sigma\rightarrow(M,g_{1})$ be a non-degenerate totally umbilic
submanifold of arbitrary codimension immersed in a pseudo-Riemannian manifold
$(M,g_{1})$. Let us consider the conformal transformation defined by

$g_{2}=e^{2\phi}\,g_{1},$ (40)

with $\phi\in C^{\infty}(M)$. Then $\Sigma$ is totally umbilical in $M$ when
is endowed with the metric $g_{2}$.

###### Proof.

Let us denote by $\nabla^{(2)}$ the Levi-Civita connection of the metric
$g_{2}$. For any $X,Y\in\mathfrak{X}(\Sigma)$, $\nabla^{(2)}$ is related to
the ambient connection $\nabla^{(1)}$ by the following formula (see for
instance [26], pg. 132):

$\nabla^{(2)}_{X}Y=\nabla^{(1)}_{X}Y+g_{1}(X,\nabla^{(1)}\phi)Y+g_{1}(Y,\nabla^{(1)}\phi)X-g_{1}(X,Y)\nabla^{(1)}\phi.$
(41)

Given any unit normal $N_{1}$ to $\Sigma$ in $(M,g_{1})$, relation (41) can be
applied to compute the covariant derivative of the associated unit normal
$N_{2}=e^{-\phi}N_{1}$ in $(M,g_{2})$ in terms of the geometry determined by
$g_{1}$,

$\nabla^{(2)}_{X}N_{2}=e^{-\phi}\left(\nabla^{(1)}_{X}N_{1}+g_{1}(N_{1},\nabla^{(1)}\phi)X\right).$

Hence, the relation between the shape operators $A_{1}$ and $A_{2}$ associated
to $N_{1}$ and $N_{2}$ becomes

$A_{2}(X)=e^{-\phi}\left(A_{1}(X)-g_{1}(N_{1},\nabla^{(1)}\phi)\right).$ (42)

The shape operator of the totally umbilic non-degenerate submanifold
$\psi:\Sigma\rightarrow(M,g_{1})$ reads $A_{1}(X)=-H_{1}\textrm{Id}$, with
$H_{1}$ the mean curvature of $\psi:\Sigma\rightarrow(M,g_{1})$ with respect
to $N_{1}$, so formula (42) becomes in this case

$A_{2}(X)=e^{-\phi}\left(-H_{1}-g_{1}(N_{1},\nabla^{(1)}\phi)\right)\textrm{Id}(X).$

∎

######  Corollary 1.

Under the conformal transformation (40), the mean curvature function $H_{2}$
of $\psi:\Sigma\rightarrow(M,g_{2}=e^{2\phi}g_{1})$ with respect to $N_{2}$
becomes

$H_{2}=e^{-\phi}\left(H_{1}+g_{1}(N_{1},\nabla^{(1)}\phi)\right).$ (43)

Moreover, if the mean curvature $H_{1}$ of the hypersurface
$\psi:\Sigma\rightarrow(M,g_{1})$ satisfies the equation

$H_{1}=-g_{1}(N_{1},\nabla^{(1)}\phi),$

then $H_{2}=0$.

The relation between conformal Laplacians can be easily found in the
literature (see for example [11]). For the sake of completeness of this text,
we add in this Appendix a proof for this result.

######  Proposition 11.

Let $(M,g_{1})$ be a pseudo-Riemannian manifold of dimension
$n=\mathrm{dim}\,M\geq 2$. Under the conformal transformation determined by
$g_{2}=e^{2\phi}g_{1}$ for some $\phi\in C^{\infty}(M)$, the associated
Laplacians $\triangle_{(1)}$ and $\triangle_{(2)}$ satisfy the relation

$\triangle_{(2)}h=e^{-2\phi}(\triangle_{(1)}h+(n-2)g_{1}(\nabla^{(1)}\phi,\nabla^{(1)}h))$
(44)

for any $h\in C^{\infty}(M)$.

###### Proof.

Let $\\{x^{i}\\}_{i=1}^{n}$ be a system of local coordinates on $M$. The
Laplacian of $h$ with respect to $g_{2}$ can be computed by the well-known
formula (read for instance [48])

$\triangle_{(2)}h=\frac{1}{\sqrt{\det
g_{2}}}\sum_{i=1}^{n}\partial_{i}\left(\sqrt{\det
g_{2}}\,(\nabla^{(2)}h)^{i}\right),$ (45)

where $(\nabla^{(2)}h)^{i}$ stands for the i-component of $\nabla^{(2)}h$ in
the associated coordinate basis $\\{\partial/\partial x^{i}\\}_{i=1}^{n}$. The
determinants of both metrics satisfy $\mathrm{det}g_{2}=e^{2\phi
n}\mathrm{det}g_{1}$, which inserted into (45) gives

$\displaystyle\triangle_{(2)}h$ $\displaystyle=$
$\displaystyle\frac{1}{e^{\phi
n}\sqrt{\mathrm{det}g_{1}}}\sum_{i=1}^{n}\partial_{i}\left(e^{\phi(n-2)}\sqrt{\mathrm{det}g_{1}}\,\,(\nabla^{(1)}h)^{i}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{e^{\phi
n}\sqrt{\mathrm{det}g_{1}}}\sum_{i=1}^{n}e^{\phi(n-2)}\left((n-2)\partial_{i}\phi\sqrt{\mathrm{det}g_{1}}\,\,(\nabla^{(1)}h)^{i}+\partial_{i}(\sqrt{\mathrm{det}g_{1}}\,\,(\nabla^{(1)}h)^{i})\right)$
$\displaystyle=$ $\displaystyle
e^{-2\phi}(\triangle_{(1)}h+(n-2)g_{1}(\nabla^{(1)}\phi,\nabla^{(1)}h)).$

∎

## References

  * [1] S. Akamine, A. Honda, M. Umehara, K. Yamada, “Bernstein-type theorems for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space”, Bull. Braz. Math. Soc., New Series 52, 175-181 (2021).
  * [2] A.L. Albujer, L.J. Alías, “Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces”, J. Geom. Phys. 59(5), 620-631) (2009).
  * [3] J.A. Aledo, L.J. Alías, “Curvature properties of compact spacelike hypersurfaces in De Sitter space”, Differential Geom. Appl. 14(2), 137-149) (2001).
  * [4] J.A. Aledo, R.M. Rubio, “Constant mean curvature spacelike hypersurfaces in Lorentzian warped products”, Advances in Math. Physics 2015, 5pp (2015).
  * [5] J.A. Aledo, R.M. Rubio, J.J Salamanca, “Complete spacelike hypersurfaces in generalized Robertson-Walker and the null convergence condition: Calabi-Bernstein problems”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 111, 115-128 (2016).
  * [6] J.A. Aledo, R.M. Rubio, J.J Salamanca, “ Space-like hypersurfaces with functionally bounded mean curvature in Lorentzian warped products and generalized Calabi-Bernstein-type problems”, Proc. Roy. Soc. Edinburgh A. 149, 849-868 (2019).
  * [7] L.J. Alías, A. Romero, M. Sánchez, “Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes”, Gen. Rel. Grav. 27, 71-84 (1995).
  * [8] J.K. Beem, P.E. Ehrlich and K.L. Easley, Global Lorentzian Geometry, second edition, Pure and Applied Math. 202, Marcel Dekker, New York (1996).
  * [9] R. Bartnik, “Remarks on cosmological spacetimes and constant mean curvature surfaces”, Commun. Math. Phys. 117, 615-624 (1988).
  * [10] R. Bartolo, E. Caponio, A. Pomponio, “Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime”, Proc. Amer. Math. Soc. 149, 5139-5151 (2021).
  * [11] A.L. Besse, Einstein Manifolds, Springer (1987).
  * [12] R.L. Bishop, “Clairaut submersions”, in Differential geometry; in honour of Kentaro Yano, Kinokuniya, Tokyo , 21-31 (1972).
  * [13] E.Bombieri, E. De Giorgi, E.Giusti , “Minimal cones and the Bernstein problem”, Invent. Math. 7, 243-268 (1969).
  * [14] D. Brill, F. Flaherty, “Isolated maximal surfaces in spacetime”, Commun. Math. Phys. 50, 157-165 (1976).
  * [15] M. Caballero, A. Romero, R.M. Rubio, “Constant mean curvature spacelike hypersurfaces in Lorentzian manifolds with a timelike gradient conformal vector field”, Class. Quantum Grav. 28, 145009 (2011).
  * [16] E.Calabi, “Examples of Bernstein problems for some nonlinear equations”, Proc. Symp. Pure Math. 15, 223-230 (1970).
  * [17] F. Camargo, A. Caminha, H. de Lima, “Bernstein-type theorems in Semi-Riemannian warped products”, Proc. Amer. Math. Soc. 139(5), 1841-1850 (2010).
  * [18] B.Y. Chen, “Totally umbilical submanifolds”, Soochow J. Math. 5, 9-37 (1979).
  * [19] B.Y. Chen, Geometry of submanifolds and applications, Science University of Tokyo (1981).
  * [20] B.Y. Chen, “A simple characterization of generalized Robertson-Walker spacetimes”, Gen. Rel. Grav. 46, 1833, 5pp (2014).
  * [21] B.Y. Chen, Differential geometry of warped product manifolds and submanifolds, World Scientific (2017).
  * [22] B.Y. Chen, “Rectifying submanifolds of Riemannian manifolds and torqued vectors”, Kragujevac Journal of Mathematics 41(1), 93-103 (2017).
  * [23] B.Y. Chen, F. Dillen, L. Verstraelen, L. Vrancken, “Lagrangian isometric immersions of a real-space-form $M^{n}(c)$ into a complex-space-form $\tilde{M}^{n}(4c)$”, Math. Proc. Cambridge Philos. Soc. 124, 107-125 (1998).
  * [24] S.Y. Cheng, S.T. Yau, “Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces”, Ann. Math. 104, 407-419 (1976).
  * [25] Y. Choquet-Bruhat, R.Geroch, “Global aspects of the Cauchy problem in General Relativity”, Commun. Math. Phys. 14, 329-335 (1969).
  * [26] M. Dajczer, Submanifolds and isometric immersions, Publish or Perish (1990).
  * [27] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ (1976).
  * [28] M.P. Do Carmo, Riemannian Geometry, Birkhäuser (1992).
  * [29] A. Fialkow, “Conformal geodesics”, Trans. Amer. Math. Soc. 45(3), 443-473 (1939).
  * [30] M. Fernández-López, E. García-Río, D.N. Kupeli, B. Ünal, “A curvature condition for a twisted product to be a warped product”, Manuscripta Mathematica 106, 213-217 (2001).
  * [31] G.J. Galloway, E. Ling, “Existence of CMC Cauchy surfaces from a spacetime curvature condition”, Gen. Rel. Grav. 50, 7pp (2018).
  * [32] N. Koike, “The decomposition of curvature netted hypersurfaces”, Geom. Dedicata 54, 1-11 (1995).
  * [33] A. Lichnerowicz, “ L’intégration des équations de la gravitation relativiste et le problème des n corps”, J. Math. pures et appl. 23, 37-62 (1944).
  * [34] M. Lohnherr, H. Reckziegel, “On ruled real hypersurfaces in complex space forms”, Geom. Dedicata 74, 267-286 (1999).
  * [35] C.A. Mantica, L.G. Molinari, “Generalized Robertson-Walker spacetimes-A survey”, Int. J. Geom. Methods Mod. Phys. 14(3), 1730001 (27pp) (2017).
  * [36] C.A. Mantica, L.G. Molinari, “Twisted Lorentzian manifolds, a characterization with torse-forming timelike unit vectors”, Gen. Rel. Grav. 49, 51(7pp) (2017).
  * [37] M. Mars, A. Soria, “Geometry of normal graphs in Euclidean space and applications to the Penrose inequality in Minkowski”, Annals Henri Poincaré 15, 1903-1918 (2014).
  * [38] J.A.S. Pelegrín, A. Romero, R.M. Rubio, “On uniqueness of the foliation by comoving observers restspaces of a Generalized Robertson-Walker spacetime”, Gen. Rel. Grav. 49(16), 14pp (2017).
  * [39] J.A.S. Pelegrín, A. Romero, R.M. Rubio, “An extension of Calabi’s correspondence between the solutions of two Bernstein problems to more general elliptic nonlinear equations”, Math Notes 105, 280-284 (2019).
  * [40] J.A.S. Pelegrín, A. Romero, R.M. Rubio, “Spacelike hypersurfaces in spatially parabolic standard static spacetimes and Calabi-Bernstein-type problems”, Mediterr. J. Math. 16(34), 15pp (2019).
  * [41] R. Ponge, H. Reckziegel, “Twisted Products in Pseudo-Riemannian Geometry”, Geom. Dedicata 48, 15-25 (1993).
  * [42] A.D. Rendall, “Constant mean curvature foliations in cosmological spacetimes”, Helvetica Physica Acta 69(4), 490-500 (1996).
  * [43] A. Romero, R.M. Rubio, “A nonlinear inequality involving the mean curvature of a spacelike surface in 3-dimensional GRW spacetimes and Calabi-Bernstein type problems”, Contemp. Math. 674, 141-152 (2016).
  * [44] A. Romero, M. Sánchez, “Completeness of Lorentz manifolds admitting a timelike conformal Killing vector field”, Proc. Amer. Math. Soc. 123, 2831-2833 (1995).
  * [45] A. Romero, M. Sánchez, “Bochner’s technique on Lorentzian manifolds and infinitesimal conformal symmetries”, Pacific J. Math. 186, 141-148 (1998).
  * [46] R. Rubio, “Calabi-Bernstein-Type Problems in Lorentzian Geometry”, in Lorentzian Geometry and Related Topics, Springer Proc. Math. Stat. 211, 213-236 (2017).
  * [47] M. Sánchez, “Structure of Lorentzian tori with a Killing vector field”, Trans. Amer. Math. Soc. 349, 1063-1080 (1997).
  * [48] G.F. Torres del Castillo, Differentiable manifolds, Birkhäuser (2012).