blue

# The equivalence of smooth and synthetic notions of timelike sectional
curvature bounds

Tobias Beran
Michael Kunzinger
Argam Ohanyan
Felix Rott<EMAIL_ADDRESS>Faculty of Mathematics, University of
Vienna<EMAIL_ADDRESS>Faculty of Mathematics,
University of Vienna<EMAIL_ADDRESS>Faculty of
Mathematics, University of Vienna<EMAIL_ADDRESS>Faculty of
Mathematics, University of Vienna, Austria.

###### Abstract

Timelike sectional curvature bounds play an important role in spacetime
geometry, both for the understanding of classical smooth spacetimes and for
the study of Lorentzian (pre-)length spaces introduced in [21]. In the smooth
setting, a bound on the sectional curvature of timelike planes can be
formulated via the Riemann curvature tensor. In the synthetic setting, bounds
are formulated by comparing various geometric configurations to the
corresponding ones in constant curvature spaces. The first link between these
notions in the Lorentzian context was established in [19], which was
instrumental in the proof of powerful results in spacetime geometry [6, 7,
17]. For general semi-Riemannian manifolds, the equivalence between sectional
curvature bounds and synthetic bounds was established in [2], however in this
approach the sectional curvatures of both timelike and spacelike planes have
to be considered. In this article, we fill a gap in the literature by proving
the full equivalence between sectional curvature bounds on timelike planes and
synthetic timelike bounds on strongly causal spacetimes. As an essential tool,
we establish Hessian comparison for the time separation and signed distance
functions.

_Keywords:_ Timelike sectional curvature bounds, triangle comparison, hinge
comparison

_MSC2020:_ 53B30, 53C23, 53B50

##  1 Introduction

Sectional curvature bounds are of great importance in both Riemannian and
Lorentzian geometry. Spaces whose sectional curvature is bounded enjoy an
abundance of geometric and topological properties which are usually quite
rigid, i.e. slight variations on the curvature bound do not break these
properties outside of some special cases.

In Riemannian geometry, Toponogov’s comparison theorem and related results
establish the equivalence between sectional curvature bounds and the
comparison of various geometric configurations in the original space with
those in the space of the corresponding constant sectional curvature. Some
geometric configurations considered in this context are (geodesic) triangles,
hinges, distances from a point to a geodesic, and so on. These descriptions of
sectional curvature bounds, due to their independence of differential
quantities, have led to a rich theory of metric spaces with sectional
curvature bounds, i.e. Alexandrov spaces (see [14] for an introductory text on
the subject, and the references therein).

In the Lorentzian setting, timelike sectional curvature bounds have been
considered as a global curvature condition stronger than the strong energy
condition from General Relativity. The Lorentzian timelike sectional curvature
comparison theory was initiated in [19], where it is shown that timelike
sectional curvature bounds from above imply a certain timelike/causal hinge
comparison result. The methods established there have been crucial in the
proofs of several important spacetime-geometric results, e.g. the proof of the
Lorentzian splitting theorem for spacetimes with nonpositive timelike
sectional curvature bounds containing a complete timelike line in [6, 7] and
the proof of Bartnik’s cosmological splitting conjecture [5] under the
stronger assumption of nonpositive timelike sectional curvature bounds in
[17].

In the general setting of semi-Riemannian manifolds with arbitrary (constant)
signature, an equivalence between bounds based on geometric
triangle/angle/hinge comparison and a certain notion of sectional curvature
comparison is available [2]. However, the methods employed in that work
involve both timelike and spacelike planes, and considering bounds on
spacelike planes in the Lorentzian context seems unnatural for the purposes of
General Relativity and spacetime geometry.

Recently, an approach analogous to that of Alexandrov spaces has been
initiated in the Lorentzian case (see [21]; see also [22] for a more recent
local approach). The literature on Lorentzian synthetic spaces with timelike
sectional curvature bounds is quite rich despite the field’s rather recent
initiation (see e.g. [18, 3, 12, 24, 11, 10, 8]).

However, there is the following gap in the literature: The proof of [19] only
establishes that if $(M,g)$ is a globally hyperbolic spacetime whose timelike
sectional curvature is bounded above by $K\in\mathbb{R}$, then it satisfies a
global bound based on timelike hinge comparison by the same constant $K$. The
converse direction as well as either direction for the opposite bounds do not
seem to be available in the literature. The aim of this article is to fill
this gap.

The paper is structured as follows: In Section 2, we recall the notion of
timelike sectional curvature bounds (Subsection 2.1) on strongly causal
spacetimes and establish a Hessian comparison result for the time separation
and signed distance from a point. Then, in Subsection 2.2, we discuss the
various notions of synthetic timelike sectional curvature bounds, but restrict
ourselves to strongly causal spacetimes (instead of general Lorentzian
synthetic spaces) for the convenience of the reader. In Section 3, we
establish the aforementioned equivalence between smooth and synthetic timelike
sectional curvature bounds, the direction synthetic $\Rightarrow$ smooth in
Subsection 3.1 using Jacobi field computations, and the direction smooth
$\Rightarrow$ synthetic in Subsection 3.2 using Hessian comparison. In
Subsection 3.3, we show via an explicit example that timelike sectional
curvature bounds are strictly weaker than the corresponding semi-Riemannian
bounds from [2]. Finally, in Section 4, we summarize our work, give an
improved version of the spacetime Reshetnyak gluing result from [12] as an
application, and discuss some related open problems.

###  1.1 Notation and conventions

The symbol $\subset$ denotes (not necessarily strict) inclusion. Spacetime
metrics are assumed to be smooth and have signature $(-,+,\dots,+)$. The
Riemann curvature tensor of a semi-Riemannian manifold $(M,g)$ is
$R_{X,Y}Z\equiv
R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z$, where
$X,Y,Z\in\mathfrak{X}(M)$. We interchangeably write $\langle.,.\rangle$ for
the metric $g$. The number $D_{K}$ is defined to be $+\infty$ if $K\geq 0$ and
$\pi/\sqrt{-K}$ if $K<0$. A vector field $J$ along a geodesic $\gamma$ is
called a Jacobi field if
$J^{\prime\prime}+R_{J,\gamma^{\prime}}\gamma^{\prime}=0$. For linear maps
$A,B$ on a semi-Euclidean vector space $(V,\langle.,.\rangle)$, we write
$A\geq B$ if $\langle Av,v\rangle\geq\langle Bv,v\rangle$ for all $v\in V$.

##  2 Various notions of timelike sectional curvature bounds

In this section, we introduce and discuss various notions of timelike
sectional curvature bounds on smooth spacetimes. Throughout, $(M,g)$ denotes a
smooth, strongly causal spacetime of dimension $\geq 2$. The reason why we
assume strong causality is because we require in an essential way the
compatibility of the global time separation function with the local time
separation on convex sets induced by the exponential map.

###  2.1 Smooth timelike sectional curvature bounds

We start with the classical smooth notion, which involves bounds on sectional
curvatures of timelike planes. Recall that if $\\{v,w\\}$ span a non-
degenerate $2$-plane in a tangent space $T_{p}M$, then the sectional curvature
of $\Pi:=\mathrm{span}(v,w)$ is

$\displaystyle K(\Pi):=\frac{\langle R(w,v)v,w\rangle}{\langle
v,v\rangle\langle w,w\rangle-\langle v,w\rangle^{2}}$ (2.1)

Note that $K(\Pi)$ does not depend on the choice of basis vectors $\\{v,w\\}$.

######  Definition 2.1 (Smooth timelike sectional curvature bounds).

We say that $(M,g)$ has smooth timelike sectional curvature bounded above
(resp. below) by $K\in\mathbb{R}$ if for all $p\in M$ and all timelike
$2$-planes $\Pi\subset T_{p}M$, we have

$\displaystyle K(\Pi)\leq K\quad(\text{resp. }K(\Pi)\geq K).$ (2.2)

######  Remark 2.2 (On smooth timelike sectional curvature bounds).

  1.   2. (i)

Usually, one works with orthonormal bases of timelike tangent planes, i.e. a
given timelike $2$-plane $\Pi\subset T_{p}M$ is spanned by $v,w\in T_{p}M$
such that $v$ is unit timelike ($\langle v,v\rangle=-1$), $w$ is unit
spacelike ($\langle w,w\rangle=+1$) and $\langle v,w\rangle=0$. In this case,
(2.1) simplifies to

$\displaystyle K(\Pi)=-\langle R(w,v)v,w\rangle.$ (2.3)

  3. (ii)

If $(M,g)$ has smooth timelike sectional curvature bounded above (resp. below)
by $K\in\mathbb{R}$, then for all $v\in TM$ timelike

$\displaystyle\mathrm{Ric}(v,v)\geq(n-1)K\langle v,v\rangle\quad(\text{resp.
}\mathrm{Ric}(v,v)\leq(n-1)K\langle v,v\rangle).$ (2.4)

Indeed: For definiteness, suppose the bound is from above by $K$ and let $v\in
T_{p}M$ be any timelike tangent vector. W.l.o.g. we may assume that $\langle
v,v\rangle=-1$. Complete $\\{v\\}$ to an orthonormal basis
$\\{v,e_{1},\dots,e_{n-1}\\}$ of $T_{p}M$, where $n:=\dim M$. Writing
$e_{0}:=v$, we have $\langle R(e_{i},v)v,e_{i}\rangle\geq-K$ ($i=1,\dots,n-1$)
by the curvature bound, hence

$\displaystyle\mathrm{Ric}(v,v)=\sum_{i=0}^{n-1}\langle
e_{i},e_{i}\rangle\langle R(e_{i},v)v,e_{i}\rangle=\sum_{i=1}^{n-1}\langle
R(e_{i},v)v,e_{i}\rangle\geq-(n-1)K.$

  4. (iii)

In [2], sectional curvature bounds on general semi-Riemannian manifolds are
defined as follows: A semi-Riemannian manifold $(M,g)$ is said to have
sectional curvature bounded below (above) by $K\in\mathbb{R}$ if for all $p\in
M$ and all $v,w\in T_{p}M$,

$\displaystyle\langle R(w,v)v,w\rangle\geq K(\langle v,v\rangle\langle
w,w\rangle-\langle v,w\rangle^{2})\quad(\text{resp. }\leq).$ (2.5)

Note that a sectional curvature bound below (above) by $K$ in this sense
implies a smooth timelike sectional curvature bound above (below) by $K$.

Due to Remark 2.2(ii), a smooth timelike sectional curvature bound from above
(resp. below) by $K\in\mathbb{R}$ corresponds geometrically (viewed from the
perspective of positive definite, i.e. Riemannian, geometry) to a “curvature
bound” from below (resp. above) by $-K\in\mathbb{R}$. This has been (at least
partially) implemented into the definitions of synthetic timelike sectional
curvature bounds, as we will discuss later. Let us now turn to an important
consequence of smooth timelike sectional curvature bounds, namely Hessian
comparison for the time separation and signed distance. Related results are
readily available in the literature (cf. [2, 26]), but we do not know of any
source which contains this exact analogue of the well-known Riemannian result.

We will need the notion of _signed distance_ from [2]: Given a convex set
$U\subset M$ and $p,q\in U$, the signed distance between $p$ and $q$ is
$|pq|_{\pm}:=\pm|\exp_{p}^{-1}(q)|_{g}$, where the sign is “$-$” if $p\ll q$
or $q\ll p$, and “$+$” otherwise.

######  Definition 2.3 (Modified distance functions).

Given $K\in\mathbb{R}$, the modified distance function
$\mathrm{md}^{K}_{\tau}:[0,D_{K})\to\mathbb{R}$ is defined as

$\displaystyle\mathrm{md}^{K}_{\tau}(t):=\begin{cases}\frac{t^{2}}{2}&K=0,\\\
\frac{\cosh(\sqrt{K}t)-1}{K}&K>0,\\\
\frac{\cos(\sqrt{-K}t)-1}{K}&K<0.\end{cases}$ (2.6)

This definition is tailored to the time separation function. For some of our
results it will be necessary to switch to the signed distance. This is why we
introduce the _extended modified distance function_
$\mathrm{md}^{K}_{S}:(-D_{K},D_{-K})\to\mathbb{R}$

$\displaystyle\mathrm{md}^{K}_{S}(t):=\begin{cases}\mathrm{md}^{-K}_{\tau}(t)&t\geq
0,\\\ -\mathrm{md}^{K}_{\tau}(-t)&t<0.\end{cases}$ (2.7)

This definition reflects the anti-isometry between de Sitter space and anti-de
Sitter space (resp. from Minkowski space to itself): it sends spacelike
vectors in de Sitter space to timelike vectors of the same length in anti-de
Sitter space and vice versa.

It is easy to check that on $(-D_{K},0]$, $\mathrm{md}^{K}_{S}$ is the unique
solution of the initial value problem

$\displaystyle\begin{cases}(\mathrm{md}^{K}_{S})^{\prime\prime}-K\mathrm{md}^{K}_{S}=-1,\\\
\mathrm{md}^{K}_{S}(0)=0,\\\ (\mathrm{md}^{K}_{S})^{\prime}(0)=0.\end{cases}$

and on $[0,D_{-K})$, $\mathrm{md}^{K}_{S}$ is the unique solution of the
initial value problem

$\displaystyle\begin{cases}(\mathrm{md}^{K}_{S})^{\prime\prime}+K\mathrm{md}^{K}_{S}=1,\\\
\mathrm{md}^{K}_{S}(0)=0,\\\ (\mathrm{md}^{K}_{S})^{\prime}(0)=0.\end{cases}$

######  Theorem 2.4 (Lorentzian Hessian comparison).

Let $(M,g)$ be a strongly causal spacetime with smooth timelike sectional
curvature bounded above (resp. below) by $K\in\mathbb{R}$, $p\in M$ and $U$ a
convex neighborhood of $p$. Then we have the following inequalities for
$(1,1)$-Hessians:

  1. (i)

On $I^{+}(p)\cap U$, for _any_ vector $v$, we have

$\left<\mathrm{Hess}\,\mathrm{md}^{K}_{S}(|p\cdot|_{\pm})(v),v\right>\leq(1-K\mathrm{md}^{K}_{S}(|p\cdot|_{\pm}))\left<v,v\right>\quad(\text{resp.
}\geq)\,,$ (2.8)

or equivalently

$\mathrm{Hess}\,\tau(p,\cdot)\geq-\frac{(\mathrm{md}^{K}_{\tau})^{\prime\prime}(\tau(p,\cdot))}{(\mathrm{md}^{K}_{\tau})^{\prime}(\tau(p,\cdot))}\pi_{\tau(p,.)}\quad(\text{resp.
}\leq)\,,$ (2.9)

where $\pi_{\tau(p,.)}$ denotes the projection onto the tangent spaces of
level sets of $\tau(p,.)$, more concretely
$\pi_{\tau(p,.)}=\mathrm{Id}+\langle\nabla\tau(p,.),.\rangle\nabla\tau(p,.)$.

If, in addition, $(M,g)$ is globally hyperbolic if $K\geq 0$ or order
($\sqrt{-K}$)-globally hyperbolic111Global hyperbolicity of order $\sqrt{-K}$
means that for any $x\ll y$ with $\tau(x,y)<D_{K}$ the causal diamond $J(x,y)$
is compact. This notion was introduced in [19]. if $K<0$, then these
inequalities hold on $I^{+}(p)\setminus\mathrm{Cut}^{+}(p)$ if $K\geq 0$, and
on $\\{q\in I^{+}(p):\tau(p,q)<D_{K}\\}\setminus\mathrm{Cut}^{+}(p)$ if $K<0$.

  2. (ii)

On all of $U$, for all _timelike_ vectors $v$, we have

$\left<\mathrm{Hess}\,\mathrm{md}^{K}_{S}(|p\cdot|_{\pm})(v),v\right>\leq(1-K\mathrm{md}^{K}_{S}(|p\cdot|_{\pm}))\left<v,v\right>\quad(\text{resp.
}\geq)\,.$ (2.10)

The time reversed version of (i) also holds.

###### Proof.

The proof proceeds by adapting arguments from [2], to which we refer for
notations used below.

For (i), note that the proof of [2, Prop. 5.2] only uses curvature comparison
in applying [2, Cor. 4.6], and the proof of that result only uses
$R_{\sigma^{\prime}}\geq\tilde{R}_{\tilde{\sigma}^{\prime}}$ where $\sigma$ is
timelike in the case we are considering.

For the part of (ii) that is not covered by (i), we only have $\langle
R_{\sigma^{\prime}}(v),v\rangle\geq\langle\tilde{R}_{\tilde{\sigma}^{\prime}}(v),v\rangle$
for timelike vectors $v$. We replace [2, Cor. 4.5] by claiming that $\langle
R_{\sigma^{\prime}}(v),v\rangle\geq\langle\tilde{R}_{\tilde{\sigma}^{\prime}}(v),v\rangle$
for all timelike vectors $v$ implies $\langle
S_{K,q}(v),v\rangle\leq\langle\tilde{S}_{K,q}(v),v\rangle$ for all timelike
vectors $v$. Note that the proof of [2, Cor. 4.5] still works up to [2, Thm.
4.3], which we again have to replace by the claim that $\langle
R_{1}(v),v\rangle\leq\langle R_{2}(v),v\rangle$ for all timelike vectors $v$
implies $\langle S_{1}(v),v\rangle\geq\langle S_{2}(v),v\rangle$ for all
timelike vectors $v$. To obtain this, in the proof of [2, Thm. 4.3] we have to
assume that the vector $x_{0}$ with
$\langle(S_{1}(t_{0})-S_{\delta}(t_{0}))x_{0},x_{0}\rangle=0$ is timelike.
Then indeed the proof of [2, Thm. 4.3] gives our desired inequality between
$S_{1}$ and $S_{2}$. ∎

######  Remark 2.5 (D’Alembertian comparison).

Suppose that $(M,g)$ has smooth timelike sectional curvature bounded above
(resp. below) by $K\in\mathbb{R}$. By taking the trace of the inequality
(2.9), one obtains the well-known d’Alembertian comparison inequality

$\displaystyle\square\tau(p,.)\geq-(n-1)\frac{(\mathrm{md}^{K}_{\tau})^{\prime\prime}(\tau(p,.))}{(\mathrm{md}^{K}_{\tau})^{\prime}(\tau(p,.))}\quad(\text{resp.\
}\leq),$ (2.11)

which is known to hold under the weaker assumption of
$\mathrm{Ric}(v,v)\geq(n-1)K\langle v,v\rangle$ (resp. $\leq$) for all $v\in
TM$ timelike (see e.g. [25, Thm. 3.3.5]).

###  2.2 Synthetic timelike sectional curvature bounds

Let us now move on to a discussion of various synthetic notions of timelike
sectional curvature bounds. The study of such notions goes back to [21], where
Lorentzian (pre-)length spaces were introduced as synthetic analogues of
spacetimes. There, synthetic timelike curvature bounds were studied via
timelike and causal triangles. Since then, notions based on angles, hinges,
four-point conditions and concavity/convexity of the time separation have been
introduced [13, 4, 9]. In [9], the authors study the interrelationship between
these various synthetic notions of timelike curvature bounds, and show the
equivalence of all these notions under mild assumptions. Since we restrict
ourselves to the study of classical spacetimes in this article, we will state
all of the definitions and results obtained in the aforementioned papers only
for spacetimes. As before, $(M,g)$ is a strongly causal spacetime of dimension
$n:=\dim M\geq 2$. Let us first introduce some preliminary notions.

######  Definition 2.6 (Constant curvature spaces).

We denote by $\mathbb{L}^{2}(K)$ the simply connected $2$-dimensional
Lorentzian space form of constant sectional curvature $K$, see [21, Def. 4.5].
The number $D_{K}$ is the timelike diameter of its maximal globally hyperbolic
subsets. We will denote by $\bar{\tau}$ the time separation function on
$\mathbb{L}^{2}(K)$.

Next, we introduce the synthetic notions needed to define synthetic timelike
sectional curvature bounds.

######  Definition 2.7 (Triangles, angles, hinges).

Let $U\subset M$ be a convex set.

  1. (i)

A timelike triangle in $U$ is a triple $(p,q,r)\in U^{3}$ such that $p\ll q\ll
r$. A causal triangle in $U$ is a triple $(p,q,r)\in U^{3}$ such that $p\ll
q\leq r$ or $p\leq q\ll r$. We will also speak of timelike or causal triangles
when we mean the connected radial geodesics between $p,q,r$, and we will write
$\Delta(p,q,r)\subset U$, in which case we refer to the connecting geodesics
as the sides of the triangle. Triangle will always refer to timelike or causal
triangles.

  2. (ii)

A comparison triangle for a triangle $\Delta(p,q,r)$ in $U$ is a triple
$(\bar{p},\bar{q},\bar{r})$ in $\mathbb{L}^{2}(K)$ with
$\tau(p,q)=\bar{\tau}(\bar{p},\bar{q})$,
$\tau(q,r)=\bar{\tau}(\bar{q},\bar{r})$ and
$\tau(p,r)=\bar{\tau}(\bar{p},\bar{r})$. When $K<0$, we suppose that $U$ is
chosen small enough such that $\sup_{p,q\in U}\tau(p,q)<D_{K}$. This ensures
that $\Delta(\bar{p},\bar{q},\bar{r})$ is a triangle in a convex subset of
$\mathbb{L}^{2}(K)$. With these conventions, for any triangle
$\Delta(p,q,r)\subset U$ there exists a unique (up to isometry) comparison
triangle in $\mathbb{L}^{2}(K)$ (cf. [2, Lem. 2.1]). Given $x\in\Delta(p,q,r)$
on a timelike side, we refer to the unique point $\bar{x}$ on the
corresponding timelike side of $\Delta(\bar{p},\bar{q},\bar{r})$ with the same
time separation to the endpoints of the side as the corresponding point.

  3. (iii)

Given a triangle $\Delta(p,q,r)\subset U$ such that $p\ll q$ and its
comparison triangle $\Delta(\bar{p},\bar{q},\bar{r})\subset\mathbb{L}^{2}(K)$,
the $K$-comparison angle at $p$ is

$\displaystyle\tilde{\measuredangle}_{p}^{K}(q,r):=\measuredangle_{\bar{p}}^{\mathbb{L}^{2}(K)}(\bar{q},\bar{r})=\mathrm{arcosh}(|\langle\gamma_{\bar{p}\bar{q}}^{\prime}(0),\gamma_{\bar{p}\bar{r}}^{\prime}(0)\rangle|),$

where $\gamma_{\bar{p}\bar{q}}$ and $\gamma_{\bar{p}\bar{r}}$ are the unique
$\bar{\tau}$-arclength parametrized timelike maximizing geodesics in
$\mathbb{L}^{2}(K)$ from $\bar{p}$ to $\bar{q}$ and $\bar{p}$ to $\bar{r}$,
respectively. The sign $\sigma$ of a $K$-comparison angle is the sign of the
inner product between the tangent vectors of the geodesics above. The signed
$K$-comparison angle is defined as the product between the $K$-comparison
angle and its sign.

  4. (iv)

Given $p\in U$ and $\alpha,\beta$ unit speed future or past directed timelike
geodesics from $p$ contained in $U$, the angle at $p$ between $\alpha$ and
$\beta$ is
$\measuredangle_{p}(\alpha,\beta):=\mathrm{arcosh}(|\langle\alpha^{\prime}(0),\beta^{\prime}(0)\rangle|)$.
It is precisely the limit of $K$-comparison angles
$\tilde{\measuredangle}_{p}^{K}(\alpha(s),\beta(t))$ as $(s,t)\to(0,0)$ with
$(s,t)$ in a set such that $\Delta(p,\alpha(s),\beta(t))$ is a timelike (or
causal) triangle, independently of $K\in\mathbb{R}$ [13]. The sign $\sigma$ of
an angle and the signed angle $\measuredangle^{S}_{p}(\alpha,\beta)$ are
defined in the analogous way.

  5. (v)

A hinge in $U$ is a point $p\in U$ together with two radial timelike geodesics
$\alpha,\beta$ from $p$. A comparison hinge in $\mathbb{L}^{2}(K)$ is a point
$\bar{p}$ together with two radial timelike geodesics
$\bar{\alpha},\bar{\beta}$ from $p$ of the same length and same time
orientation as $\alpha,\beta$, such that
$\measuredangle_{p}(\alpha,\beta)=\measuredangle_{\bar{p}}(\bar{\alpha},\bar{\beta})$.

######  Definition 2.8 (Synthetic timelike sectional curvature bounds).

We say that $(M,g)$ has synthetic timelike sectional curvature bounded below
(resp. above) by $K\in\mathbb{R}$ in any of the following senses if $M$ is
covered by convex sets $U$ (if $K<0$, suppose in addition
$\sup_{q_{1},q_{2}\in U}\tau(q_{1},q_{2})<D_{K}$) such that the corresponding
properties listed below are satisfied:

  1. (i)

Timelike triangle comparison: Given any timelike triangle
$\Delta(p,q,r)\subset U$ and its comparison triangle
$\Delta(\bar{p},\bar{q},\bar{r})\subset\mathbb{L}^{2}(K)$, for any $x,y$ on
the sides and corresponding points $\bar{x},\bar{y}$, we have

$\displaystyle\tau(x,y)\leq\bar{\tau}(\bar{x},\bar{y})\quad(\text{resp.\
}\geq).$ (2.12)

  2. (ii)

Monotonicity comparison: Given two radial timelike geodesics $\alpha:[0,a]\to
U$ and $\beta:[0,b]\to U$ starting at $x:=\alpha(0)=\beta(0)$, the function
$\theta(s,t):=\tilde{\measuredangle}_{x}^{K,S}(\alpha(s),\beta(t))$, defined
for all $(s,t)\in(0,a]\times(0,b]$ such that $\alpha(s)\leq\beta(t)$, is
monotonically increasing in both $s$ and $t$ (resp. decreasing).

  3. (iii)

Angle comparison: Given two radial timelike geodesics $\alpha:[0,a]\to U$ and
$\beta:[0,b]\to U$ starting at $x:=\alpha(0)=\beta(0)$ such that
$\alpha(a)\leq\beta(b)$, we have

$\displaystyle\measuredangle^{S}_{x}(\alpha,\beta)\leq\tilde{\measuredangle}_{x}^{K,S}(\alpha(a),\beta(b))\quad(\text{resp.\
}\geq).$ (2.13)

  4. (iv)

Hinge comparison: Given two radial timelike geodesics $\alpha:[0,a]\to U$ and
$\beta:[0,b]\to U$ starting at $x:=\alpha(0)=\beta(0)$, consider a comparison
hinge $(\bar{\alpha},\bar{\beta})$ in $\mathbb{L}^{2}(K)$. Then

$\displaystyle\tau(\alpha(a),\beta(b))\geq\bar{\tau}(\bar{\alpha}(a),\bar{\beta}(b))\quad(\text{resp.\
}\leq).$ (2.14)

  5. (v)

Causal triangle comparison: Given any causal triangle $\Delta(p,q,r)\subset U$
and its comparison triangle
$\Delta(\bar{p},\bar{q},\bar{r})\subset\mathbb{L}^{2}(K)$, for any $x,y$ on
the sides of $\Delta(x,y,z)$ and corresponding points
$\bar{x},\bar{y}\in\Delta(\bar{p},\bar{q},\bar{r})$ we have

$\displaystyle\tau(p,q)\leq\bar{\tau}(\bar{x},\bar{y})\quad(\text{resp.\
}\geq).$ (2.15)

  6. (vi)

Convexity/concavity of the time separation: Given any $p\in U$ and any
timelike geodesic segment $\gamma:[a,d]\to U$, the function
$f:D\to\mathbb{R}$, where $D:=\\{t\in[a,d]:\gamma(t)\in J^{+}(p)\cup
J^{-}(p)\\}$, defined by

$\displaystyle
f(t):=\begin{cases}\mathrm{md}^{K}_{\tau}(\tau(p,\gamma(t)))&p\leq\gamma(t),\\\
\mathrm{md}^{K}_{\tau}(\tau(\gamma(t),p))&\gamma(t)\leq p,\end{cases}$ (2.16)

can be extended to all of $[a,d]$ in such a way that it satisfies the
following differential inequality in the distributional sense:

$\displaystyle f^{\prime\prime}-Kf\geq 1\quad(\text{resp.\ }\leq).$ (2.17)

######  Theorem 2.9 (Equivalence of synthetic bounds).

Let $(M,g)$ be a strongly causal spacetime. Then all of the notions of
synthetic timelike sectional curvature bounds below (resp. above) by
$K\in\mathbb{R}$ introduced in Definition 2.8 are equivalent.

###### Proof.

This follows from [9, Thm. 5.1] by noting that the Lorentzian pre-length space
structure induced by a strongly causal spacetime is regular, locally causally
closed and locally causally $D_{K}$-geodesic in the sense of the definitions
appearing in that reference. ∎

######  Remark 2.10 (On synthetic bounds).

For simplicity of presentation, we did not include all notions of synthetic
timelike sectional curvature bounds discussed in [9]. There the authors
consider (in addition to what we have presented in Definition 2.8) one-sided
versions of the various conditions, e.g. in the triangle comparisons the time
separation from an arbitrary point to a vertex, four-point configurations, as
well as strict versions of curvature bounds, which amount to comparing the
extended time separation function $l$ which is defined as $l(x,y):=\tau(x,y)$
if $x\leq y$ and $l(x,y):=-\infty$ otherwise. All of these versions are again
equivalent among each other and to the ones we have discussed on strongly
causal spacetimes. We refer to [9] for details.

Let us close our discussion on synthetic timelike sectional curvature bounds
by addressing the question of globalization of synthetic timelike curvature
bounds by $K\in\mathbb{R}$. To simplify the discussion, for the remainder of
this section we restrict our attention to globally hyperbolic spacetimes if
$K\geq 0$ and order ($\sqrt{-K}$)-globally hyperbolic spacetimes if $K<0$. We
are interested in the following question: When are the comparison conditions
in Definition 2.8 satisfied for arbitrarily large synthetic objects instead of
local ones in a convex neighborhood? This question has been answered for upper
curvature bounds in [10] and for lower curvature bounds in [8] in the more
general context of Lorentzian (pre-)length spaces. We summarize the main
results in the following Theorem. By global synthetic timelike sectional
curvature bounds above resp. below by $K\in\mathbb{R}$ we will mean that any
of the conditions in Definition 2.8 hold for arbitrarily large objects, e.g.
global timelike triangle comparison means for any global timelike triangle
(i.e. choice of points $p\ll q\ll r$, together with a choice of maximizing
timelike geodesics between them, such that there exists a comparison triangle
in $\mathbb{L}^{2}(K)$) the comparison inequality holds, and so on. All of the
global versions of the notions described in Definition 2.8 are again
equivalent due to [9, Thm. 5.1] (although that result only refers to local
bounds, the arguments used in its proof apply to global comparison situations
just as well, given our global hyperbolicity assumptions on $M$).

######  Theorem 2.11 (Globalization of synthetic timelike sectional curvature
bounds, [10, 8]).

Let $K\in\mathbb{R}$ and let $(M,g)$ be a globally hyperbolic spacetime if
$K\geq 0$ and an order ($\sqrt{-K}$)-globally hyperbolic spacetime if $K<0$.

  1. (i)

Suppose $(M,g)$ has a synthetic timelike sectional curvature bound from above
by $K\in\mathbb{R}$ in any of the equivalent senses of Definition 2.8. In
addition, suppose the following conditions hold:

     * $\bullet$

For any $(x,y)\in\tau^{-1}((0,D_{K}))$ there exists a _unique_ timelike
maximizing geodesic $\gamma_{xy}$ connecting them (parametrized proportional
to $\tau$-arclength on $[0,1]$).

     * $\bullet$

The geodesic map $\tau^{-1}((0,D_{K}))\times[0,1]\to M$,
$(x,y,t)\mapsto\gamma_{xy}(t)$ is continuous.

Then $(M,g)$ has global synthetic timelike sectional curvature bounded above
by $K$.

  2. (ii)

Suppose $(M,g)$ has synthetic timelike sectional curvature bounded below by
$K\in\mathbb{R}$ in any of the equivalent senses of Definition 2.8. Then
$(M,g)$ has global synthetic timelike sectional curvature bounded below by
$K$.

##  3 Equivalence of timelike sectional curvature bounds

In this section, we give the proof of the equivalence between smooth and
synthetic timelike sectional curvature bounds in full detail. We also show by
an explicit example that smooth timelike sectional curvature bounds are
strictly weaker than the semi-Riemannian bounds described in [2].

###  3.1 From synthetic to smooth bounds

######  Theorem 3.1.

Let $(M,g)$ be a strongly causal spacetime with synthetic timelike sectional
curvature bounded below (resp. above) by $K\in\mathbb{R}$ in any of the
equivalent senses of Definition 2.8. Then $(M,g)$ has smooth timelike
sectional curvature bounded above (resp. below) by $K\in\mathbb{R}$ in the
sense of Definition 2.1.

###### Proof.

The proof follows ideas in [2], but we have to modify the setup to use
timelike hinge comparison. Let $p\in M$ and let $\Pi\subset T_{p}M$ be a
timelike plane. Let $\\{v,w\\}$ be an orthonormal basis of $\Pi$, with $v$
timelike. Consider the future directed timelike unit speed geodesic $\gamma$
defined by the initial conditions $\gamma(0)=p$, $\gamma^{\prime}(0)=v$. We
extend it to a geodesic variation as follows: It is easy to check that $2v-w$
is a future directed timelike vector, and we define

$h(s,t):=\exp_{p}(t(v+s(2v-w)))$

Clearly, $h$ is a geodesic variation of $\gamma$ by future directed timelike
geodesics, thus the variation field $J(t):=\partial_{s}|_{s=0}h(s,t)$ is a
Jacobi field along $\gamma$ satisfying $J(0)=0$, $J^{\prime}(0)=2v-w$.

Similarly, in $\mathbb{L}^{2}(K)$, consider a future directed timelike
$\tau$-unit speed geodesic $\tilde{\gamma}$ with
$\tilde{p}:=\tilde{\gamma}(0)$, $\tilde{v}:=\tilde{\gamma}^{\prime}(0)$, and
let $\tilde{w}\in T_{\tilde{p}}\mathbb{L}^{2}(K)$ be a unit spacelike vector
orthogonal to $\tilde{v}$. We define the analogous geodesic variation in
$\mathbb{L}^{2}(K)$ via

$\tilde{h}(s,t)=\exp_{\tilde{p}}(t(\tilde{v}+s(2\tilde{v}-\tilde{w})))$

As before, $\tilde{h}$ is a geodesic variation of $\tilde{\gamma}$ by future
directed timelike geodesics, hence its variation field
$\tilde{J}(t):=\partial_{s}|_{s=0}\tilde{h}(s,t)$ is a Jacobi field along
$\tilde{\gamma}$ satisfying $\tilde{J}(0)=0$,
$\tilde{J}^{\prime}(0)=2\tilde{v}-\tilde{w}$. Note that for each fixed $s$,
$h(s,\cdot)$ and $\tilde{h}(s,\cdot)$ have the same (not necessarily unit)
speed. Due to this, as well as the orthonormality of $\\{v,w\\}$ and
$\\{\tilde{v},\tilde{w}\\}$, it is easy to see that
$\measuredangle_{\tilde{p}}(\tilde{\gamma},\tilde{h}(s,\cdot))=\measuredangle_{p}(\gamma,h(s,\cdot))$
and thus $(\tilde{\gamma}|_{[0,a]},\tilde{h}(s,\cdot)|_{[0,b]})$ forms a
comparison hinge to $(\gamma|_{[0,a]},h(s,\cdot)|_{[0,b]})$ (for any $a,b>0$
such that the geodesics in question exist up to these parameters).
Additionally, one may verify explicitly in $\mathbb{L}^{2}(K)$ that for small
enough $s,t>0$ we have that $\tilde{h}(s,t)\ll\tilde{\gamma}(t)$. We may thus
apply hinge comparison to obtain

$\tau(h(s,t),\gamma(t))\geq\bar{\tau}(\tilde{h}(s,t),\tilde{\gamma}(t))\quad(\text{resp.\
}\leq).$ (3.1)

By positivity of the right hand side also $h(s,t)\ll\gamma(t)$ if the
curvature is bounded from below. In the case of curvature bounded above we
cannot conclude $h(s,t)\ll\gamma(t)$, but the reverse triangle inequality
guarantees $\gamma(t)\not\ll h(s,t)$: Indeed, we have

$\underbrace{\bar{\tau}(\tilde{p},\tilde{\gamma}(t))}_{=\tau(p,\gamma(t))}\geq\underbrace{\bar{\tau}(\tilde{p},\tilde{h}(s,t))}_{=\tau(p,h(s,t))}+\underbrace{\bar{\tau}(\tilde{h}(s,t),\tilde{\gamma}(t))}_{>0}\,,$

contradicting the reverse triangle inequality
$\tau(p,h(s,t))\geq\tau(p,\gamma(t))+\tau(\gamma(t),h(s,t))$ which would need
to hold if $\gamma(t)\ll h(s,t)$ were true.

As we know $\tilde{h}(s,t)\ll\tilde{\gamma}(t)$, and either
$h(s,t)\ll\gamma(t)$ if the curvature bound is from below by $K$ or
$\gamma(t)\not\ll h(s,t)$ if it is from above by $K$, we may formulate the
above considerations via the signed distance as

$|h(s,t)\gamma(t)|_{\pm}\leq|\tilde{h}(s,t)\tilde{\gamma}(t)|_{\pm}\quad(\text{resp.
}\geq).$

This leads to the analogous inequality on the signed norm of the Jacobi
fields:

$|J(t)|_{\pm}=\lim_{s\searrow
0}\frac{|\gamma(t)h(s,t)|_{\pm}}{s}\leq\lim_{s\searrow
0}\frac{|\tilde{\gamma}(t)\tilde{h}(s,t)|_{\pm}}{s}=|\tilde{J}(t)|_{\pm}\quad(\text{resp.
}\geq)$

and the same inequality for $\langle J(t),J(t)\rangle$ and
$\langle\tilde{J}(t),\tilde{J}(t)\rangle$, i.e. $\langle
J(t),J(t)\rangle\leq\langle\tilde{J}(t),\tilde{J}(t)\rangle$.

Note that $J$ satisfies the following set of equalities:

$\displaystyle J^{\prime}(0)$ $\displaystyle=2v-w,$ $\displaystyle
J^{\prime\prime}$ $\displaystyle=-R_{J,\gamma^{\prime}}\gamma^{\prime},$
$\displaystyle J^{\prime\prime}(0)$ $\displaystyle=0,$ $\displaystyle
J^{\prime\prime\prime}$
$\displaystyle=-R_{J,\gamma^{\prime}}\gamma^{\prime}-R_{J^{\prime},\gamma^{\prime}}\gamma^{\prime},$
$\displaystyle J^{\prime\prime\prime}(0)$ $\displaystyle=-R_{2v-w,v}v.$

The analogous equalities hold for $\tilde{J}$ in $\mathbb{L}^{2}(K)$.

If we develop the inequality $\langle
J(t),J(t)\rangle\leq\langle\tilde{J}(t),\tilde{J}(t)\rangle$ in orders of $t$,
all of the terms up to order $t^{3}$ agree, and the term of order $t^{4}$
yields the following inequality, where the Riemann curvature tensor
$\tilde{R}$ on $\mathbb{L}^{2}(K)$ is of the form
$\tilde{R}(u_{1},u_{2})u_{3}=K(\langle u_{2},u_{3}\rangle u_{1}-\langle
u_{1},u_{3}\rangle u_{2})$ due to the constant sectional curvature:

$\displaystyle\langle-R_{2v-w,v}v,2v-w\rangle$
$\displaystyle\leq\langle-\tilde{R}_{2\tilde{v}-\tilde{w},\tilde{v}}\tilde{v},2\tilde{v}-\tilde{w}\rangle\quad(\text{resp.
}\geq)$ $\displaystyle=-K(\langle\tilde{v},\tilde{v}\rangle\langle
2\tilde{v}-\tilde{w},2\tilde{v}-\tilde{w}\rangle-\langle
2\tilde{v}-\tilde{w},\tilde{v}\rangle^{2}).$

Note that $\langle u_{1},u_{2}\rangle^{2}>\langle u_{1},u_{1}\rangle\langle
u_{2},u_{2}\rangle$ for any pair of linearly independent timelike vectors by
the reverse Cauchy-Schwarz inequality. Due to this and the fact that we may
replace the products on the right hand side by the corresponding ones of $v,w$
in $T_{p}M$, we conclude

$\displaystyle\frac{\langle R_{2v-w,v}v,2v-w\rangle}{\langle v,v\rangle\langle
2v-w,2v-w\rangle-\langle 2v-w,v\rangle^{2}}\leq K\quad(\text{resp. }\geq).$

Since $\\{v,2v-w\\}$ is a basis of $\Pi$, we conclude that the sectional
curvature of $\Pi$ is bounded above (resp. below) by $K$, which gives the
claim. ∎

###  3.2 From smooth to synthetic bounds

######  Theorem 3.2.

Let $(M,g)$ be a strongly causal spacetime with smooth timelike sectional
curvature bounded above (below) by $K\in\mathbb{R}$ in the sense of Definition
2.1. Then $(M,g)$ has synthetic timelike sectional curvature bounded below
(above) by $K$ in any of the equivalent senses of Definition 2.8.

###### Proof.

We verify (vi) in Definition 2.8. The proof is a consequence of Hessian
comparison for the signed distance, cf. Theorem 2.4(ii). For definiteness,
suppose that the smooth timelike sectional curvature is bounded above by $K$.
Fix $p\in M$ and let $U$ be a convex neighborhood of $p$. Let $\gamma:[0,b]\to
U$ be the unique $\tau$-arclength parametrized timelike maximizer from
$q_{-}:=\gamma(0)$ to $q_{+}:=\gamma(b)$.

Consider the function $f(t):=\mathrm{md}^{K}_{S}(|p\gamma(t)|_{\pm})$, which
is smooth on $[0,b]$: Indeed it can be written as a convergent power series in
$\langle\exp_{p}^{-1}(\gamma(t)),\exp_{p}^{-1}(\gamma(t))\rangle$, cf. [1, Eq.
(1.6)]. Since $-f$ is an extension of the function in Definition 2.8 (vi), we
are done if we can show that $f^{\prime\prime}-Kf\leq-1$. We conclude our
desired result from Theorem 2.4(ii):

$\displaystyle f^{\prime\prime}(t)$
$\displaystyle=\langle\mathrm{Hess}\,\mathrm{md}^{K}_{S}(|p\cdot|_{\pm})(\gamma^{\prime}(t)),\gamma^{\prime}(t)\rangle$
$\displaystyle\leq(1-K\mathrm{md}^{K}_{S}(|p\gamma(t)|_{\pm}))\underbrace{\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle}_{=-1}$
$\displaystyle=-1+Kf\,.$

In the case of lower timelike sectional curvature bounds, Hessian comparison
goes in the other direction and the remaining calculations can be done in the
same way. ∎

######  Remark 3.3 (Previous work).

  1.   2. (i)

In [19, Thm. 3.4], the author proves (in the language we use) that if $(M,g)$
is a globally hyperbolic (resp. order ($\sqrt{-K}$)-globally hyperbolic)
spacetime with smooth timelike sectional curvature bounded above by
$K\in\mathbb{R}$, then it satisfies a global synthetic timelike sectional
curvature bound below by $K\in\mathbb{R}$ in the sense of hinge comparison.
The restriction on sidelengths required there is included in the notion of
size bounds: configurations so large that comparison configurations in
$\mathbb{L}^{2}(K)$ do not exist are not considered. We obtain this result
from Theorem 3.2 and the globalization Theorem 2.11(ii).

  3. (ii)

In [2], the authors prove compatibility of smooth and synthetic sectional
curvature bounds for general semi-Riemannian manifolds by studying a modified
distance function and the corresponding Riccati comparison theory for its
Hessian. However, the sectional curvature bounds they use also include bounds
for spacelike planes, and their Jacobi field comparison methods use spacelike
distances, so our purely Lorentzian results are not a consequence of their
semi-Riemannian ones.

###  3.3 Timelike sectional curvature bounds and non-timelike planes

We now show via an explicit example that smooth timelike sectional curvature
bounds are strictly weaker than the semi-Riemannian sectional curvature bounds
introduced in [2]. To this end, let $(M,g)$ be an FLRW spacetime satisfying
the Einstein equations for the perfect fluid with energy density $\rho$ and
pressure $p$. Then due to the isotropy of spatial slices, the sectional
curvature of an arbitrary timelike $2$-plane $\Pi$ can be expressed as (see
[17, Sec. 3])

$\displaystyle K(\Pi)=-\frac{8\pi}{3}\left(C\frac{3(\rho+p)}{2}-\rho\right),$

where $C=C(\Pi)\geq 1$ is some constant (which can be arbitrarily large). Let
now $\rho<0$ and $p:=-\tilde{C}\rho$ (for some fixed constant $\tilde{C}\geq
1$), then $K(\Pi)\leq 0$. By arbitrariness of $\Pi$, $(M,g)$ thus has a smooth
timelike sectional curvature bound from above by $0$. However, if $e_{i}$
($i=1,2,3$) denotes an orthonormal basis of the spatial slice, then
$\tilde{\Pi}:=\mathrm{span}(e_{i},e_{j})$ is a spacelike $2$-plane, and (see
again [17, Sec. 3])

$\displaystyle K(\tilde{\Pi})=\frac{8\pi}{3}\rho<0.$

We conclude that $(M,g)$ has smooth timelike sectional curvature bounded above
by $0$, but it does not have sectional curvature bounded below by $0$ in the
sense of [2]. For an explicit example of an FLRW spacetime with $\rho<0$ and
$p=-\rho$, let $f$ be a solution of the initial value problem
$f^{\prime\prime}f-f^{\prime 2}-2=0$, $f(0)=1$, $f^{\prime}(0)=1/4$. Let
$(-\varepsilon,\varepsilon)$ be such that $f^{\prime}(t)>0$ and
$|f^{\prime}(t)|\leq 1/2$ for all $t\in(-\varepsilon,\varepsilon)$ and
consider the FLRW spacetime $(-\varepsilon,\varepsilon)\times_{f}H^{3}$. It is
then easy to check (see [23, Thm. 12.11]) that $\rho<0$ and $p=-\rho$. Note
that our example is not in contradiction to the equivalence of the semi-
Riemannian bounds for some nonpositive constant (not necessarily $0$, as the
example shows) in the sense of [2] and the strong energy condition on FLRW
spacetimes [2, Ex. 7.3]. The latter, in turn, is known to be equivalent to a
smooth timelike sectional curvature bound from above by $0$ [17, Sec. 3]. We
conclude that, on FLRW spacetimes, the strong energy condition is equivalent
to a smooth timelike sectional curvature bound from above by $0$ and a lower
bound by some nonpositive (not necessarily $0$) constant on the sectional
curvatures of spacelike tangent $2$-planes.

##  4 Conclusion & outlook

In this work, we have established the equivalence of smooth timelike sectional
curvature bounds from above (resp. below) with (local) synthetic timelike
curvature bounds below (above), see Theorems 3.1 and 3.2. We recalled that
local synthetic bounds globalize under appropriate conditions, see Theorem
2.11, thus obtaining the hinge comparison result from [19] as a consequence.
As an essential tool, we utilized the Hessian comparison estimate for the time
separation and signed distance functions, see Theorem 2.4. Finally, in
Subsection 3.3, we saw that timelike sectional curvature bounds do not imply
the corresponding semi-Riemannian sectional curvature bounds from [2].

One immediate consequence of Theorem 3.2 is a relaxation in the assumptions of
the Lorentzian Reshetnyak gluing theorem [12, Thm. 5.2.1], which states that
when gluing spacetimes with upper curvature bounds in the sense of [2] one
obtains a Lorentzian pre-length space with upper curvature bounds in the sense
of triangle comparison. In fact, due to results from [24], this can be
improved even further, one even obtains a strongly causal Lorentzian length
space. We refer to [12] for notions below not introduced in this article.

######  Theorem 4.1 (Improved version of the Lorentzian Reshetnyak gluing
theorem).

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be strongly causal spacetimes with
$\dim(M_{1})=:n\geq m:=\dim(M_{2})\geq 2$. Let $A_{1}$ and $A_{2}$ be two
closed non-timelike locally isolating subsets of $M_{1}$ and $M_{2}$,
respectively. Let $f:A_{1}\to A_{2}$ be a $\tau$-preserving and
$\leq$-preserving locally bi-Lipschitz homeomorphism which locally preserves
the signed distance. Suppose $A_{1}$ and $A_{2}$ are convex in the sense of
[12, Remark 5.1.1(iii)]. If $M_{1}$ and $M_{2}$ have smooth timelike sectional
curvature bounded _below_ 222Recall that due to the convention of reversing
the curvature inequality for timelike planes, it is now justified to talk
about lower timelike curvature bounds, as this is the correct inequality
corresponding to upper sectional curvature bounds in the sense of [2]. by
$K\in\mathbb{R}$, then the Lorentzian amalgamation $M_{1}\sqcup_{A}M_{2}$ is a
_strongly causal Lorentzian length space_ with timelike curvature bounded
above by $K$.

###### Proof.

Due to Theorem 3.2 and [21, Prop. 3.5, Ex. 3.24, Thm. 3.26 & Ex. 4.9], $M_{1}$
and $M_{2}$ are strongly causal regular (SR)-localizable Lorentzian length
spaces with timelike curvature bounded above by $K$. We want to apply [12,
Thm. 5.2.1], then [24, Thm. 4.9 & Thm. 5.11(iv)] yield the claim. The proof of
[12, Thm. 5.2.1] only uses non-timelike curvature bounds in its reference to
[12, Lem. 4.3.3], whose proof is analogous to that of [12, Lem. 4.3.1], except
that it uses one sided triangle comparison in triangles with one spacelike
side, i.e. a spacelike geodesic $\gamma$ from $p_{2}$ to $p_{3}$ in the
timelike future (or past) of a point $p_{1}$, and comparing
$\tau(p_{1},\gamma(t))$ to the corresponding length in a comparison triangle
for $p_{1}p_{2}p_{3}$.

So we only need to prove this triangle comparison. This works via Theorem
2.4(i): consider
$f(t)=\mathrm{md}^{K}_{S}(|p\gamma(t)|_{\pm})=-\mathrm{md}^{K}_{\tau}(\tau(p,\gamma(t)))$,
then

$f^{\prime\prime}(t)=\langle\mathrm{Hess}\,|p\cdot|_{\pm}(\gamma^{\prime}(t)),\gamma^{\prime}(t)\rangle\leq(1-Kf)\underbrace{\langle\gamma^{\prime}(t),\gamma^{\prime}(t)\rangle}_{=1}$

This corresponds to [20, Eq. (5.6)] (note that $h=f$ here), and from there on,
we can follow the rest of the proof of [20, Thm. 5.2.4]. ∎

Even in the setting of smooth spacetimes with timelike sectional curvature
bounds, there still remain interesting open questions, many of which would be
even more impactful under timelike Ricci bounds due to the connection to
General Relativity, but are more reasonably achievable under the stronger
sectional curvature condition. An example would be the analogue of
Cheeger–Colding’s quantitative almost splitting result [15], a proof of which
would also indicate the “correct” notion of Lorentzian Gromov–Hausdorff
convergence, which is a very present topic of conversation in the research of
Lorentzian synthetic spaces. Efforts in this direction are currently under
way. Another interesting question is whether one can find the soul of a
spacetime under timelike sectional curvature bounds, similar to the famous
Cheeger–Gromoll result [16].

## Acknowledgments

This research was funded in part by the Austrian Science Fund (FWF) [Grant DOI
10.55776/PAT1996423 and 10.55776/P33594]. For open access purposes, the
authors have applied a CC BY public copyright license to any author accepted
manuscript version arising from this submission. Argam Ohanyan is also
supported by the ÖAW-DOC scholarship of the Austrian Academy of Sciences.

## References

  * [1] S. Alexander and R. Bishop.  A cone splitting theorem for Alexandrov spaces.  Pac. J. Math., 218:1–15, 2005.
  * [2] S. B. Alexander and R. L. Bishop.  Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds.  Communications in Analysis and Geometry, 16(2):251–282, 2008\.
  * [3] S. B. Alexander, M. Graf, M. Kunzinger, and C. Sämann.  Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems.  Communications in Analysis and Geometry, to appear, arXiv:1909.09575, 2024.
  * [4] W. Barrera, L. M. de Oca, and D. A. Solis.  Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds.  General Relativity and Gravitation, 54(9):107, 2022.
  * [5] R. Bartnik.  Remarks on cosmological spacetimes and constant mean curvature surfaces.  Communications in Mathematical Physics, 117(4):615–624, 1988.
  * [6] J. Beem, P. Ehrlich, S. Markvorsen, and G. Galloway.  A Toponogov splitting theorem for Lorentzian manifolds.  In Global Differential Geometry and Global Analysis 1984, pages 1–13. Springer, 1985.
  * [7] J. K. Beem, P. E. Ehrlich, S. Markvorsen, and G. J. Galloway.  Decomposition theorems for Lorentzian manifolds with nonpositive curvature.  Journal of Differential Geometry, 22(1):29–42, 1985.
  * [8] T. Beran, J. Harvey, L. Napper, and F. Rott.  A Toponogov globalisation result for Lorentzian length spaces.  arXiv preprint arXiv:2309.12733, 2023.
  * [9] T. Beran, M. Kunzinger, and F. Rott.  On curvature bounds in Lorentzian length spaces.  Submitted, arXiv:2309.12062, 2023.
  * [10] T. Beran, L. Napper, and F. Rott.  Alexandrov’s patchwork and the Bonnet-Myers theorem for Lorentzian length spaces.  Submitted, arXiv:2302.11615, 2023.
  * [11] T. Beran, A. Ohanyan, F. Rott, and D. A. Solis.  The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.  Letters in Mathematical Physics, 113(2):48, 2023.
  * [12] T. Beran and F. Rott.  Gluing constructions for Lorentzian length spaces.  Manuscripta Mathematica, pages 1–44, 2023.
  * [13] T. Beran and C. Sämann.  Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds.  Journal of the London Mathematical Society, 2:1–58, 2023.
  * [14] D. Burago, Y. Burago, and S. Ivanov.  A Course in Metric Geometry, volume 33.  American Mathematical Soc., 2001.
  * [15] J. Cheeger and T. H. Colding.  Lower bounds on Ricci curvature and the almost rigidity of warped products.  Annals of Mathematics, 144(1):189–237, 1996.
  * [16] J. Cheeger and D. Gromoll.  On the structure of complete manifolds of nonnegative curvature.  Annals of Mathematics, 96(3):413–443, 1972.
  * [17] G. J. Galloway and E. Ling.  Existence of CMC Cauchy surfaces from a spacetime curvature condition.  General Relativity and Gravitation, 50:1–7, 2018.
  * [18] J. D. E. Grant, M. Kunzinger, and C. Sämann.  Inextendibility of spacetimes and Lorentzian length spaces.  Annals of Global Analysis and Geometry, 55(1):133–147, 2019.
  * [19] S. Harris.  A triangle comparison theorem for Lorentz manifolds.  Indiana University Mathematics Journal, 31(3):289–308, 1982.
  * [20] M. Kirchberger.  Lorentzian comparison geometry.  Master’s thesis, University of Vienna, 2018.  https://phaidra.univie.ac.at/open/o:1351312.
  * [21] M. Kunzinger and C. Sämann.  Lorentzian length spaces.  Annals of Global Analysis and Geometry, 54(3):399–447, 2018.
  * [22] E. Minguzzi and S. Suhr.  Lorentzian metric spaces and their Gromov-Hausdorff convergence.  arXiv:2209.14384, 2022.
  * [23] B. O’Neill.  Semi-Riemannian geometry with applications to relativity.  Academic press, 1983.
  * [24] F. Rott.  Gluing of Lorentzian length spaces and the causal ladder.  Classical and Quantum Gravity, 40(17), 2023.
  * [25] J.-H. Treude.  Ricci curvature comparison in Riemannian and Lorentzian geometry.  Master’s thesis, University of Freiburg, 2011.  https://freidok.uni-freiburg.de/data/8405.
  * [26] J.-H. Treude and J. D. Grant.  Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems.  Annals of Global Analysis and Geometry, 43:233–251, 2013.