Geometry of weighted Lorentz–Finsler manifolds II:
A splitting theorem
Yufeng LU∗

Ettore MINGUZZI†

Shin-ichi OHTA∗,‡

December 24, 2022

Abstract
We show an analogue of the Lorentzian splitting theorem for weighted Lorentz–
Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci
curvature satisﬁes certain completeness and metrizability conditions and includes
a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector ﬁeld of a Busemann function.
Moreover, our formulation in terms of the ϵ-range introduced in our previous work
enables us to unify the previously known splitting theorems for weighted Lorentzianmanifolds by Case and Woolgar–Wylie into a single framework.

1

Introduction

The purpose of this work is to develop an analogue of the Lorentzian splitting theorem forweighted Lorentz–Finsler manifolds. The Lorentzian version reads as follows.
Theorem 1.1 (Lorentzian splitting theorem) Let (M, g) be a connected, timelike geodesically complete spacetime which satisﬁes the strong energy condition, i.e., Ric(v) ≥ 0 for
all timelike vectors v. If (M, g) includes a timelike straight line, then it is isometric to(R × Σ, −dt2 + h), where (Σ, h) is a complete Riemannian manifold.
This splitting theorem was conjectured by Yau [Ya] in 1982 and established by Newman[Ne] in 1990. We refer to [GH] for a simpliﬁed proof (see also [BEE, Chapter 14]), [Ga2]for a variant splitting theorem assuming the global hyperbolicity instead of the geodesic
completeness, and to [Fl, Ga4] for surveys including some similarities and diﬀerences
between Theorem 1.1 and the Cheeger–Gromoll splitting theorem [CG] in Riemannian
geometry. Some generalizations to weighted Lorentzian manifolds can be found in [Ca,WW].
∗

Department of Mathematics, Osaka University, Osaka 560-0043, Japan (yufenglu.math@gmail.com,
s.ohta@math.sci.osaka-u.ac.jp)†
Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Via S. Marta
3, I-50139 Firenze, Italy (ettore.minguzzi@uniﬁ.it)‡
RIKEN Center for Advanced Intelligence Project (AIP), 1-4-1 Nihonbashi, Tokyo 103-0027, Japan1

Lorentz–Finsler manifolds generalize Lorentzian manifolds in the same manner as
Finsler manifolds generalize Riemannian manifolds (see Section 2 for the precise deﬁnition). This class of space(time)s has been attracting growing interest from the viewpoint of the investigation of less regular spacetimes as well as the synthetic geometricstudy of Lorentzian manifolds. The latter study was motivated by the fruitful theory
in the positive-deﬁnite case, namely the curvature-dimension condition as a synthetic
notion of lower (weighted) Ricci curvature bound. Finsler manifolds of weighted Ricci
curvature bounded below provide an important “non-Riemannian” class of spaces satisfying the curvature-dimension condition (see [Oh1, Oh5]), and this fact motivated theintroduction of a reinforced condition called the Riemannian curvature-dimension condition. We refer to a survey [Am] for the successful theory of metric measure spaces
satisfying the Riemannian curvature-dimension condition. In the recent developmentof a Lorentzian counterpart to the curvature-dimension condition (see, e.g., [CM, Mc]),(weighted) Lorentz–Finsler manifolds will have a similar signiﬁcance to Finsler manifolds,
and then it is important to know what results in Lorentzian geometry can be generalized
to the Lorentz–Finsler setting, which is an aim of our work.
In our previous papers [LMO1, LMO2], we have investigated time oriented Lorentz–
Finsler manifolds equipped with weight functions on causal vectors, that we call weighted
Finsler spacetimes (see Section 3 for the precise setting). We established several singularity theorems in [LMO1] and comparison theorems including the Laplacian comparison
theorem in [LMO2] (see Theorem 3.5). On one hand, the Laplacian comparison theorem
is a key ingredient in the proof of the Lorentzian splitting theorem. On the other hand,
the Cheeger–Gromoll splitting theorem was generalized to (weighted) Finsler manifolds
in [Oh3] (with the help of the Laplacian comparison theorem in [OS]). Hence it is naturalto proceed to splitting theorems for weighted Finsler spacetimes.
Our main result is the following.
Theorem 1.2 (Splitting theorem; Theorem 7.3) Let (M, L, Ψ) be a connected, timelike geodesically complete, weighted Berwald spacetime of RicN ≥ 0 on Ω with N ∈(−∞, 0) ∪ [n, +∞], where dim M = n + 1. Suppose also that(1) the ϵ-completeness holds for some ϵ in the corresponding ϵ-range (3.1),(2) there is a timelike straight line η : R −→ M ,(3) there is a timelike geodesically complete Lorentzian structure g of M whose Levi-Civita
connection coincides with the Chern connection of L such that η is timelike for g,f, g̃) of (M, g), a lift of η is a timelike straight line.
(4) in the universal cover (MThen the Lorentzian manifold (M, g∇bη ) isometrically splits in the sense that there exists
an n-dimensional complete Riemannian manifold (Σ, h) such that (M, g∇bη ) is isometricto (R × Σ, −dt2 + h).
Furthermore, for every x ∈ Σ, the image ζ(t) of (t, x) is a straight line bi-asymptotic
to η and Ψ is constant on ζ.
Let us quickly explain the terms in the theorem. A Berwald spacetime is a special
kind of Finsler spacetime having various ﬁne properties (see Deﬁnition 2.3), Ψ : M −→ R2

is a weight function and RicN is the associated weighted Ricci curvature (Deﬁnition 3.1),
and Ω is the set of future-directed timelike vectors. The Lorentzian structure g∇bη (x 7−→g∇bη (x) ) is the second order approximation of the original Lorentz–Finsler structure L in
the direction of the gradient vector ﬁeld ∇bη of the Busemann function bη for the rayη̄ in the past direction (see (2.2), (2.6) and Subsection 5.2). The existence of g as in (3)can be regarded as a Lorentzian analogue of Szabó’s metrizability theorem. On one hand,
it is not known if it holds true in our setting (see Remark 2.4). On the other hand, themetrizability is a necessary condition for the splitting as above (one can take g = g∇bη ,while g in Theorem 1.2 may not coincide with g∇bη ). The last hypothesis (4) is needed
merely for technical reasons and likely redundant.
In addition to the generalization from Lorentzian to Lorentz–Finsler manifolds, our approach using the ϵ-range (introduced in [LMO1]; see (3.1)) is more general than [Ca, WW]
for weighted Lorentzian manifolds. The ϵ-completeness (Deﬁnition 3.7) in the hypothesis (1) is an appropriate generalization of the timelike geodesic completeness involving
the weight function Ψ, and the use of the ϵ-range enables us to unify the splitting theorems in [Ca, WW] into a single framework (N ∈ [n, +∞) with ϵ = 1 recovers [Ca], andN ∈ (−∞, 0) ∪ {+∞} with ϵ = 0 corresponds to [WW]). We remark that RicN in this+1
paper corresponds to RicN
in [WW] (see Remark 3.2).
f
The main idea of the splitting theorem is extending the line η into a congruence oflines (precisely, asymptotes of η). Such lines in fact coincide with integral curves of∇bη . A distinct feature of the Lorentz–Finsler version is that the splitting is done forthe Lorentzian structure g∇bη and the original Lorentz–Finsler structure L itself does
not possess a product structure. This is actually a similar phenomenon to the Finsler
case; note that normed spaces do not isometrically split. Nonetheless, we can show thefollowing in the same spirit as [Oh3, §5].
Corollary 1.3 (Isometric translations; Corollary 7.4) Let (M, L, Ψ) be as in Theorem 1.2. Then, in the product structure M = R × Σ, the translations(s, x) ∈ R × Σ, t ∈ R,

Φt (s, x) := (s + t, x),
are isometric transformations of (M, L).

As a weight, one can also employ a function ψm on causal vectors induced from ameasure m on M in the same manner as [Oh1] in the Finsler setting (see Remarks 3.3, 6.5
and 7.5 for details). In this case, Φt is measure-preserving (see Remark 7.5). We remarkthat the Berwald condition is essential at least in Corollary 1.3, due to the extreme
ﬂexibility of general Lorentz–Finsler structures. The timelike congruence extending ηprobes the indicatrix only near the vector ﬁeld ∇bη . Thus, from the ensuing data ofg∇bη , one cannot recover the indicatrix far from ∇bη (see Remark 7.6 for details).
As for what concerns our strategy, we ﬁrst analyze rays and associated Busemannfunctions along the lines of [GH]. Then, to avoid the use of Bartnik’s existence theorem [Ba, Theorem 4.1] of maximal spacelike hypersurfaces which has no analogue in theLorentz–Finsler setting at present, we follow a more direct argument given in [Es] to show
the splitting theorem. We remark that our argument will be indebted to various ﬁneproperties of Berwald spacetimes, and it is unclear if Theorem 1.2 can be generalized to
non-Berwald Finsler spacetimes (cf. [Oh3, §4] in the Finsler setting).
3

This article is organized as follows. We review necessary notions for Lorentz–Finsler
manifolds in Section 2 and discuss the weighted situation in Section 3. We study rays
and associated Busemann functions in Sections 4 and 5, respectively. Then Section 6 isdevoted to a key step concerning the behavior of the Busemann functions near the straightline. We ﬁnally prove the splitting theorem in Section 7.

2

Preliminaries for Finsler spacetimes

We brieﬂy recall necessary notions for Lorentz–Finsler manifolds (see also [LMO1, LMO2]).
One can in fact discuss similarly to the positive-deﬁnite case (see, e.g., [BCS, Oh5, Sh])to some extent. We refer to [BEE, ON] for the basics of Lorentzian geometry and to[Mi2, Mi6, Mi7] for some generalizations including Lorentz–Finsler manifolds.
Throughout the article, let M be a connected C ∞ -manifold without boundary of
α ndimension n + 1 (≥ 2). Given a local coordinate system)α=0 on an open set U ⊂ M ,Pn (xαwe will use the ﬁber-wise linear coordinates v = α=0 v (∂/∂xα )|x , x ∈ U . We follow
Beem’s deﬁnition [Be] of a Finsler version of Lorentzian manifolds as follows.
Deﬁnition 2.1 (Lorentz–Finsler structures) A Lorentz–Finsler structure of M is afunction L : T M −→ R satisfying the following conditions:(1) L ∈ C ∞ (T M \ 0), where 0 denotes the zero section;(2) L(cv) = c2 L(v) for all v ∈ T M and c > 0;
(3) For any v ∈ T M \ 0, the symmetric matrix
 2
n
n
∂ L
gαβ (v) α,β=0 :=(v)∂v α ∂v β
α,β=0

(2.1)

is non-degenerate with signature (−, +, . . . , +).
Then we call (M, L) a (C ∞ -)Lorentz–Finsler manifold.
We say that (M, L) is reversible if L(−v) = L(v) for all v ∈ T M . For v ∈ Tx M \ {0},we deﬁne a Lorentzian metric gv of Tx M by using (2.1) as
!
n
n
n
X
X
X
∂
∂
aα α ,gvbβ β:=
gαβ (v)aα bβ .
(2.2)
∂x
∂x
x
x
α=0
β=0
α,β=0
Then we have gv (v, v) = 2L(v) by Euler’s homogeneous function theorem.
A tangent vector v ∈ T M is said to be timelike (resp. null ) if L(v) < 0 (resp. L(v) = 0).
We say that v is lightlike if it is null and nonzero, and causal (or non-spacelike) if it istimelike or lightlike (i.e., L(v) ≤ 0 and v 6= 0). Spacelike vectors are those S
for which′′L(v) > 0 or v = 0. We denote by Ωx ⊂ Tx M the set of timelike vectors and Ω := x∈M Ω′x .
For later use, we deﬁne, for causal vectors v,
pp
F (v) := −2L(v) = −gv (v, v).
(2.3)
4

Note that Ω′x 6= ∅, every connected component of Ω′x is a convex cone (see [Be], [LMO1,
Lemma 2.3]), and that the closures of diﬀerent components intersect only at 0 (see [Mi3,Proposition 1]). In general, the number of connected components of Ω′x may be larger than
2 (see [Be], [LMO1, Example 2.4]). This fact will not aﬀect our discussion because we shalldeal with only future-directed (timelike or causal) vectors. We also remark that Ωx′ hasexactly two connected components in reversible Lorentz–Finsler manifolds of dimension≥ 3 (see [Mi3, Theorem 7]).
Deﬁnition 2.2 (Finsler spacetimes) If a Lorentz–Finsler manifold (M, L) admits asmooth timelike vector ﬁeld X, then (M, L) is said to be time oriented (by X). A timeoriented Lorentz–Finsler manifold will be called a Finsler spacetime.
In a Finsler spacetime oriented by X, a causal vector v ∈ Tx M is said to be futuredirected if it lies in the same connected component of Ω′x \ {0} as X(x). We denote byΩx ⊂ Ω′x the set of future-directed timelike vectors, and deﬁne[[[(Ωx \ {0}).
Ω :=
Ωx ,
Ω :=
Ωx ,
Ω \ 0 :=
x∈M

x∈M

x∈M

A C 1 -curve in (M, L) is said to be timelike (resp. causal ) if its tangent vector is always
timelike (resp. causal). All causal curves will be future-directed.
Given x, y ∈ M , we write x  y (resp. x < y) if there is a future-directed timelike(resp. causal) curve from x to y, and x ≤ y means that x = y or x < y. Then we deﬁne
the chronological past and future of x by
I − (x) := {y ∈ M | y  x},I + (x) := {y ∈ M | x  y}.

We also deﬁne the Lorentz–Finsler distance d(x, y) for x, y ∈ M byZ 1
d(x, y) := sup ℓ(η),ℓ(η) :=F η̇(t) dt,
η0

where η : [0, 1] −→ M runs over all causal curves from x to y (recall (2.3) for F ). We
set d(x, y) := 0 if there is no causal curve from x to y (namely x 6< y). A constant
speed causal curve attaining the above supremum, which is a causal geodesic, is said to
be maximizing. In general, causal geodesics are locally maximizing in the same way as
geodesics are locally minimizing in Riemannian or Finsler geometry.
In general, the distance function d is only lower semi-continuous and can be inﬁnite(see, e.g., [Mi4, Proposition 6.7]). In globally hyperbolic Finsler spacetimes (see [LMO1]
for the deﬁnition), d is ﬁnite and continuous, and any pair of points x, y ∈ M with x < yadmits a maximizing geodesic from x to y (see [Mi4, Propositions 6.8, 6.9]). In what
follows, however, we will not assume the global hyperbolicity.
Next we introduce the covariant derivative and the Ricci curvature. Deﬁnen
1 X αλ
∂gλδ
∂gβλ
∂gβδ
α
γβδ (v) :=
g (v)
(v) +
(v) −
(v)
2 λ=0
∂xβ
∂xδ
∂xλ

5

for α, β, δ = 0, 1, . . . , n and v ∈ T M \ 0, where (g αβ (v)) is the inverse matrix of (gαβ (v)),
1 X α
G (v) :=γβδ (v)v β v δ ,2 β,δ=0nα∂GααNβ (v) :=(v)
β∂v

for v ∈ T M \ 0 (Gα (0) = Nβα (0) := 0 by convention), and∂gλδ µ ∂gβλ µ ∂gβδ µ
g (v)
N +
N −
N (v)
2 λ,µ=0∂v µ β∂v µ δ
∂v µ λ1αΓαβδ (v) := γβδ
(v) −

n
X

αλ

(2.4)

Pn
α
α
on T M \ 0. Then the covariant derivative of a vector ﬁeld X =α=0 X (∂/∂x ) isdeﬁned as
)
(
n
n
α
X
X
∂
∂X
Dvw X :=
v β β (x) +Γαβδ (w)v β X δ (x)
∂x
∂xα x
α,β=0
δ=0
for v ∈ Tx M with reference vector w ∈ Tx M \ {0}. We remark that the functions Γαβδ in(2.4) are the coeﬃcients of the Chern(–Rund ) connection.
α
In the Lorentzian case, gαβ is constant in each tangent space (thus Γαβδ = γβδ) andthe covariant derivative does not depend on the choice of a reference vector. In theLorentz–Finlser setting, the following class is worth considering.
Deﬁnition 2.3 (Berwald spacetimes) A Finsler spacetime (M, L) is said to be of
Berwald type (or called a Berwald spacetime) if Γαβδ is constant on the slit tangent space
Tx M \ {0} for any x in the domain of every local coordinate system.
The Berwald condition is strong but provides a reasonable (nontrivial) class where
we can mimic Lorentzian techniques. By the very deﬁnition, the covariant derivativeon a Berwald spacetime is deﬁned independently from the choice of a reference vector.
Examples of Berwald spacetimes include Lorentzian manifolds and ﬂat Lorentz–Finsler
structures of Rn+1 (every tangent space Tx Rn+1 is canonically isometric to T0 Rn+1 and weαhave Γαβδ = γβδ= 0). We refer to [FHPV, FP, FPP, GT, HPV] for some investigations
on Berwald spacetimes, and to [BCS, Chapter 10] for the positive-deﬁnite case.
Remark 2.4 (Metrizability) In the positive-deﬁnite case, Szabó showed that a Berwald
space (M, F ) admits a Riemannian metric g whose Levi-Civita connection coincides with
the Chern connection, i.e., the Christoﬀel symbols of g coincide with Γijk of F (see [Sz]and [BCS, Exercise 10.1.4]). This is called the metrizability theorem. It is not known
whether the metrizability can be generalized to the Lorentz–Finsler setting. In [FHPV],
some counter-examples were constructed for Lorentz–Finsler structures deﬁned only on asubset of T M . Their discussion is not applicable to Lorentz–Finsler structures deﬁned onwhole T M as in Deﬁnition 2.1.
The geodesic equation for a causal curve η : [0, 1] −→ M is written as Dη̇η̇ η̇ ≡ 0. The
exponential map is deﬁned in the same way as the Riemannian case, which is C ∞ on aneighborhood of the zero section only in Berwald spacetimes (see, e.g., [Mi5] as well as[BCS, Exercise 5.3.5] in the positive-deﬁnite case).
6

Next we introduce the Ricci curvature. A C ∞ -vector ﬁeld J along a geodesic η iscalled a Jacobi ﬁeld if it satisﬁes Dη̇η̇ Dη̇η̇ J + Rη̇ (J) = 0, wherenX

Rv (w) :=Rβα (v)wβ

α,β=0

∂
∂xα

for v, w ∈ Tx M and
 X
n 
n
X
∂Nβα
∂Nβα
∂Gα
α
δ
δ
Rβ (v) := 2 β (v) −(v)v − 2 δ (v)G (v) −Nδα (v)Nβδ (v)
δ
∂x
∂x
∂v
δ=0
δ=0
is the curvature tensor. A Jacobi ﬁeld is also characterized as the variational vector ﬁeldof a geodesic variation.
Note that Rv (w) is linear in w, thereby Rv : Tx M −→ Tx M is an endomorphism foreach v ∈ Tx M . For v ∈ Ωx , we deﬁne the Ricci curvature (or Ricci scalar ) of v as thetrace of Rv : Ric(v) := trace(Rv ). We remark that Ric(cv) = c2 Ric(v) for c > 0.
In order to introduce the spacetime Laplacian, we consider the dual structure to L
and the Legendre transform (see [Mi3], [Mi6, §3.1], [LMO2, §4.4] for further discussions).
Deﬁne the polar cone to Ωx byΩ∗x := ω ∈ Tx∗ M | ω(v) < 0 for all v ∈ Ωx \ {0} .
This is an open convex cone in Tx∗ M . For ω ∈ Ω∗x , we deﬁne
1
L (ω) := −
2
∗

2


sup

ω(v)

v∈Ωx ∩F −1 (1)

=−

2
1
ω(v) .
inf−1
2 v∈Ωx ∩F (1)Then we have the reverse Cauchy–Schwarz inequality 4L∗ (ω)L(v) ≤ (ω(v))2 for v ∈ Ωxand ω ∈ Ω∗x , and arrive at the following variational deﬁnition of the Legendre transform.
Deﬁnition 2.5 (Legendre transform) Deﬁne the Legendre transform L ∗ : Ω∗x −→ Ωxas the map sending ω ∈ Ω∗x to the unique element v ∈ Ωx satisfying L(v) = L∗ (ω) =ω(v)/2. We also deﬁne L ∗ (0) := 0.
A coordinate expression of the Legendre transform is given by
n
X
∂L∗

∂
L (ω) =(ω) α ,
∂ωα
∂x
α=0
∗

where ω =nX

ωα dxα .

(2.5)

α=0

We refer to [LMO2, Mi3, Mi6] for some basic properties of the Legendre transforms.
A continuous function f : M −→ R is called a time function if f (x) < f (y) for all
x, y ∈ M with x < y. A C 1 -function f : M −→ R is said to be temporal if −df (x) ∈ Ω∗x
for all x ∈ M . Observe that temporal functions are time functions.
For a temporal function f : M −→ R, deﬁne the gradient vector of −f at x ∈ M by
(2.6)∇(−f )(x) := L ∗ −df (x) ∈ Ωx .
7

Note that, thanks to (2.5), we have g∇(−f ) (∇(−f )(x), v) = −df (v) for any v ∈ Tx M .
For a C 2 -temporal function f : M −→ R and x ∈ M , we deﬁne the Hessian ∇2 (−f ) :Tx M −→ Tx M by
∇2 (−f )(v) := Dv∇(−f ) ∇(−f ) .
(2.7)
This spacetime Hessian has the following symmetry (see [LMO2, Lemma 4.12]):g∇(−f ) ∇2 (−f )(v), w = g∇(−f ) v, ∇2 (−f )(w)

(2.8)

for all v, w ∈ Tx M . Then we deﬁne the spacetime Laplacian (or d’Alembertian) as thetrace of the Hessian:∆(−f ) := trace ∇2 (−f ) .
(2.9)
We remark that this Laplacian is not elliptic but hyperbolic, and is nonlinear (since theLegendre transform is nonlinear).
Remark 2.6 (Berwald case) For v ∈ Tx M and the geodesic η : (−ε, ε) −→ M with
η̇(0) = v, the second order derivative (−f ◦η)′′ (0) does not coincide with g∇(−f ) (∇2 (−f )(v), v)
in general. They coincide in Berwald spacetimes thanks to the ﬁber-wise constancy of theconnection coeﬃcients Γαβδ (see [Oh5, §12.1] for the positive-deﬁnite case).

3

Weighted Finsler spacetimes

Next, as a weight on a Finsler spacetime (M, L), we employ a C ∞ -function Ψ : M −→ R.
The following deﬁnition generalizes the weighted Ricci curvature for Finsler manifolds
introduced in [Oh1] (see also [Oh4] for the case of N < 0).
Deﬁnition 3.1 (Weighted Ricci curvature) Given v ∈ Ω \ 0, let η : (−ε, ε) −→ M
be the causal geodesic with η̇(0) = v. Then we deﬁne the weighted Ricci curvature by
RicN (v) := Ric(v) + (Ψ ◦ η)′′ (0) −(Ψ ◦ η)′ (0)2
N −nfor N ∈ R\{n}. We also deﬁne Ric∞ (v) := Ric(v)+(Ψ◦η)′′ (0), Ricn (v) := limN ↓n RicN (v),and RicN (0) := 0.
+1
Remark 3.2 Since dim M = n + 1, RicN in this paper corresponds to RicN
in [WW].
fRemark 3.3 (Weight functions associated with measures) Instead of a function
Ψ on M as above, one can alternatively employ a function ψm : Ω \ 0 −→ R on causal
vectors associated with a measure m on M . To be precise, ψm is deﬁned byq−ψm ◦η̇− det gαβ (η̇) dx0 dx1 · · · dxn
m(dx) = ealong causal geodesics η. One way to unify these two cases is to consider a general 0homogeneous function ψ : Ω \ 0 −→ R as in [LMO1, LMO2] (we remark that, in general,such a function may not be associated with any measure; see [Oh2, Oh5] for the Finslercase). However, since a weighted Laplacian corresponding to general ψ is yet to be
developed (even for Riemannian manifolds), we restrict ourselves to Ψ or ψm .
8

We will say that RicN ≥ K holds in timelike directions for some K ∈ R if we have
RicN (v) ≥ KF 2 (v) = −2KL(v) for all v ∈ Ω (recall (2.3)). Observe that
Ricn (v) ≤ RicN (v) ≤ Ric∞ (v) ≤ RicN ′ (v)holds for n < N < ∞ and −∞ < N ′ < n. Associated with the real parameter N , thefollowing notion was introduced in [LMO1].
Deﬁnition 3.4 (ϵ-range) Given N ∈ (−∞, 0] ∪ [n, +∞], we will consider ϵ ∈ R in thefollowing ϵ-range:rN
ϵ = 0 for N = 0,|ϵ| <for N 6= 0, n,ϵ ∈ R for N = n.
(3.1)
N −n
The associated constant c = c(N, ϵ) is deﬁned as
12N − nc(N, ϵ) :=1−ϵ> 0 for N 6= 0,nNc(0, 0) :=

1
.
n

(3.2)

Note that ϵ = 1 is admissible only for N ∈ [n, +∞) and ϵ = 0 is always admissible. By
the ϵ-range, we can unify results for constant and variable curvature bounds into a single
framework. See [LMO1] for singularity theorems and [LMO2] for comparison theorems.
A Laplacian comparison theorem with ϵ-range was established in [LMO2] as follows.
Given z ∈ M and any maximizing timelike geodesic η : [0, T ) −→ M with η(0) = z, thefunction u(x) := d(z, x) satisﬁes −du(x) ∈ Ω∗x for x ∈ η((0, T )) and hence ∆(−u)(x) asin (2.9) is well-deﬁned. We deﬁne the Ψ-Laplacian (or the weighted Laplacian) of −u by∆Ψ (−u)(x) := ∆(−u)(x) − dΨ ∇(−u)(x) .
Then the Laplacian comparison theorem ([LMO2, Theorem 5.8]) for nonnegatively curved
spacetimes asserts the following.
Theorem 3.5 (Laplacian comparison theorem) Let (M, L, Ψ) be a weighted Finsler
spacetime of dimension n + 1 ≥ 2, N ∈ (−∞, 0] ∪ [n, +∞], and ϵ ∈ R belong to the
ϵ-range (3.1). Suppose that RicN ≥ 0 holds in timelike directions. Then, for any z ∈ M ,the distance function u(x) := d(z, x) satisﬁes
 Z t
−1

2(ϵ−1)
2(ϵ−1)
Ψ(η(t))
Ψ(η(s))e n∆ (−u) η(t) ≤ e n
c
ds
Ψ

(3.3)

0

for any maximizing timelike geodesic η : [0, T ) −→ M of unit speed (i.e., F (η̇) ≡ 1)emanating from z and any t ∈ (0, T ), where c is as in (3.2).
Precisely, we considered a general 0-homogeneous function ψ : Ω \ 0 −→ R in [LMO2](recall Remark 3.3). Note also that (3.3) is actually an intermediate estimate in the proofof [LMO2, Theorem 5.8]. We refer to [Ca, WW] for the weighted Lorentzian case.
We also explain the reverse version of Theorem 3.5 for thoroughness. The reverse←−
Lorentz–Finsler structure of L is deﬁned as L (v) := L(−v), and we put an arrow ←
9

←
−
←
−
on a quantity associated with L . The Lorentz–Finsler manifold (M, L ) is time oriented←−←−
by −X, so that Ω = −Ω. The corresponding weighted Laplacian is given as ∆ Ψ f =←−−∆Ψ (−f ) for temporal functions f (as for ψm , we have ψ m (v) = ψm (−v)). We remark
←−
←
−
←−
that, for the weighted Ricci curvature Ric of (M, L , Ψ), we have RicN (v) = RicN (−v) for←−←−v ∈ Ω , and hence the timelike curvature bound RicN ≥ 0 is equivalent to RicN ≥ 0.
Corollary 3.6 (Reverse version) Let (M, L, Ψ), N and ϵ be as in Theorem 3.5. Then,for any z ∈ M , the function ū(x) := d(x, z) satisﬁes
 Z t
−1


2(ϵ−1)
2(ϵ−1)←−ΨΨΨ(η̄(t))Ψ(η̄(s))nn∆ (−ū) η̄(t) = −∆ ū η̄(t) ≤ e
c
e
ds
(3.4)
0

for any maximizing timelike geodesic η : (−T, 0] −→ M of unit speed with η(0) = z andany t ∈ (0, T ), where η̄(s) := η(−s).
Note that η is a maximizing timelike geodesic of unit speed with respect to L and so←−is η̄ with respect to L . In the application of the Laplacian comparison theorem, we needto require that the right-hand side of (3.3) (and (3.4)) diverges to inﬁnity. To be precise,
we will assume the following completeness condition introduced in [LMO1].
Deﬁnition 3.7 (ϵ-completeness) A weighted Finsler spacetime (M, L, Ψ) is said to be
future timelike ϵ-complete if, for any future inextendible timelike geodesic η : [0, T ) −→ M ,Z t2(ϵ−1)lime n Ψ(η(s)) ds = ∞
t→T

0

←
−
holds. We say that (M, L, Ψ) is timelike ϵ-complete if both (M, L, Ψ) and (M, L , Ψ) are
future timelike ϵ-complete.
The 1-completeness corresponds to the usual geodesic completeness (see Deﬁnition 4.1
below) and the 0-completeness recovers the Ψ-completeness introduced in [Wy] in the
Riemannian case (see [WW] for the Lorentzian case). A typical situation is that ϵ < 1,
Ψ is bounded above and (M, L) is (future) timelike geodesically complete.

4

Rays and generalized co-rays

In this and the next sections, we shall follow the argument in [GH] to study rays andassociated Busemann functions (see also [BEE, Chapter 14]). We ﬁrst recall the geodesic
completeness conditions. See [Ga3] for some connections between the global hyperbolicity
and the geodesic completeness.
Deﬁnition 4.1 (Geodesic completeness) A Finsler spacetime (M, L) is said to be
future timelike geodesically complete if any timelike geodesic η : [0, 1] −→ M can beextended to a geodesic η̃ : [0, ∞) −→ M . We say that (M, L) is timelike geodesically←−complete if both (M, L) and (M, L ) are future timelike geodesically complete (in otherwords, the above η is extended to a geodesic η̃ : R −→ M ).
10

A future inextendible causal geodesic η : [0, ∞) −→ M is called a ray if each ofits segments is maximizing, i.e., ℓ(η|[a,b] ) = d(η(a), η(b)) for all 0 ≤ a ≤ b. For later
convenience, we deﬁne
[


(4.1)
I − (η) :=I − η(t) ,I(η) := I + η(0) ∩ I − (η).
t>0Note that I(η) is an open set including η((0, ∞)). Observe also that, for x, y ∈ I(η) withx ≤ y, we have d(x, y) < ∞ since the reverse triangle inequality yieldsd η(0), x + d(x, y) + d y, η(t) ≤ d η(0), η(t)(4.2)
for suﬃciently large t.
Let (M, L) be future timelike geodesically complete in the remainder of the section.
Recall that, without the global hyperbolicity, connecting maximizing curves may not exist.
Thus the following notion will play a role.
Deﬁnition 4.2 (Limit maximizing sequences) A sequence ηk : [ak , bk ] −→ M of
causal curves is said to be limit maximizing if ℓ(ηk ) ≥ d(ηk (ak ), ηk (bk )) − δk holds forsome δk → 0.
Observe that, if x < y and d(x, y) < ∞, then there exists a limit maximizing sequenceof causal curves from x to y by the deﬁnition of d.
We make use of the limit curve theorems of Lorentzian geometry in both the “oneendpoint” and “two endpoints” cases (see [BEE, EG, Ga1, GH] and more recent [Mi1,Mi7]). As observed in [Mi4], these limit curve theorems generalize to the Lorentz–Finsler
setting thanks to the existence of convex neighborhoods (see also [Mi6] for the Finslerianstatements of these results under weak diﬀerentiability conditions).
What is relevant for us is the fact that, due to the lower semi-continuity of d andthe upper semi-continuity of the length ℓ (with respect to the uniform convergence oncompact intervals), any limit maximizing sequence of causal curves admits a subsequence
converging, in a suitable parametrization, to a maximizing geodesic. For instance, wehave the following analogue of [GH, Proposition 2.3].
Lemma 4.3 Let ηk : [ak , bk ] −→ M be a limit maximizing sequence
T of causal curves
converging to η : [a, b] −→ M uniformly on some interval [a, b] ⊂ k [ak , bk ]. Then we
have ℓ(η) = d(η(a), η(b)), and hence η is a maximizing curve from η(a) to η(b).
The next lemma is an application of the limit curve theorem (see also [GH, Lemma 2.4]).
Lemma 4.4 Let a sequence (xk )k∈N in M converge to x and take yk ∈ I + (xk ) such thatd(xk , yk ) < ∞. Let ηk : [0, ak ] −→ M be a limit maximizing sequence of causal curves
with ηk (0) = xk and ηk (ak ) = yk . If d(xk , yk ) → ∞, then there exists a subsequence of(ηk )k∈N with suitable reparametrizations converging uniformly on compact intervals to aray η : [0, ∞) −→ M emanating from x.
We remark that the above ray η may not be timelike. This fact is at the origin of
several complications in the proof of the splitting theorem.
11

Rays asymptotic to a given ray, called co-rays, play an important role in splitting
theorems. In the non-globally hyperbolic case, due to the possible absence of maximizing
curves, we need to consider also generalized co-rays. They are deﬁned as follows (co-rays
and generalized co-rays were introduced in [BEMG] and [EG], respectively).
Deﬁnition 4.5 (Generalized co-rays) Let η : [0, ∞) −→ M be a timelike ray andtake x ∈ I(η). If a limit maximizing sequence of causal curves ζk : [0, ak ] −→ M satisﬁesζk (0) → x and ζk (ak ) = η(tk ) for some tk > 0 with tk → ∞, then its limit curveζ : [0, ∞) −→ M is called a generalized co-ray of η. If each ζk is maximizing, then we call
ζ a co-ray of η. A co-ray ζ with ζk (0) = x for all k is called an asymptote of η.
Lemma 4.4 ensures that the limit curve ζ is indeed a ray. As another instance of thelimit curve theorem, we have the following analogue of [GH, Proposition 3.1].
Proposition 4.6 Let x, y ∈ M satisfy y ∈ I + (x) and d(x, y) < ∞, and let ηk : [0, ak ] −→M be a limit maximizing sequence of causal curves from x to y. Then (ηk )k∈N admits asuitably reparametrized subsequence converging uniformly on compact intervals to either
a lightlike ray or a maximizing curve from x to y.
To avoid the convergence to a lightlike ray, we introduce the following condition.
Deﬁnition 4.7 (Generalized timelike co-ray condition (GTCC)) Let η be a timelike ray and x ∈ I(η). We say that the generalized timelike co-ray condition (GTCC for
short) for η holds at x if any generalized co-ray of η emanating from x is timelike.
Under the GTCC, we can prove the following existence result of maximizing timelike
geodesics (by using Proposition 4.6 and Lemma 4.4).
Lemma 4.8 Let η : [0, ∞) −→ M be a timelike ray and suppose that the GTCC for η
holds at some x ∈ I(η). Then there exist a neighborhood U of x and T > 0 such that, for
any z ∈ U and t > T , there exists a maximizing timelike geodesic from z to η(t).
See [GH, Lemma 3.3] for a proof (roughly speaking, if there do not exist such U and
T , then we ﬁnd a lightlike generalized co-ray of η emanating from x, a contradiction to
the GTCC). A similar argument shows that the GTCC is an open condition in I(η).
Furthermore, taking smaller U and larger T if necessary, we can show that there exists
a compact set K ⊂ Ω ∩ F −1 (1) such that any maximizing timelike geodesic ζ from z ∈ U
to η(t) with t > T of unit speed satisﬁes ζ̇(0) ∈ K (we refer to [GH, Lemma 3.4] for a
proof). Therefore, we ﬁnd the following.
Lemma 4.9 (Existence of timelike asymptotes) Let η : [0, ∞) −→ M be a timelike
ray and suppose that the GTCC for η holds at some x ∈ I(η). Then there exists aneighborhood U of x such that every z ∈ U admits a timelike asymptote of η emanatingfrom z.
Finally, the GTCC holds on a neighborhood of η as follows.

12

Proposition 4.10 (GTCC near rays) Let η : [0, ∞) −→ M be a timelike ray. Then,any generalized co-ray emanating from η(a) with a > 0 necessarily coincides with η. In
particular, the GTCC holds on an open set including η((0, ∞)).
The ﬁrst assertion is shown in the same manner as [GH, Proposition 5.1] (and could
be compared with [Es, Lemma 3.4]). The second assertion follows from the fact that the
GTCC is an open condition (see [GH, Corollary 5.2]).

5

Busemann functions

In this section, we continue assuming the future timelike geodesic completeness.

5.1

For rays

Let η : [0, ∞) −→ M be a timelike ray of unit speed. We introduce a central ingredientof the proof of the splitting theorem.
Deﬁnition 5.1 (Busemann functions) Deﬁne the Busemann function bη : M −→[−∞, +∞] associated with η bybη (x) := lim bηt (x),where bηt (x) := t − d x, η(t) .
t→∞

The limit above always exists in R∪{±∞}. Precisely, if x 6∈ I − (η), then d(x, η(t)) = 0
for all t and hence bη (x) = ∞ (recall (4.1) for the deﬁnition of I − (η)). If x ∈ I − (η), thenthe reverse triangle inequality implies that, for large s < t,bηt (x) ≤ t − d x, η(s) + d η(s), η(t) = bηs (x).
Hence bηt (x) is non-increasing in t and converges to bη (x) ∈ R ∪ {−∞}. Moreover, sinced is lower semi-continuous, bη is upper semi-continuous on I − (η).
When x ∈ I(η), it follows from (4.2) (with y = x) that bη (x) ≥ d(η(0), x) ≥ 0.
Therefore bη is well-posed in the domain I(η) of our interest. We also observe that, forx, y ∈ M with x ≤ y,bη (y) ≥ bη (x) + d(x, y)(5.1)
by the reverse triangle inequality. In the Riemannian or Finsler setting, the reverse
inequality to (5.1) holds for all x, y and it implies that the Busemann function is 1Lipschitz continuous. In the Lorentzian case, however, bη is not necessarily continuous.
Next we introduce useful upper support functions for bη on I(η). We give a proof (inthe same manner as [GH, Lemma 2.5]) for completeness.
Lemma 5.2 (Support functions for bη ) Let ζ : [0, ∞) −→ M be a timelike asymptoteof η of unit speed with z := ζ(0) ∈ I(η).
(i) For each t > 0,
ρ(x) := bη (z) + bζt (x) = bη (z) + t − d x, ζ(t)



(5.2)

is an upper support function for bη at z in the sense that ρ(x) ≥ bη (x) on aneighborhood of z and ρ(z) = bη (z).
13

(ii) We havebη ζ(t) = bη (z) + t(5.3)

for all t ≥ 0. In particular, I(ζ) ⊂ I(η) holds.
Recall from Lemma 4.9 and Proposition 4.10 that points in a neighborhood of η((0, ∞))admit timelike asymptotes.
Proof. (i) Let ζk : [0, ak ] −→ M be a sequence of maximizing timelike geodesics withζk (0) = z converging to ζ. By deﬁnition (Deﬁnition 4.5), we have ζk (ak ) = η(tk ) for some
tk and limk→∞ ak = limk→∞ tk = ∞. Fix x ∈ I + (η(0)) ∩ I − (ζ(t)) and take large k sothat z ∈ I − (η(tk )), x ∈ I − (ζk (t)) and ak > t. Then we deduce from the reverse triangle
inequality thatbηtk (x) − bηtk (z) = d z, η(tk ) − d x, η(tk )≤ d z, η(tk ) − d x, ζk (t) − d ζk (t), η(tk )= d z, ζk (t) − d x, ζk (t) .
Letting k → ∞ yieldsbη (x) − bη (z) ≤ d z, ζ(t) − d x, ζ(t) = t − d x, ζ(t)with the help of Lemma 4.3. This completes the proof of (i).
(ii) For s < t, we have ζ(s) ∈ I + (η(0)) ∩ I − (ζ(t)) and hence (i) impliesbη ζ(s) ≤ ρ ζ(s) = bη (z) + s.
Combining this with (5.1), we ﬁnd that bη (ζ(s)) = bη (z) + s holds for all s ≥ 0. This
also shows that bη < ∞ on ζ and hence ζ([0, ∞)) ⊂ I(η). Thus we have I(ζ) ⊂ I(η). □The continuity of the Busemann function can be shown in the same way as [GH,Theorem 3.7] (see also [Es, Lemma 3.3]).
Theorem 5.3 (Continuity of bη ) Assume that the GTCC for η holds at x ∈ I(η).
Then bη is Lipschitz continuous on a neighborhood of x.
We remark that the Lipschitz continuity is understood with respect to an auxiliary
Riemannian metric (the choice of a metric costs no generality since the assertion is local).
For the sake of further analyzing support functions as in Lemma 5.2, we would like tohave some control over the past timelike cut locus. Given z ∈ M , we say that x ∈ I − (z)is a past timelike cut point to z if there is a maximizing timelike geodesic ξ : [0, 1] −→ Mfrom x to z such that its extension ξ˜ : [−δ, 1] −→ M is not maximizing for any δ > 0.
The past timelike cut locus of z is the set of all past timelike cut points to z. (We remark←−that the past timelike cut locus for L is the future timelike cut locus for L .)Proposition 5.4 Let ζ : [0, ∞) −→ M be a timelike asymptote of η with ζ(0) ∈ I(η)
and assume that the GTCC for η holds at ζ(0). Then, for each t > 0, there exists aneighborhood of ζ([0, t]) which does not meet the past timelike cut locus of ζ(t).
14This is proved along the lines of [GH, Proposition 3.9]. We remark that a past-directed←−causal curve means a future-directed causal curve with respect to L in our setting. (To be
precise, Proposition 5.4 is an analogue of [BEE, Proposition 14.23]. There is a small gap
in the proof of [GH, Lemma 3.10] which can be ﬁxed assuming the continuity of d(·, α(r))
at p, and the continuity of d in turn follows from the continuity of the Busemann function(Theorem 5.3) as proved in [BEE, Sublemma 14.24].)
It follows from Proposition 5.4 that d(·, ζ(t)) is smooth near ζ(0) and the Laplaciancomparison theorem (Corollary 3.6) can be used to analyze ρ(x) as in (5.2).

5.2

For straight lines

In splitting theorems, we assume the existence of a timelike straight line η : R −→ M ,which is a globally maximizing timelike geodesic of unit speed, i.e., d(η(s), η(t)) = t − sholds for all s < t. We will denote by bη the Busemann function associated with the ray
η|[0,∞) . Moreover, the curve η̄(t) := η(−t), t ∈ [0, ∞), is a timelike ray of unit speed with←−respect to the reverse structure L (v) = L(−v) (recall Section 3). Hence we can introduce
the corresponding Busemann function asbη (x) := lim t − d η(−t), x
t→∞(d is with respect to L). It follows from the reverse triangle inequality thatbη (x) + bη (x) ≥ lim 2t − d η(−t), η(t) = 0,
t→∞

(5.4)

and equality holds on η.
For straight lines we modify the deﬁnition of (4.1) into
!
!
[
[


I(η) :=
I + η(−t) ∩I − η(t) .
S

t>0

t>0

Note that I(η) = s<0 I(η|[s,∞) ), therefore the results in this section are available in I(η).
We deduce from (5.1) and (5.3) the following (see [Es, Lemma 4.1]).
Lemma 5.5 Assume that x ∈ I(η) satisﬁes bη (x) + bη (x) = 0, and let ζ + and ζ − betimelike asymptotes of η and η̄ emanating from x, respectively. Then the concatenationof ζ + and ζ − provides a timelike straight line.
←−
Precisely, ζ − is an asymptote of η̄ with respect to L , and the concatenation ζ is givenby ζ(t) := ζ + (t) for t ≥ 0 and ζ(t) := ζ − (−t) for t < 0. We say that such a straight lineζ is bi-asymptotic to η. We also observe from Lemma 5.2 that bζ + bζ ≥ bη + bη holdson I(ζ) ⊂ I(η) with equality on ζ.

6

A key step

With the analysis of rays and Busemann functions in the previous sections in hand, weshall generalize Eschenburg’s argument [Es] toward the splitting theorem. Proposition 6.3
15

below, which generalizes [Es, Proposition 6.1], is a crucial step. Recall that the spacetime Laplacian is not elliptic but hyperbolic. Therefore we need to consider a spacelike
hypersurface and the Laplacian restricted on it, which is elliptic. The metrizability will
be assumed in Proposition 6.3 to overcome a diﬃculty arising in our Lorentz–Finsler situation. For the sake of clarifying where we need the metrizability, we do not intend toutilize it fully but minimize the use of it in our discussion.
Remark 6.1 (Positive-deﬁnite case) In the positive-deﬁnite case, the Laplacian comparison theorem implies the subharmonicity of the Busemann function bη , namely ∆bη ≥0. We similarly obtain the subharmonicity of bη with respect to the reverse Finsler struc←−←−ture ( ∆bη ≥ 0), and hence ∆bη ≥ − ∆bη = ∆(−bη ). Comparing this with bη + bη ≤ 0
by the triangle inequality and bη + bη ≡ 0 on η, we deduce from the strong comparison←−
principle that bη + bη ≡ 0 as well as ∆bη = ∆bη ≡ 0. We refer to [Oh3] for details.
We begin with a bound of the Hessian of the distance function. Henceforth, we assume
the Berwald condition. Recall (2.7) for the deﬁnition of the Hessian, and Deﬁnition 2.3
for Berwald spacetimes.
Lemma 6.2 (Hessian bound) Let (M, L) be a future timelike geodesically complete
Berwald spacetime, η : [0, ∞) −→ M be a timelike ray of unit speed, and take x ∈ I(η) at
where the GTCC for η holds. Then, given a small neighborhood U of x and t0 > 0, there
exists a constant C = C(U, t0 ) > 0 such that, for each timelike asymptote ζ : [0, ∞) −→ Mof η of unit speed with z := ζ(0) ∈ U , we haveg∇dt (z) ∇2 dt (v), v ≥ −C · gζ̇(0) (v ⊥ , v ⊥ )(6.1)
for all v ∈ Tz M and t ≥ t0 , where dt := d(·, ζ(t)) and v ⊥ is the projection of v to thegζ̇(0) -normal space to ζ̇(0) in Tz M .
We take U on which the GTCC for η holds. Note that −dt is a temporal function and
C on a neighborhood of z thanks to Proposition 5.4, thereby ∇2 dt (v) is well-deﬁned.
Observe also that gζ̇(0) (v ⊥ , v ⊥ ) ≥ 0 and v ⊥ = v + gζ̇(0) (ζ̇(0), v)ζ̇(0) since





gζ̇(0) ζ̇(0), v + gζ̇(0) ζ̇(0), v ζ̇(0) = gζ̇(0) ζ̇(0), v 1 + gζ̇(0) ζ̇(0), ζ̇(0) = 0∞

by gζ̇(0) (ζ̇(0), ζ̇(0)) = 2L(ζ̇(0)) = −1.
Proof. Note that (v, w) 7−→ g∇dt (z) (∇2 dt (v), w) is a quadratic form by the symmetry(2.8) and that
∇2 dt ζ̇(0) = Dζ̇ζ̇ ζ̇(0) = 0
since ∇dt (ζ(s)) = ζ̇(s) for s ∈ [0, t). Hence it is suﬃcient to show (6.1) for all v in thegζ̇(0) -normal space to ζ̇(0) (i.e., v = v ⊥ ).
As we mentioned before Lemma 4.9, the set of initial vectors of timelike asymptotes
of unit speed emanating from a (small) neighborhood U is included in a compact set ofunit timelike vectors (see [GH, Corollary 3.5] as well). Therefore, given t0 > 0, thanks tothe smoothness of dt0 , we ﬁnd C > 0 such that (6.1) holds at t = t0 .
16

For extending (6.1) to t > t0 , it suﬃces to show that g∇dt (z) (∇2 dt (v), v) is nondecreasing in t. To this end, recall that(6.2)
g∇dt (z) ∇2 dt (v), v = (dt ◦ ξv )′′ (0)holds for the geodesic ξv : (−δ, δ) −→ M with ξ˙v (0) = v (see Remark 2.6). Observe fromthe reverse triangle inequality thatdt ξv (r) ≥ ds ξv (r) + d ζ(s), ζ(t) = ds ξv (r) + t − s
for small r > 0 and t0 ≤ s < t, while dt (z) = ds (z) + t − s. Combining these with(dt ◦ ξv )′ (0) = (ds ◦ ξv )′ (0) = gζ̇(0) ζ̇(0), v
from the ﬁrst variation formula, we ﬁnd that dt is more convex than ds at z. Therefore
we have(dt ◦ ξv )′′ (0) ≥ (ds ◦ ξv )′′ (0),□which completes the proof by (6.2).

We are ready to show the main result in this section. Proposition 4.10 will play a
crucial role (in place of [Es, Lemma 3.4]).
Proposition 6.3 (Key step) Let (M, L, Ψ) be a weighted, timelike geodesically complete
Berwald spacetime of RicN ≥ 0 on Ω for some N ∈ (−∞, 0) ∪ [n, +∞]. Suppose also that(1) the ϵ-completeness for some ϵ in the corresponding ϵ-range (3.1) holds,(2) there is a timelike straight line η : R −→ M ,(3) there is a timelike geodesically complete Lorentzian structure g of M whose Levi-Civita
connection coincides with the Chern connection of L such that η is timelike for g,f, g̃) of (M, g), a lift of η is a timelike straight line.
(4) in the universal cover (MThen there exists a neighborhood W of η(R) on which we have bη + bη = 0.
Proof. We will suppress η in bη and bη . The proof is divided into ﬁve steps. We follow
the lines of [Es, Proposition 6.1] as far as possible, and the metrizability (3) comes intoplay in the fourth step.
Step 1 Observe ﬁrst that b and b are continuous on a neighborhood W0 of η(R) by
Theorem 5.3 and Proposition 4.10. Now, for deriving a contradiction, suppose that there
is no neighborhood W satisfying the claim. Then we ﬁnd that the open set
Q := {x ∈ W0 | b(x) + b(x) > 0}
has a boundary point on η(R), denoted by x0 (recall that b + b ≥ 0 by (5.4)). We may
assume by translation that η(0) = x0 . Moreover, we can take a small coordinate domain
17

B in Q such that x0 ∈ ∂B (possibly replacing η with a timelike straight line given by
Lemma 5.5; see [Es, Proposition 6.1] for details).
We take Fermi(–Walker) coordinates (xα )nα=0 on a suﬃciently small tubular neighborhood U around η([−R, R]), where R ∈ (0, 1/3]. Precisely, given gη̇ -parallel vector ﬁelds
Pα (α = 0, 1, . . . , n) along η such that {Pα (0)}nα=0 is gη̇ -orthonormal with P0 = η̇, we
consider a local coordinate system given by
!
n
n
X
X
(xi )2 < R2 .
|x0 | < R,(xα )nα=0 7−→ expη(x0 )xi Pi (x0 ) ,
i=1

i=1

We remark that the exponential map is C ∞ on a neighborhood of the zero section by the
Berwald condition.
By taking smaller B if necessary, B ∩ U may be represented as()
nX
B ∩ U = (xα )nα=0 ∈ U(xα )2 − 6Rxn < 0α=0in the coordinates above. Now we consider the set()
nX
A := (xα )nα=0 ∈ U(xi )2 > (1 + 2R)(x0 )2
i=1

including the spacelike hypersurface {x0 = 0} \ {x0 }. Note that the Fermi coordinates
enjoy Γαβδ (η̇) = 0 along η (see, e.g., [Mi5, Proposition 3.3] in a more general non-Berwald
situation).
Step 2 We consider the ﬂat Minkowski structure
g0 = −(dx0 )2 + (dx1 )2 + · · · + (dxn )2
associated with the Fermi coordinates (xα )nα=0 above. Note that g0 = g∂/∂x0 along η.
We can construct a smooth function ϑ on U satisfying the following properties:(a) ϑ(x0 ) = 0;(b) ϑ > 0 on A \ B.
This is done along the lines of [Es, Lemma 6.2] as follows. We ﬁrst consider a function
1X i 2(x ) − xn − (x0 )2
R i=1
n−1

ϕ(x) =

in the above coordinates, and deﬁne ϑ := 1 − e−σϕ for large σ > 0. Then (a) is clear and(b) is seen by the argument of Claim in the proof of [Es, Lemma 6.2] as follows.
It suﬃces to show that x ∈ A satisfying ϕ(x) ≤ 0 necessarily lies in B. We deducefrom x ∈ A and ϕ(x) ≤ 0 that
(1 + 2R)(x0 )2 < R xn + (x0 )2 + (xn )2 ,18

which implies


2
R2
R
n
x +
−
> (1 + R)(x0 )2 ≥ 0.
2
4

Noticing xn > −R, we have xn > 0 and
!
r
r
2
R
R
4(1
+
R)
R
xn > (1 + R)(x0 )2 +− =(x0 )2 + 1 − 1 .
4
2
2
R2
Since 4(1 + R)(x0 )2 /R2 < 4(1 + R) < 8 and the slope of the function
1/6 for s ∈ (1, 9), we observer4(1 + R) 0 2
1 4(1 + R) 0 2(x ) + 1 > 1 +(x ) .
2R
6 R2
Thus we havexn >√s is larger than

1+R 0 2(x ) ,
3R

and hence x ∈ B sincenX(xα )2 − 6Rxn ≤ (1 + R)(x0 )2 + (xn )2 − 5Rxn < (xn )2 − 2Rxn < 0.
α=0

Step 3 We put(
Ur :=expη(x0 )nX!
xi Pi (x0 )nX|x0 | < R,(xi )2 < r2

)
⊂U

i=1

i=1

for r ∈ (0, R] and Sr := (∂Ur ) ∩ {b ≤ 0}. Then b ≥ 0 on Sr by (5.4). Moreover, taking
smaller r if necessary, we have
Sr ⊂ {b > 0} ∪ {ϑ > 0}
since b < 0 on Sr \ A (by the ﬁrst variation formula for bηt for large t), ϑ > 0 on A \ B
by (b), and b > 0 on B ∩ {b ≤ 0}.
This implies that inf Sr ∩{ϑ≤0} b > 0, therefore, for small τ > 0 (depending on r),
f := b + τ ϑ is positive on Sr . Set Ur− := Ur ∩ {b ≤ 0} and observe that∂Ur− ⊂ Sr ∪◦
−1
U r ∩ b (0) .

Since f (x0 ) = 0 by (a) and f |Sr > 0, f |Ur− takes its minimum value at some point
◦

◦
−1
z ∈ U−
∪
U
∩
b
(0)
.
r
r
◦

In fact, by construction, we ﬁnd df (z) 6= 0 (for small τ ) and hence z ∈ U r ∩ b−1 (0).
19For large s > 0, let ζ − : [0, ∞) −→ M be an asymptote of η̄(t) = η(−t) of unit speed←−
emanating from z with respect to L , and we considerρ̄(x) := b(z) + s − d ζ − (s), x ,which is an upper support function for b at z in the same manner as (5.2). Then,fs := ρ̄ + τ ϑ
is an upper support function for f at z and minUr− fs is attained again at z.
Next, let ζ + be an asymptote of η|[0,∞) of unit speed with ζ + (0) = z and putρ(x) := s − d x, ζ + (s) ,which is an upper support function for b at z by Lemma 5.2. Then the level surface
Hs := ρ−1 (0) ∩ Ur passing through z is a smooth g∇(−ρ) -spacelike hypersurface. Sinceb ≤ ρ = 0 on Hs , we have Hs ⊂ Ur− . Hence fs |Hs takes its minimum at z and, sincedfs (z) 6= 0, we have dfs (z) = −Λ dρ(z) for some Λ > 0 close to 1 (both are close to −dx0
by virtue of Proposition 4.10).
Step 4 Now, we make use of the metrizability (3). Note that geodesics are common
between L and g. In this step we consider the case where η is a timelike straight line
also for g. Since asymptotes of η with respect to g has tangent vectors close to η̇ by
Proposition 4.10, they are timelike also for L and RicN ≥ 0 holds along them in (M, g, Ψ).
Thus we can apply the splitting theorem in [WW] to obtain a local isometric splittingc , g) = (R × Σ,b −dt2 + ĥ) of a neighborhood W
c of η, and Ψ is constant on each line(W
c is decomposed into those on R and Σ.
bt 7−→ (t, x). Note that the geodesic equation on W
Since parallel transports preserve L in Berwald spacetimes (see, e.g., [BCS, Propoc , L) inherits the product structure of (Wc, g) (whereas Σb may not besition 10.1.1]), (Wc including η if necessary, we have aspacelike for gη̇ or L). Precisely, taking smaller W
c = R × Σ such that the map (s, x) 7−→ (s + t, x) preserves L for all
modiﬁed splitting W(s, x) ∈ R × Σ and t ∈ R and that Σ is spacelike and perpendicular to V with respect togV , where V is the parallel vector ﬁeld extending η̇.
In this product structure, we ﬁnd that the asymptotes ζ − and ζ + (for L) together
form a straight line parallel to η. This implies that ζ̇ + (0) = −ζ̇ − (0), and hencedρ̄(z) = −dρ(z).

(6.3)

Then we have
τ dϑ(z) = dfs (z) − dρ̄(z) = (1 − Λ)dρ(z).
This is, however, a contradiction since dϑ(z) is close to −σ dxn by construction. Therefore,b + b = 0 holds on a neighborhood of η.
Step 5 Finally, we consider the remaining case where η is not a straight line for g. Thene the Lorentz–Finsler structure of M
f induced
we need the hypothesis (4). Denote by L
f of η is a timelike straight line for g̃ by (4), we can split
from L. Since a lift η̃ : R −→ M
20

f, g̃, Ψ)e and the Busemann functions bη̃ and bη̃ with respect to L
e satisﬁes bη̃ + bη̃ = 0(M
on a neighborhood of η̃ by the above argument.
f −→ M be the projection and observe thatLet π : M
dLe p, η̃(t) ≤ dL π(p), η(t)f close to η̃ and large t > 0. Therefore we obtainfor p ∈ M
0 ≤ (b + b) ◦ π ≤ bη̃ + bη̃ = 0,
which implies that b + b = 0 holds on a neighborhood of η. This completes the proof. □A typical situation where η is not a straight line for g is that M is a cylinder:(M, g) = R × (R/Z), −dt2 + ds2 ,η(t) = (t, ct)
for small c > 0. By modifying g, we can construct a ﬂat Lorentzian metric for which η isa timelike straight line. Two more remarks on the above proof are in order.
Remark 6.4 The above proof relying on the splitting theorem of g is obviously not a
standard one. The strategy in [Es] is to show that the Laplacian of fs |Hs is negative at z
by using the Laplacian comparison theorem (Theorem 3.5, Corollary 3.6) and the Hessian
bound (Lemma 6.2). One can follow this line to some extent by using the linearized
Laplacian ∆∇ρ̄ with respect to g∇ρ̄ and ﬁne properties of the Fermi coordinates. In
general, however, there is a diﬃculty arising from the absence of (6.3) (precisely, we do
not know how to estimate ∆∇ρ̄ ρ). This is the reason why we assumed the metrizability
and digressed from [Es] in Steps 4 and 5.
Remark 6.5 (The case of ψm ) When we consider the weight function ψm associatedwith a measure m on M as in Remark 3.3, along any geodesic ξ, volgξ̇ is a constantmultiplication of volg since parallel transports preserve L in Berwald spacetimes. Hence
RicN for (M, L, m) coincides with that for (M, g, Ψm ), where Ψm ∈ C ∞ (M ) is given bym = e−Ψm volg , and we can apply the above proof with (M, g, Ψm ).

7

Proof of the splitting theorem

This last section is devoted to the proof of the main theorem. We need to recall somestandard concepts in Lorentzian geometry and their counterparts in the weighted setting;see [LMO1, LMO2] for details.
Let ζ : [0, l) −→ M be a timelike geodesic of unit speed and denote by Nζ (t) ⊂ Tζ(t) M
the n-dimensional subspace gζ̇(t) -orthogonal to ζ̇(t). Given a Lagrange tensor ﬁeld J(t) :Nζ (t) −→ Nζ (t) along ζ, we set B := J′ J−1 . Then we have the Riccati equation
B′ + B2 + R = 0,
where R(t) := Rζ̇(t) : Nζ (t) −→ Nζ (t) is the curvature endomorphism.
21

(7.1)

Taking the weight function Ψ on M into account, we deﬁne2(1−ϵ)(Ψ ◦ ζ)′ (t)Ψ(ζ(t))nBϵ (t) := eB(t) −In (t) ,n(7.2)

where In (t) : Nζ (t) −→ Nζ (t) is the identity. We also deﬁneno2(1−ϵ)
Ψ(ζ(t))′
ntrace B(t) − (Ψ ◦ ζ) (t) ,θϵ (t) := trace Bϵ (t) = eσϵ (t) := Bϵ (t) −θϵ (t)In (t)nas the expansion scalar and the shear tensor. Then, for N ∈ (−∞, 0) ∪ [n, +∞], it
was established in [LMO1, Theorem 5.6] the following timelike weighted Raychaudhuri
equation:2
N (N − n) ϵθϵ (Ψ ◦ ζ)∗∗2θϵ + cθϵ +++ trace(σϵ2 ) + RicN (ζ ∗ ) = 0,
(7.3)
n
N
N −n
where c = c(N, ϵ) > 0 in (3.2),
Z t
φζ (t) :=e2(ϵ−1)
Ψ(ζ(s))
n

ds,

0


2(1−ϵ)′
Ψ(ζ(t)) ′nθϵ∗ (t) := (θϵ ◦ φ−1
θϵ (t),ζ ) φζ (t) = e
and similarly (Ψ ◦ ζ)∗ (t) := e7.12(1−ϵ)
Ψ(ζ(t))n(Ψ ◦ ζ)′ (t) and ζ ∗ (t) := e2(1−ϵ)
Ψ(ζ(t))nζ̇(t).

Local splitting

We ﬁrst build a local splitting on top of Proposition 6.3.
Proposition 7.1 (Local isometric splitting) Let (M, L, Ψ) be a weighted Berwald spacetime as in Proposition 6.3. Then a neighborhood W ′ of η(R) isometrically splits in thesense that there exists an n-dimensional Riemannian manifold (Σ, h) such that (W ′ , g∇bη )is isometric to (R × Σ, −dt2 + h).
Furthermore, for every x ∈ Σ, the image ζ(t) of (t, x) is a straight line bi-asymptotic
to η and Ψ is constant on ζ.
The latter assertion is interpreted as the splitting of the weight function Ψ.
Proof. We again suppress η in bη and bη . As is common with splitting theorems, weshall show that b is aﬃne (see [Es, Proposition 6.3] for the Lorentzian case and [EH,
Section 4] for the Riemannian case), for which the Berwald condition is essential (we refer
to [Oh3, Lemma 5.1] for the Finsler case).
Fix z ∈ W and let ζ : R −→ M be the timelike straight line of unit speed bi-asymptotic
to η with ζ(0) = z (given by Lemma 5.5). Since there is no conjugate point along ζ, we22

have R(t) = 0 on Nζ (t) for any t ∈ R by [LMO1, Proposition 7.6]. Similarly to the proofof Proposition 6.3, we ﬁx (large) s > 0 and considerρ̄s (x) := b(z) + s − d ζ(−s), x ,which is an upper support function for b at z. Now, let Js (t) : Nζ (t) −→ Nζ (t) be theLagrange tensor ﬁeld along ζ with Js (−s) = 0 and J′s (−s) = In (see [LMO1, Proposition 5.13] for a construction). We set Bs := J′s J−1
s , and observe that it is the Hessian ofthe distance function ds := d(ζ(−s), ·) as we saw in the proof of [LMO2, Theorem 5.8].
Precisely, we have [Bs (t)](w) = ∇2 (−ds )(w) for all w ∈ Nζ (t).
Next we deﬁne2(1−ϵ)(Ψ ◦ ζ)′sΨ◦ζs
Bϵ := e nB −In ,θϵs := trace(Bsϵ ),nsimilarly to (7.2). Since ζ contains no conjugate point and is ϵ-complete, it follows from
[LMO1, Proposition 5.8] that θϵs ≥ 0. Combining this lower bound and the fact that Bs (0)is non-increasing in s as in the proof of Lemma 6.2, we can take the limit B := lims→∞ Bs(we remark that only Bs depends on s in the deﬁnition of Bsϵ ). Note that B is smoothby the bootstrap argument and satisﬁes the Riccati equation (7.1) with R ≡ 0, i.e.,B′ + B2 = 0. Then we deﬁne Bϵ in the same way as (7.2) and observe from the absenceof conjugate points and [LMO1, Proposition 5.8] that θϵ ≡ 0. Therefore it follows from
(7.3) and RicN (ζ̇) ≥ 0 that (Ψ ◦ ζ)′ ≡ 0, trace(σϵ2 ) ≡ 0 and RicN (ζ̇) ≡ 0. In particular,
we have B ≡ 0.
We deduce from B ≡ 0 that the Hessian of ρ̄s at z can be arbitrarily close to zeroas s goes to inﬁnity. We similarly ﬁnd that ρs (x) := b(z) + s − d(x, ζ(s)) is an uppersupport function for b with the same property. Hence, for any geodesic ξ : [0, 1] −→ W ,the functions b ◦ ξ and b ◦ ξ have upper support functions whose second order derivatives
arbitrarily close to zero (here we use the Berwald condition; recall Remark 2.6). Thisimplies that b ◦ ξ and b ◦ ξ are concave. Combining this with b + b = 0 on W , we ﬁndthat b ◦ ξ and b ◦ ξ are aﬃne functions along every geodesic in W . Therefore b|W issmooth and every level set of b in W is totally geodesic. Moreover, the gradient vector
ﬁeld ∇b is parallel in the sense that ∇2 b ≡ 0.
Now we setΣ := b−1 (0) ∩ W = b−1 (0) ∩ W
and deﬁne a map Θ : R × Σ −→ M by Θ(t, x) := ζx (t), where ζx : R −→ M is thetimelike straight line of unit speed bi-asymptotic to η with ζx (0) = x. By construction
ζx is an integral curve of ∇b and satisﬁes b(ζx (t)) = −b(ζx (t)) = t for all t ∈ R (recall(5.3)). In particular, b + b = 0 holds on the image of Θ. Since every integral curve of∇b is a geodesic, ∇b is parallel also with respect to g∇b (see [LMO1, Proposition 3.2]).
Hence one can apply the de Rham–Wu decomposition theorem [Wu] to see that Θ is a local
isometry, where Σ and Θ(R×Σ) are equipped with the Riemannian metric g∇b |T Σ and theLorentzian metric g∇b , respectively. Furthermore, Θ is injective since asymptotes of η do
not intersect. This completes the proof of the former assertion by setting W ′ := Θ(R×Σ).
The latter assertion (Ψ ◦ ζx )′ ≡ 0 was already seen above.
□23

One can see from the above proof the following corollary on the behavior of the original
Lorentz–Finsler structure L (compare this with [Oh3, Proposition 5.2] in the Finsler
setting).
Corollary 7.2 (Local isometric translations) Let (M, L, Ψ) be as in Proposition 7.1.
Then, in the product structure W ′ = R × Σ, the translations(s, x) ∈ R × Σ, t ∈ R,

Φt (s, x) := (s + t, x),
are isometric transformations of (W ′ , L).

Proof. It suﬃces to show that Φt preserves L. Given a straight line ζ(t) = Θ(t, x) andv ∈ Tx M , the vector ﬁeld V (t) := dΦt (v) along ζ is parallel with respect to g∇b byg
Proposition 7.1. Since Dζ̇ V = Dζ̇ ∇b V ≡ 0 by the fact that all integral curves of ∇b are
geodesic and ζ̇ = ∇b(ζ) (see [LMO1, Proposition 3.2]), we obtain 1 dd
L(V ) =gV (V, V ) = gV (Dζ̇ V, V ) ≡ 0
dt
2 dt
as desired (see [LMO1, (3.2)] for the second equality).

7.2□

Global splitting

We can extend the local splitting in Proposition 7.1 by the open-and-closed argument (see[Es, Section 7] and [BEE, §14.4]).
Theorem 7.3 (Splitting theorem) Let (M, L, Ψ) be a connected, timelike geodesicallycomplete, weighted Berwald spacetime of RicN ≥ 0 on Ω with N ∈ (−∞, 0) ∪ [n, +∞].
Suppose also that(1) the ϵ-completeness holds for some ϵ in the corresponding ϵ-range (3.1),(2) there is a timelike straight line η : R −→ M ,(3) there is a timelike geodesically complete Lorentzian structure g of M whose Levi-Civita
connection coincides with the Chern connection of L such that η is timelike for g,f, g̃) of (M, g), a lift of η is a timelike straight line.
(4) in the universal cover (MThen the Lorentzian manifold (M, g∇bη ) isometrically splits in the sense that there exists
an n-dimensional complete Riemannian manifold (Σ, h) such that (M, g∇bη ) is isometricto (R × Σ, −dt2 + h).
Furthermore, for every x ∈ Σ, the image ζ(t) of (t, x) is a straight line bi-asymptotic
to η and Ψ is constant on ζ.
Proof. We put Z := (bη )−1 (0) for brevity. We shall show that the local splitting as in
Proposition 7.1 can be extended to cover whole M . To this end, we consider the supremum
r0 of r > 0 such that there is an isometric embeddingΘ : (R × Σr , −dt2 + h) −→ (M, g∇bη )24

as in the proof of Proposition 7.1, where Σr ⊂ Z is the open ball with center η(0) andradius r with respect to g∇bη |T Z and h is the restriction of g∇bη |T Z to Σr . Proposition 7.1
ensures r0 > 0. Note that, for x ∈ Σr0 , ζx = Θ(·, x) is a timelike straight line of unit
speed and I(ζx ) = I(η) ⊃ Θ(R × Σr0 ). Moreover, bζx = bη = −bη holds on Θ(R × Σr0 )since it follows from Lemma 5.2(i) that bζx ≥ bη and similarly bη ≥ bζx .
If Θ(R × Σr0 ) 6= M , then r0 < ∞. For x ∈ ∂Σr0 , take a sequence (xk )k∈N in Σr0
converging to x. Then the straight line ζxk converges to a straight line ζx passing throughx (thanks to Lemma 4.3), and we have I(ζx ) = I(η) and bζx = bη as above. Applying
Z
Proposition 7.1 to ζx , one can extend Θ to R × (Σr0 ∪ Bx,τ) for some τ > 0, whereZ
Bx,τ
⊂ Z is the open ball with center x and radius τ with respect to g∇bη |T Z . Note that,Z
for z ∈ Bx,τ\ Σr0 , ξz = Θ(·, z) satisﬁes I(ξz ) = I(ζx ) = I(η) and bξz = bζx = bη . We also
observe that, for any y ∈ Σr0 , ζy and ξz do not intersect since bζy = bξz .
Applying the above argument to each x ∈ ∂Σr0 , we can extend Θ to R × Σr for somer > r0 , a contradiction to the choice of r0 . Therefore we conclude that Θ(R × Σr ) = M
for some r ∈ (0, ∞] and Θ gives the desired isometry (in particular, we have Σ = Z).
The completeness of Σ follows from the timelike geodesic completeness of (M, L). The□
last assertion (Ψ ◦ ζ)′ ≡ 0 has been shown in Proposition 7.1.
Similarly to Corollary 7.2, we obtain the following.
Corollary 7.4 (Isometric translations) Let (M, L, Ψ) be as in Theorem 7.3. Then,in the product structure M = R × Σ, the translations(s, x) ∈ R × Σ, t ∈ R,

Φt (s, x) := (s + t, x),
are isometric transformations of (M, L).
We close the article with some remarks.

Remark 7.5 (The case of ψm ) If we consider the weight funtion ψm induced from ameasure m on M (recall Remarks 3.3 and 6.5), then we have (ψm ◦ ζ̇)′ ≡ 0 which implies
that the push-forward of m by Φt coincides with m. Hence, Φt : (M, L, m) −→ (M, L, m)
is measure-preserving.
Remark 7.6 (Non-Berwald case) In the non-Berwald case, one cannot obtain isometric translations as in Corollary 7.4. This is because, since our hypotheses are imposed
only in future-directed timelike directions, one can modify L in spacelike directions toviolate the isometry of Φt .
Remark 7.7 (The case of N = 0) In the extremal case of N = 0, it was established in[WW] that a Lorentzian manifold (M, g, Ψ) has a warped product splitting (see [Wy] for
the Riemannian case). It is unclear if we can generalize it to the Lorentz–Finsler setting.
In our proof of Proposition 6.3, there is a diﬃculty in the construction of Σ in Step 4
b is a warped product.
when R × Σ
Acknowledgements. YL and SO were supported by JSPS Grant-in-Aid for ScientiﬁcResearch (KAKENHI) 19H01786. EM thanks the Department of Mathematics of Osaka
University for kind hospitality.
25References[Am]

L. Ambrosio, Calculus, heat ﬂow and curvature-dimension bounds in metric measurespaces. Proceedings of the International Congress of Mathematicians—Rio de Janeiro
2018. Vol. I. Plenary lectures, 301–340, World Sci. Publ., Hackensack, NJ, 2018.

[BCS]

D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann–Finsler geometry.
Springer-Verlag, New York, 2000.

[Ba]

R. Bartnik, Regularity of variational maximal surfaces. Acta Math. 161 (1988), 145–
181.

[Be]

J. K. Beem, Indeﬁnite Finsler spaces and timelike spaces. Can. J. Math. 22 (1970),
1035–1039.

[BEE]

J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry. Marcel
Dekker Inc., New York, 1996.

[BEMG] J. K. Beem, P. E. Ehrlich, S. Markvorsen and G. J. Galloway, A Toponogov splitting
theorem for Lorentzian manifolds. Global diﬀerential geometry and global analysis
1984 (Berlin, 1984), 1–13, Lecture Notes in Math., 1156, Springer, Berlin, 1985.
[Ca]

J. S. Case, Singularity theorems and the Lorentzian splitting theorem for the Bakry–
Emery–Ricci tensor. J. Geom. Phys. 60 (2010), 477–490.

[CM]

F. Cavalletti and A. Mondino, Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications. Preprint (2020). Available at arXiv:2004.08934[CG]

J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci
curvature. J. Diﬀerential Geometry 6 (1971/72), 119–128.

[Es]

J.-H. Eschenburg, The splitting theorem for space-times with strong energy condition.
J. Diﬀerential Geometry 27 (1988), 477–491.

[EG]

J.-H. Eschenburg and G. J. Galloway, Lines in space-times. Comm. Math. Phys. 148(1992), 209–216.

[EH]

J. Eschenburg and E. Heintze, An elementary proof of the Cheeger–Gromoll splitting
theorem. Ann. Global Anal. Geom. 2 (1984), 141–151.

[Fl]

J. L. Flores, The Riemannian and Lorentzian splitting theorems. Topics in modern
diﬀerential geometry, 1–20, Atlantis Trans. Geom., 1, Atlantis Press, [Paris], 2017.

[FHPV] A. Fuster, S. Heefer, C. Pfeifer and N. Voicu,
ability of Berwald Finsler spacetimes. Universe 6
https://doi.org/10.3390/universe6050064

On the non metriz(2020), no. 5:
64.

[FP]

A. Fuster and C. Pabst, Finsler pp-waves. Phys. Rev. D 94 (2016), no. 10, 104072, 5
pp.

[FPP]

A. Fuster, C. Pabst and C. Pfeifer, Berwald spacetimes and very special relativity.
Phys. Rev. D 98 (2018), no. 8, 084062, 14 pp.

26

[GT]

R. Gallego Torromé, On singular generalized Berwald spacetimes and the equivalenceprinciple. Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 6, 1750091, 21 pp.

[Ga1]

G. J. Galloway, Curvature, causality and completeness in space-times with causallycomplete spacelike slices. Math. Proc. Camb. Phil. Soc. 99 (1986), 367–375.

[Ga2]

G. J. Galloway, The Lorentzian splitting theorem without the completeness assumption. J. Diﬀerential Geometry 29 (1989), 373–387.

[Ga3]

G. J. Galloway, Some connections between global hyperbolicity and geodesic completeness. J. Geom. Phys. 6 (1989), 127–141.

[Ga4]

G. J. Galloway, The Lorentzian version of the Cheeger–Gromoll splitting theorem andits application to general relativity. Diﬀerential geometry: geometry in mathematicalphysics and related topics, 249–257, Proc. Sympos. Pure Math., 54, Part 2, Amer.
Math. Soc., Providence, RI, 1993.

[GH]

G. J. Galloway and A. Horta, Regularity of Lorentzian Busemann functions. Trans.
Amer. Math. Soc. 348 (1996), 2063–2084.

[HPV]

M. Hohmann, C. Pfeifer and N. Voicu, Cosmological Finsler spacetimes. Universe 6,
no. 5: 65. https://doi.org/10.3390/universe6050065[LMO1] Y. Lu, E. Minguzzi and S. Ohta, Geometry of weighted Lorentz–Finsler manifolds I:
Singularity theorems. J. Lond. Math. Soc. 104 (2021), 362–393.
[LMO2] Y. Lu, E. Minguzzi and S. Ohta, Comparison theorems on weighted Finsler manifolds
and spacetimes with ϵ-range. Anal. Geom. Metr. Spaces 10 (2022), 1–30.
[Mc]

R. J. McCann, Displacement convexity of Boltzmann’s entropy characterizes the strong
energy condition from general relativity. Camb. J. Math. 8 (2020), 609–681.

[Mi1]

E. Minguzzi, Limit curve theorems in Lorentzian geometry. J. Math. Phys. 49, 092501(2008).

[Mi2]

E. Minguzzi, Convex neighborhoods for Lipschitz connections and sprays. Monatsh.
Math. 177 (2015), 569–625.

[Mi3]

E. Minguzzi, Light cones in Finsler spacetimes. Comm. Math. Phys. 334 (2015), 1529–
1551.

[Mi4]

E. Minguzzi, Raychaudhuri equation and singularity theorems in Finsler spacetimes.
Class. Quantum Grav. 32, 185008 (2015).

[Mi5]

E. Minguzzi, Special coordinate systems in pseudo-Finsler geometry and the equivalence principle. J. Geom. Phys. 114 (2017), 336–347.

[Mi6]

E. Minguzzi, Causality theory for closed cone structures with applications. Rev. Math.
Phys. 31 (2019), 1930001, 139pp.

[Mi7]

E. Minguzzi, Lorentzian causality theory. Living Reviews in Relativity 22, 3 (2019).
https://doi.org/10.1007/s41114-019-0019-x.

27

[Ne]

R. P. A. C. Newman, A proof of the splitting conjecture of S.-T. Yau. J. Diﬀerential
Geometry 31 (1990), 163–184.

[Oh1]

S. Ohta, Finsler interpolation inequalities. Calc. Var. Partial Diﬀerential Equations 36(2009), 211–249.

[Oh2]

S. Ohta, Vanishing S-curvature of Randers spaces. Diﬀerential Geom. Appl. 29 (2011),
174–178.

[Oh3]

S. Ohta, Splitting theorems for Finsler manifolds of nonnegative Ricci curvature. J.
Reine Angew. Math. 700 (2015), 155–174.

[Oh4]

S. Ohta, (K, N )-convexity and the curvature-dimension condition for negative N . J.
Geom. Anal. 26 (2016), 2067–2096.

[Oh5]

S. Ohta, Comparison Finsler geometry. Springer Monographs in Mathematics.
Springer, Cham, 2021.

[OS]

S. Ohta and K.-T. Sturm, Heat ﬂow on Finsler manifolds. Comm. Pure Appl. Math.
62 (2009), 1386–1433.

[ON]

B. O’Neill, Semi-Riemannian geometry: With applications to relativity. Academic
Press, Inc., New York, 1983.

[Sh]

Z. Shen, Lectures on Finsler geometry. World Scientiﬁc Publishing Co., Singapore,
2001.

[Sz]

Z. I. Szabó, Positive deﬁnite Berwald spaces. Structure theorems on Berwald spaces.
Tensor (N.S.) 35 (1981), 25–39.

[WW]

E. Woolgar and W. Wylie, Curvature-dimension bounds for Lorentzian splitting theorems. J. Geom. Phys. 132 (2018), 131–145.

[Wu]

H. Wu, On the de Rham decomposition theorem. Illinois J. Math. 8 (1964), 291–311.

[Wy]

W. Wylie, A warped product version of the Cheeger–Gromoll splitting theorem. Trans.
Amer. Math. Soc. 369 (2017), 6661–6681.

[Ya]

S. T. Yau, Problem section. Seminar on Diﬀerential Geometry, pp. 669–706, Ann. of
Math. Stud., 102, Princeton Univ. Press, Princeton, N.J., 1982.

28