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# Differential Geometric Aspects of Causal Structures

Omid Makhmali Department of Mathematics and Statistics, McGill University
805 Sherbrooke Street West, Montreal, Quebec,Canada,
H3A 0B9<EMAIL_ADDRESS>
###### Abstract.

This article in concerned with causal structures, which are defined as a field
of tangentially non-degenerate projective hypersurfaces in the projectivized
tangent bundle of a manifold. The local equivalence problem of causal
structures on manifolds of dimension at least four is solved using Cartan’s
method of equivalence, leading to an $\\{e\\}$-structure over some principal
bundle. It is shown that these structures correspond to parabolic geometries
of type $(D_{n},P_{1,2})$ and $(B_{n},P_{1,2})$ when $n\geq 4$ and
$(D_{3},P_{1,2,3}).$ The essential local invariants are determined and
interpreted geometrically. Several special classes of causal structures are
considered including those that are a lift of pseudo-conformal structures and
those referred to as causal structures with vanishing Wsf curvature. A
rigidity theorem for causal structures of vanishing Wsf curvature over compact
manifolds is obtained. A notion of half-flatness is defined for four
dimensional causal structures whose null cones are ruled projective surfaces,
which generalizes the notion of half-flat conformal structures of split
signature. A twistorial construction for causal structures with vanishing Wsf
curvature is given. Finally, the unique causal structure of split signature in
dimension four with submaximal symmetry which does not descend to a pseudo-
conformal structure is introduced.

###### Key words and phrases:

Causal geometry, Conformal geometry, Equivalence method, Cartan connection,
Parabolic geometry

###### 1991 Mathematics Subject Classification:

53B55

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation and history
    2. 1.2 Main results
  2. 2 Local description of causal structures
    1. 2.1 Definitions and notational conventions
    2. 2.2 The flat model
    3. 2.3 The first order structure equations
    4. 2.4 Causal structures as parabolic geometries
    5. 2.5 Essential invariants and the Raychaudhuri equation
    6. 2.6 Examples
  3. 3 Causal structures with vanishing Wsf curvature
    1. 3.1 A compatibility condition
    2. 3.2 A rigidity theorem
    3. 3.3 Semi-integrable Lie contact structures
  4. 4 Four dimensions
    1. 4.1 Causal structures with ruled fibers
    2. 4.2 Submaximal causal structures of split signature
  5. A Full structure equations
    1. A.1 Prolongation and an {e}-structure
    2. A.2 Quantities in (A.17)
    3. A.3 Bianchi identities for $W_{ijkl}$

##  1\. Introduction

###  1.1. Motivation and history

In the General Theory of Relativity, space-time is represented by a four
dimensional manifold $M$ equipped with a Lorentzian metric $g$. An important
feature of a Lorentzian structure is its associated field of null cones in
$TM$ given by the set of vectors $v\in TM$ satisfying $g(v,v)=0.$ As a result,
it is possible to define a relation of _causality_ between the points of $M,$
i.e., a partial ordering $x\prec y$ defined by the property that $x$ can be
joined to $y$ by a curve that is either time-like or null. If $x$ and $y$
represent two events in space-time then $x\prec y$ means that the occurrence
of $x$ has an effect on $y.$ In [KP67] Kronheimer and Penrose examined
causality in a manifold $M$ of dimension $n$ on an axiomatic basis. The
starting point of their study is a continuous assignment of null cones in the
tangent space of each point of $M$ so that it is possible to define the set of
points that are inside, outside and on the null cone in each tangent
space.111It should be said that in [KP67] it is assumed that each null cone is
given by the vanishing locus of a quadratic form of signature $(n,1).$ The
importance of the causal relationship between points of space-time became more
significant in the context of the singularity theorems of Penrose and Hawking
[Pen65, Pen72, HE73]. The study of causal relations in spaces equipped with a
field of convex null cones in $TM$ received a major contribution from the
Alexandrov school in Russia under the name of _chronogeometry_ [Ale67, Gut95],
i.e., a geometry determined by the chronological relations between the points
of the manifold. The objective was to use the principles of causality and
symmetry to give an axiomatic treatment of special relativity. In [Seg76],
Segal attempted to give such a treatment for general relativity (see also
[DS01, Lev93, Wor84, Tau77, EPS72].) Subsequently, structures defined by
assigning a convex cone in each tangent space, termed as causal structures,
came to play a role in the Lie theory of semi-groups [HÓ97], causal boundaries
[GPS05], and hyperbolic systems of partial differential equations [Sch13].

Rather than to investigate causality relations between the points of a
manifold with a given causal structure, the main idea of this article is to
gain a deeper understanding of the geometry of causal structures through the
solution of the Cartan equivalence problem. This approach enables one to
construct a complete set of local invariants and also to obtain some results
on symmetries of causal structures. A global rigidity result can also be
obtained through this approach.

Before going any further, a more precise definition of causal structures is in
order. Given an $(n+1)$-dimensional manifold $M,$ with $n\geq 3,$ a causal
structure $(M,\mathcal{C})$ of signature $(p+1,q+1)$ with $p,q\geq 0,$ and
$p+q=n-1$ is given by a sub-bundle of the projectivized tangent bundle
$\pi:\mathcal{C}\subset\mathbb{P}TM\rightarrow M,$ whose fibers
$\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$ are projective hypersurfaces222Here,
a hypersurface is always taken to be a codimension one submanifold. with
projective second fundamental form of signature $(p,q)$ (see Definition 2.4).
Note that in this definition the convexity of the null cones $\mathcal{C}_{x}$
is not assumed.

Two causal structures $(M,\mathcal{C})$ and
$(M^{\prime},\mathcal{C}^{\prime})$ are locally equivalent around the points
$p\in M$ and $p^{\prime}\in M^{\prime}$ if there exists a diffeomorphism
$\psi:U\longrightarrow U^{\prime},$ where $p\in U\subset M,$
$p^{\prime}=\psi(p)\in U^{\prime}\subset M^{\prime},$ satisfying
$\psi_{*}(\mathcal{C}_{x})=\mathcal{C}_{\psi(x)}$ for all $x\in U.$

In order to describe a causal structure in an open set $U\subset TM,$ one can
use a defining function $L:TM\rightarrow\mathbb{R},$ $(x^{i};y^{j})\mapsto
L(x^{i};y^{j}),$ where $(x^{0},\dots,x^{n})$ are coordinates for $M$ and
$(y^{0},\dots,y^{n})$ are fiber coordinates. The function $L$ is assumed to be
homogeneous of some degree $r_{1}$ in the fiber variables and is referred to
as a Lagrangian for the causal structure. Then, the vanishing set of $L$ over
$U$ coincides with $U\cap\widehat{\mathcal{C}}$ where
$\widehat{\mathcal{C}}\subset TM$ is the cone over
$\mathcal{C}\subset\mathbb{P}TM.$ The functions $L(x^{i};y^{i})$ and
$S(x^{i};y^{i})L(x^{i};y^{i}),$ where $S(x^{i};y^{i})$ is nowhere vanishing
and homogeneous of degree $r_{2}$ in $y^{i}$’s, define the same causal
structure.

As an example, if the fibers $\mathcal{C}_{x}$ are hyperquadrics defined by a
pseudo-Riemannian metric $g$ of signature $(p+1,q+1),$ then the causal
structure $(M,\mathcal{C})$ given by the family of null cones of $g$
corresponds to the pseudo-conformal structure induced by the metric $g.$333A
pseudo-conformal structure refers to the conformal class of a pseudo-
Riemannian metric [AG96]. In other words, there exists a local defining
function of this causal structure of the form
$F(x^{i};y^{j})=g_{ij}(x)y^{i}y^{j}.$

The above description makes it clear that the relation of causal structures to
pseudo-conformal structures is an analogue of what Finsler structures are to
Riemannian structures. Recall that a Finsler structure
$(M^{n+1},\Sigma^{2n+1})$ is given by a codimension one sub-bundle
$\Sigma\subset TM,$ whose fibers $\Sigma_{x}\subset T_{x}M$ are strictly
convex affine hypersurfaces, i.e., the second fundamental form of each fibers
is positive definite. The sub-bundle $\Sigma$ is called the indicatrix bundle
(see [Bry02]).444 Similarly, it is possible to define pseudo-Finsler
structures of signature $(p,q),p+q=n+1$ by imposing the condition that the
second fundamental form of the fibers $\Sigma_{x}$ have signature $(p-1,q).$

The main motivation behind the study of causal structures comes from a variety
of perspectives. As a part of the geometric study of differential equations,
it was shown by Holland and Sparling [HS10] that there is locally a one-to-one
correspondence between contact equivalence classes of third order ODEs
$y^{\prime\prime\prime}=F(x,y,y^{\prime},y^{\prime\prime})$ and causal
structures on their locally defined three dimensional solution space $M$. A
causal structure is given in this setting by a family of curves
$\pi:\mathcal{C}\rightarrow M$ where
$\mathcal{C}_{x}:=\pi^{-1}(x)\subset\mathbb{P}T_{x}M$ is a _non-degenerate_
curve in the sense that it admits a well-defined projective Frenet frame. This
causal structure descends to a conformal Lorentzian structure on $M$ if a
certain contact invariant of the third order ODE known as the Wünschmann
invariant vanishes. In other words, it is shown in [HS10] that $\mathcal{C}$
is locally isomorphic to $J^{2}(\mathbb{R},\mathbb{R})$555 In this article
$J^{k}(\mathbb{R}^{l},\mathbb{R}^{m})$ denotes the space of $k$-jets of
functions from $\mathbb{R}^{l}$ to $\mathbb{R}^{m}$. endowed with a foliation
by lifts of contact curves on $J^{1}(\mathbb{R},\mathbb{R})$ and that the
Wünschmann invariant at each point of $J^{2}(\mathbb{R},\mathbb{R})$ coincides
with the projective curvature of the curve $\mathcal{C}_{x}$. As a result, the
vanishing of the Wünschmann invariant implies that the curves
$\mathcal{C}_{x}$ are conics, i.e., the causal structure is conformal
Lorentzian.

The work of Holland and Sparling is a part of the program of studying
geometries arising from differential equations, which has a long history going
back to Monge, Jacobi, Lie, Darboux, Goursat, Cartan and others. In the case
of third order ODEs under contact transformations, Chern [Che40] used Cartan’s
method to solve the equivalence problem and showed that the solution depends
on two essential relative invariants, namely the Wünschmann invariant $I(F)$
and the invariant $C(F)=\frac{\partial^{4}}{\partial q^{4}}F(x,y,p,q).$
Furthermore, he observed that if $I(F)=0,$ then the space of solutions of the
ODE can be endowed with a conformal Lorentzian structure. Using Tanaka theory
[Tan79], Sato and Yashikawa [SY98] showed how one can use Chern’s
$\\{e\\}$-structure to associate a normal Cartan connection on some
10-dimensional principal bundle to the contact equivalence class of a third
order ODE. In particular, they expressed the relative contact invariants of
the ODE in terms of the curvature of the Cartan connection. Godlinski and
Nurowski [GN09] explicitly computed the normal Cartan connection and explored
other geometric structures that arise from a third order ODE. In particular,
they used a result of Fox [Fox05b] to show if $C(F)=0,$ then the ODE induces a
_contact projective structure_ 666A contact path geometry over a contact
manifold $N$ is given by a family of contact paths with the property that at
each point $x\in N$, a unique curve of the family passes through each contact
direction in $T_{x}M.$ A contact projective geometry is a contact path
geometry whose contact paths are geodesics of some affine connection.  on
$J^{1}(\mathbb{R},\mathbb{R}).$

One can realize the works mentioned above on third order ODEs in terms of
geometries arising from the double fibration

(1.1)
$J^{1}(\mathbb{R},\mathbb{R})\overset{\tau}{\longleftarrow}J^{2}(\mathbb{R},\mathbb{R})\overset{\pi}{\longrightarrow}M$

where the map $\pi$ is the quotient map of $J^{2}(\mathbb{R},\mathbb{R})$ by
the foliation defined by the 2-jets of the solutions of the ODE. This map
endows $M$ with a causal structure. The null geodesics of $M$ are given by
$\pi\circ\tau^{-1}(p)$ where $p\in J^{1}(\mathbb{R},\mathbb{R}),$ while the
contact geodesics are given by $\tau\circ\pi^{-1}(q)$ where $q\in M.$

Equivalently, given a causal structure $(M,\mathcal{C})$ in dimension three,
Holland and Sparling showed that the double fibration above is equivalent to

$\mathcal{N}\overset{\tau}{\longleftarrow}\mathcal{C}\overset{\pi}{\longrightarrow}M$

where $\mathcal{N}$ is the space of _characteristic curves_ (or _null
geodesics_)777In this article the term characteristic curves is used to refer
to the lift of null geodesics to the null cone bundle. of the causal
structure. The precise definition of these curves is given in Section 2.3.5.
It turns out that starting with a causal structure $(M,\mathcal{C}),$ one can
show that locally $\mathcal{C}\cong J^{2}(\mathbb{R},\mathbb{R})$ and
$\mathcal{N}\cong J^{1}(\mathbb{R},\mathbb{R}).$ Therefore, all the results
mentioned above can be translated in terms of local invariants of the causal
structure in three dimensions.

The correspondence above motivates the study of the local invariants of causal
geometries in higher dimensions and differential equations that can be
associated to them. As will be explained, this study turns out to be related
to a number of current themes of research.

In [HS11], Holland and Sparling studied causal structures in higher dimensions
and showed that an analogue of the Weyl sectional curvature can be defined.
Also, they showed for causal structures the Raychaudhuri - Sachs equations
from General Relativity remains valid for higher dimensional causal structures
(also see [AJ16, Min15]).

Before describing the nature of the complete set of invariants for causal
structures it is helpful to make parallels with what happens in Finsler
geometry. On an $(n+1)$-dimensional manifold $M$ the indicatrix bundle
$\Sigma$ of a Finsler structure is $(2n+1)$-dimensional equipped with a
contact 1-form, i.e., a 1-form $\omega\in T^{*}\Sigma$ satisfying
$\omega{\,{\wedge}\;}(\mathrm{d}\omega)^{n}\neq 0.$ The 1-form $\omega$ is
called the Hilbert form of the Finsler structure. As indicated by Cartan
[Car34, Car30] and Chern [Che48], in Finsler geometry there are two essential
invariants comprising the total obstruction to local flatness, namely the flag
curvature and the centro-affine cubic form of the fibers $\mathcal{C}_{x}$
(also referred to as the Cartan torsion). The vanishing of the Cartan torsion
implies that the Finsler structure is a Riemannian structure, in which case
the flag curvature coincides with the sectional curvature.

Regarding causal structures, the main step is to realize that the
$2n$-dimensional space $\mathcal{C}$ carries a quasi-contact structure (also
known as even contact structure), i.e., a 1-form $\omega^{0}$ satisfying
$\omega^{0}{\,{\wedge}\;}(\mathrm{d}\omega^{0})^{n-1}\neq 0$ and
$(\mathrm{d}\omega^{0})^{n}=0.$ The 1-form $\omega^{0}$ is referred to as the
_projective Hilbert form_.888This terminology is introduced in analogy with
the Hilbert form in Finsler geometry and the calculus of variations [BCS00].
It is shown that, for $n\geq 3,$ there are two essential local invariants
which form the total obstruction to local flatness. As one would expect, the
Fubini cubic form of each fiber $\mathcal{C}_{x}$ as a projective
hypersurfaces in $\mathbb{P}T_{x}M$ constitutes one of these obstructions
whose vanishing implies that the causal structure is a pseudo-conformal
structure. The second essential invariant is a generalization of the sectional
Weyl curvature of a pseudo-conformal structure restricted to the _shadow
space_ 999The terminology is due to a physical interpretation made by Sachs
[Sac61] according to which if an object follows a congruence of null
geodesics, its infinitesimal shadow casts onto the pull-back of this space.
This term was initially used by Holland and Sparling in [HS11]. as found by
Holland and Sparling in [HS11] (see Section 2.5). By analogy with Finsler
geometry, the second essential invariant is called the Weyl shadow flag
curvature (referred to as Wsf curvature.)

Furthermore, from the first order structure equations for causal geometries it
follows that a causal structure $(M,\mathcal{C})$ can be equivalently
formulated as having a manifold $\mathcal{C}^{2n}$ equipped with a
distribution $D$ of rank $n$ which contains an involutive corank one
subdistribution $\Delta\subset D$ and satisfies a certain non-degeneracy
condition. The (small) growth vector of $D$ is $(n,2n-1,2n)$, i.e.,
$\operatorname{rank}[D,D]=2n-1,$ and
$\operatorname{rank}[D,[D,D]]=2n.$101010The sections of the distribution
$[D,D]$ are given by $\Gamma([D,D])=[\Gamma(D),\Gamma(D)].$ (see Section
2.4.2.) In this case, $M^{n+1}$ is the space of integral leaves of
$\Delta.$111111In [ANN15], these structures are referred to as the $B_{n}$ and
$D_{n}$ classes of what they call parabolic geometries of Monge type. This
description identifies causal geometries as parabolic geometries modeled on
$B_{n}/P_{1,2}$ and $D_{n}/P_{1,2}$ for $n\geq 4$ and $D_{3}/P_{1,2,3},$ with
their natural $|3|$-grading. As will be shown, they have some similar features
to path geometries [Gro00] and contact path geometries [Fox05a], which
correspond respectively to parabolic geometries modeled on $A_{n}/P_{1,2}$ and
$C_{n}/P_{1,2}$.

Another essential part of any causal structure is given by its set of
_characteristic curves._ Note that as a quasi-contact manifold, $\mathcal{C}$
is equipped with the characteristic line field of the projective Hilbert form,
i.e., the unique degenerate direction of $\mathrm{d}\omega^{0}$ lying in
$\textrm{Ker}\,\,\omega^{0},$ defined by vector fields $\mathbf{v}$ satisfying

(1.2) $\omega^{0}(\mathbf{v})=0,\qquad\mathbf{v}\im\mathrm{d}\omega^{0}=0.$

The integral curves of a characteristic vector field $\mathbf{v}$ is called a
characteristic curve. In the case of pseudo-conformal structures the
projection of these curves to $M$ coincide with null geodesics.

The description above shows that, unlike Finsler geometry, the definition of
characteristic curves does not depend on a choice of metric. Recall that in
Finsler geometry geodesics can be defined as the extremals of the arc-length
functional

$\Phi(\gamma)=\int_{I}\gamma^{*}\phi,$

where $\gamma:(a,b)=:I\rightarrow\mathcal{C}$ is a curve in $\mathcal{C}$ and
$\gamma^{*}\phi=L(x^{i};y^{i})\mathrm{d}t.$ The function $L(x^{i};y^{i})$ is
called the Finsler metric which in the calculus of variations is referred to
as the Lagrangian. The Finsler metric is homogeneous in $y^{i}$’s and has non-
degenerate vertical Hessian121212It is usually assumed that the Finsler metric
when restricted to $T_{x}M$ only vanishes at the origin and is positive
elsewhere [BCS00]. However, as was mentioned previously, the condition can be
relaxed [She01].. When addressing the geodesics of a Finsler structure, or
more generally extremal manifolds in the calculus of variations, it is
generally assumed that the Lagrangian does not vanish along them [Gri83]. The
study of extremal curves along which the Lagrangian vanishes seems to have
received much less attention in the literature (see [Run83, Run84]). The
exception is the case of Lagrangians that are quadratic in $y^{i}$’s. This is
due to the fact that they arise in the context of pseudo-conformal structures.
For instance, this problem is briefly discussed by Guillemin [Gui89], where he
studies the deformations of the product metric
$\pi^{*}_{1}\mathrm{d}t^{2}\oplus-\pi_{2}^{*}g$ on $S^{1}\times S^{2}$ where
$\pi_{1}$ and $\pi_{2}$ are projections onto $S^{1},$ and $S^{2}$ and $g$ is
the round metric on $S^{2},$ which is Zollfrei, i.e., all of its null
geodesics are closed with the same period. In Section 2.5.3 it is discussed
how causal structures provide a geometric setup for the variational problems
involving certain classes of Lagrangians $L(x^{i};y^{i})$ which are
homogeneous in $y^{i}$’s and have vertical Hessian of maximal rank over their
vanishing set.

The foliation of $\mathcal{C}$ defined by the characteristic curves allows one
to define the quotient map $\tau:\mathcal{C}\rightarrow\mathcal{N}$ where
$\mathcal{N}$ is $(2n-1)$-dimensional and is called the space of
characteristic curves of $\mathcal{C}$. A priori, $\mathcal{N}$ is only
locally defined around generic points and can be used to give a geometric
description of the theory of correspondence spaces for causal structure in the
sense of Čap [Čap05] (see Section 3.3). Note that the space $\mathcal{N},$
when globally defined, is of interest in Penrose’s twistor theory [LeB90] and
a twistorial desciption of causal relations among the points of the space-time
[Low06].

Finally, an interesting instance of studying causal structure can be found in
[Hwa13] where Hwang considers complex causal structures arising on certain
uniruled projective manifolds, $M^{n+1},$ satisfying the property that through
a generic point $x\in M$ there passes an $(n-1)$-parameter family of rational
curves of minimal degree. The hypersurface
$\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$ obtained from the tangent directions
to such curves at the point $x$ is called the variety of minimal rational
tangent (or VMRT). Under the assumption that $\mathcal{C}_{x}$ is smooth of
degree $\geq 3,$ he shows that the causal structure defined by them is locally
isotrivially flat (see Definition 2.18)131313Hwang refers to such structures
as flat cone structures of codimension one.. Furthermore, it turns out that
the characteristic curves for such causal structures coincide with rational
curves of minimal degree141414Motivated by the notion of _conformal torsion_
introduced by LeBrun [LeB83] and _contact torsion_ introduced by Fox [Fox05a],
one can define _causal torsion_ whose vanishing implies that the rational
curves of minimal degree coincide with the characteristic curves of the causal
structure they define. The precise definition of causal torsion is not given
in this article. . Finally, assuming that $M$ has Picard number 1 and $n\geq
2,$ he shows that $M$ is biregular to a hyperquadric equipped with its natural
causal structure (see Section 2.2). This theorem is a generalization of what
has been considered previously in [Bel01, Ye94] for complex conformal
structures.

More broadly, Hwang’s study of causal structures is a part of the differential
geometric characterization of uniruled varieties initiated by Hwang and Mok
[Hwa12] via geometric structures referred to as _cone structures_ (see
Definition 2.1). The starting point for Hwang and Mok is a given family of
rational curves of minimal degree whose tangent directions at a generic point
$x\in M$ give rise to the VMRT $\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$. For
instance, if $\mathcal{C}_{x}=\mathbb{P}T_{x}M$ one obtains the complex
analogue of path geometry [Gro00] and if $\mathcal{C}_{x}$ is a hyperplane
which induces a contact distribution on $M,$ one obtains a complex contact
path geometry [Fox05a]. The program of Hwang and Mok seems interesting and
calls for further exploration from the perspective of the method of
equivalence, especially through the lens of flag structures as developed in
[DZ12].

###  1.2. Main results

In all the statements of this article smoothness of underlying manifolds and
maps is always assumed and the dimension of $M$ is always taken to be at least
four unless otherwise specified.

In Section 2, after reviewing the flat model for causal structures, the first
order structure equations for a causal geometry $(M^{n+1},\mathcal{C})$ of
signature $(p+1,q+1)$ are derived. It is shown that the bundle of null cones
$\mathcal{C}$ is foliated by characteristic curves and is endowed with the
conformal class of a quadratic form which is degenerate along the vertical
directions of the fibration $\mathcal{C}\rightarrow M.$ Subsequently, the
first order structure equations are used in the Appendix to obtain structure
equations leading to an $\\{e\\}$-structure.

The essential theorem of this article that is stated in Section 2.3.13 is the
following.

######  Theorem 2.12.

To a causal structure $(M^{n+1},\mathcal{C}^{2n}),$ one can associate an
$\\{e\\}$-structure on a principal bundle
$P_{1,2}\rightarrow\mathcal{P}\rightarrow\mathcal{C}$ of dimension
$\frac{(n+3)(n+2)}{2}$ where $P_{1,2}\subset\mathrm{O}(p+2,q+2)$ is the
parabolic subgroup defined in (2.5). The essential invariants of a causal
structure are the Fubini cubic form of the fibers and the Wsf curvature. The
vanishing of the essential invariants implies that the $\\{e\\}$-structure
coincides with the one defined by the Maurer-Cartan forms of
$\mathfrak{o}(p+2,q+2)$.

The proof of the above theorem takes up most of the Appendix. Using the
{e}-structure thus obtained, it is straightforward to deduce that the
essential invariants are the Fubini cubic form and the Wsf curvature. Section
2.4 deals with the parabolic nature of causal structures. In Section 2.5 some
geometrical interpretations of the essential invariants are given. The first
Section ends with examples of causal structures.

In Section 3.1, the structure equations for causal geometries with vanishing
Wsf curvature are examined. It is shown that if the Lie derivative of the
Fubini cubic form along the vector fields tangent to the characteristic curves
is proportional to itself, then the Wsf curvature has to vanish, unless the
causal geometry is a pseudo-conformal structure. Such spaces are the causal
analogues of Landsberg spaces in Finsler geometry151515See [BCS00] for an
account.. Moreover, having defined the scalar shadow flag curvature (referred
to as the ssf curvature), and the notion of completeness of characteristic
curves for a causal structure, the following theorem is proved in Section 3.2.

######  Theorem 3.5.

Let $(M,\mathcal{C})$ be a causal structure with vanishing Wsf curvature over
a compact manifold with non-zero Fubini cubic form. If the characteristic
curves of $\mathcal{C}$ are complete, then the scalar flag curvature of
$(M,\mathcal{C})$ is non-negative.

The above theorem implies the following corollary.

######  Corollary 3.6.

Let $(M,\mathcal{C})$ be a compact causal structure with vanishing Wsf
curvature and complete characteristic curves. If it has negative ssf
curvature, then it is a flat pseudo-conformal structure.

The corollary above can be taken as a rigidity theorem which is a causal
analogue of the seminal work of Akbar-Zadeh in Finsler geometry [AZ88].

Section 3.3 involves the induced structure on the space of characteristic
curves, $\mathcal{N},$ of a causal structure with vanishing Wsf curvature. The
$(2n-1)$-dimensional manifold $\mathcal{N}$ is shown to have a contact
structure and is endowed with an $(n+1)$-parameter family of Legendrian
submanifolds with the property that through each point $\gamma\in\mathcal{N}$
there passes a 1-parameter family of them. Moreover, the sub-bundle of
$T\mathcal{N}$ defined by the contact distribution contains a Segré cone of
type $(2,n-1).$ The theorem below is proved in which the property of being
_semi-integrable_ means that at every point $\gamma\in\mathcal{N},$ the Segré
cone is ruled by the tangent spaces of the corresponding 1-parameter family of
Legendrian submanifolds passing through $\gamma.$

######  Theorem 3.10.

Given a causal structure with vanishing Wsf curvature $(\mathcal{C},M),$ its
space of characteristic curves admits a semi-integrable Lie contact structure.
Conversely, any semi-integrable Lie contact structure on a manifold
$\mathcal{N}$ induces a causal structure with vanishing Wsf curvature on the
space of its corresponding Legendrian submanifolds.

It should be noted that the theorem above is reminiscent of Grossman’s work
[Gro00] on torsion-free path geometries and the term semi-integrability is
borrowed from there. He showed that given a torsion-free path structure on
$N^{n+1}$, the space of paths, $\mathcal{S}^{2n},$ has a Segré structure,
i.e., each tangent space is equipped with a Segré cone of type $(2,n).$
Furthermore, $\mathcal{S}$ contains an $(n+1)$-parameter family of
$n$-dimensional submanifolds such that through each point
$\gamma\in\mathcal{S}$ there passes a 1-parameter family of them. Grossman
showed that at each point $\gamma\in\mathcal{S},$ the Segré cone is ruled by
the tangent spaces of the 1-parameter family of submanifolds passing through
$\gamma.$ He called such Segré structures _semi-integrable._ Furthermore,
twistorial constructions such as that of Theorem 3.10 fit exactly into Čap’s
theorem of correspondence spaces [Čap05]. However, Čap’s theory does not
contain the geometric interpretation given here in terms of existence of
submanifolds that rule the null cone or the Segré cone. Such interpretations
are clear from the constructions presented here.

In Section 4.1 a notion of half-flat causal structures of signature $(2,2)$ is
introduced using the existence of a 3-parameter family of surfaces whose
tangent planes at each point of $M$ give a ruling of the projective surfaces
$\mathcal{C}_{x}$. This notion of half-flatness coincides with the notion of
half-flatness for conformal structures of signature $(2,2)$.

Finally, in Section 4.2 a study of the symmetry algebra of certain non-flat
causal structures of split signature that do not induce a conformal structure
on $M$ is given from which the proposition below follows.

######  Proposition 4.8.

The submaximal causal structure of split signature that does not descend to a
pseudo-conformal structure has 8 dimensional symmetry algebra and is locally
equivalent to the locally isotrivially flat causal structure whose null cones
are projectively equivalent to Cayley’s cubic scroll.

Locally isotrivially flat causal structures, defined in 2.18, are the causal
analogue of locally Minkowskian Finsler spaces.

The Appendix contains the details of the equivalence problem calculations.
Since causal structures are examples of parabolic geometries of type
$(B_{n},P_{1,2})$ and $(D_{n},P_{1,2})$ if $n\geq 4$ and $(D_{3},P_{1,2,3})$,
they admit a regular and normal Cartan connection ([ANN15, ČS09]). Using the
machinery of Tanaka, the correct change of coframe can be given recursively,
in order to introduce a Cartan connection on the structure bundle of causal
geometries which is normal in the sense of Tanaka. However, for computational
reasons, writing down the precise form of the Cartan connection leads to very
long expressions which are not particularly enlightening (see [Mak16]).
Furthermore, the first order structure equations suffice for the purposes of
this article.

### Acknowledgments

This work was carried out under the guidance of Professor N. Kamran, whom I
would like to thank greatly for his clear explanation of Cartan’s method of
equivalence and introducing me to the work of Holland and Sparling [HS10]. His
encouragement, help and support during my PhD made this work possible. I am
also very much indebted to Dennis The for the many helpful conversations that
we had and for patiently explaining some aspects of parabolic geometry. The
funding for this work was provided by McGill University.

##  2\. Local description of causal structures

In this section, after introducing the necessary definitions and notations,
the first order structure equations of causal structures are derived. It is
assumed that the reader is familiar with Cartan’s method of equivalence for
$G$-structures (see [Gar89, Olv95] for an account). The essential torsion
terms appearing in the first structure equations are interpreted
geometrically. The derivation of the full structure equations is carried out
in the Appendix A. In Section 2.5 a discussion on the fundamental invariants
of a causal structure is given. Finally, the section ends with a few examples
of causal structures.

###  2.1. Definitions and notational conventions

Since the main purpose of this article is the local study of causal
geometries, the smoothness of manifolds and maps that are considered is always
assumed. In other words, the manifolds considered are assumed to be the
maximal open sets over which the necessary smoothness conditions are
satisfied.

Let $M$ be a smooth $(n+1)$-dimensional (real or complex) manifold, $n\geq 3$
and $\mathbb{P}TM$ denote its projectivized tangent bundle with projection
$\pi:\mathbb{P}TM\rightarrow M$, and fibers $\pi^{-1}(x)=\mathbb{P}T_{x}M.$
Given a tangent vector $y\in T_{x}M$, its projective equivalence class is
denoted by $[y]\in\mathbb{P}T_{x}M.$ For $S\subset\mathbb{P}T_{x}M$, the cone
over $S$ is defined as

$\widehat{S}:=\left\\{y\in T_{x}M\ \vline\ [y]\in S\right\\}\subset T_{x}M.$

Throughout this article, the indices range as

$0\leq i,j,k\leq n,\qquad 1\leq a,b,c\leq n-1,\qquad 0\leq\alpha,\beta\leq
n-1,$

and are subjected to the summation convention. When a set of 1-forms is
introduced as $\mathcal{I}=I\\{\omega^{i}\\}_{i=0}^{n},$ it is understood that
$\mathcal{I}$ is the ideal algebraically generated by
$\\{\omega^{0},\dots,\omega^{n}\\}$ in the exterior algebra of $M$.

The symmetrization and anti-symmetrization operations for tensors are denoted
using Penrose’s abstract index notation. For example, consider a tensor with
coefficients $A_{abcd}.$ The tensor $A_{(ab|c|d)}$ is symmetric with respect
to all indices except the third, and $A_{[a|bc|d]}$ is anti-symmetric in the
first and last indices. When dealing with symmetric forms such as
$g=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b},$ or
$F_{abc}\theta^{a}\circ\theta^{b}\circ\theta^{c}$ with $F_{abc}=F_{(abc)},$
the symbol $\circ$ denotes the symmetric tensor product. Bold-face letters
such as $\mathbf{i},\mathbf{j},\mathbf{a},\mathbf{b}$ are used to denote a
specific index, e.g. the set $\\{\omega^{\mathbf{a}}\\}$ only contains the
1-form $\omega^{\mathbf{a}}$ . The summation convention is not applied to
bold-face letters.

At this stage, before stating the definition of a causal structure, a more
general geometric structure, namely that of a _cone structure_ , is defined
[Hwa12]. The reason for this is that in the process of coframe adaption for
cone structures of codimension one, causal structures are distinguished as a
sub-class of such geometries via the characterizing property of having
_tangentially non-degenerate_ fibers (see Section 2.3.3).

######  Definition 2.1.

A cone structure of codimension $k\leq n$ on $M^{n+1}$ is given by an
immersion $\iota:\mathcal{C}\rightarrow\mathbb{P}TM$ where $\mathcal{C}$ is a
connected, smooth manifold of dimension $2n+1-k$ with the property that the
fibration $\pi\circ\iota:\mathcal{C}\rightarrow M$ is a submersion whose
fibers $\mathcal{C}_{x}:=(\pi\circ\iota)^{-1}(x)$ are mapped, via the
immersion $\iota_{x}:\mathcal{C}_{x}\rightarrow\mathbb{P}T_{x}M,$ to connected
projective submanifolds of codimension $k$ in $\mathbb{P}T_{x}M.$

Two cone structures $\iota:\mathcal{C}\rightarrow\mathbb{P}TM$ and
$\iota^{\prime}:\mathcal{C}^{\prime}\rightarrow\mathbb{P}TM^{\prime}$ are
locally equivalent around points $x\in M,x^{\prime}\in M^{\prime},$ if there
exists a diffeomorphism $\phi:U\rightarrow U^{\prime}$ where $x\in U\subset M$
and $x^{\prime}\in U^{\prime}\subset M^{\prime},$ such that
$x^{\prime}=\phi(x)$ and
$\phi_{*}\left(\iota(\mathcal{C})\right)=\iota^{\prime}(\mathcal{C}^{\prime}_{\phi(y)})$
for all $y\in M$

In this article, only cone structures of codimension one are considered i.e.,
the fibers $\iota(\mathcal{C}_{x})\subset\mathbb{P}T_{x}M$ are projective
hypersurfaces. Since the map $\iota:\mathcal{C}\rightarrow\mathbb{P}TM$ is
assumed to be an immersion and this article concerns the local nature of
geometric structures, it is safe to drop the symbol $\iota$ in
$\iota(\mathcal{C})$ and $\iota(\mathcal{C}_{x})$ and denote them as
$\mathcal{C}$ and $\mathcal{C}_{x}$, respectively. As a result, by restricting
to small enough neighborhoods, one can safely assume
$\mathcal{C}\subset\mathbb{P}TM$ is a sub-bundle defined in an open set
$U\subset\mathbb{P}TM,$ where the fibers $\mathcal{C}_{x}$ are smooth
submanifolds. Consequently, a cone structure of codimension one over a
manifold $M$ can be represented by $(\mathcal{C},M).$

In local coordinates, a point of $TM$ is represented by $(x;y)$ where
$x=(x^{0},\dots,x^{n})$ are the base coordinates and $y=(y^{0},\dots,y^{n})$
are the fiber coordinates. In a sufficiently small neighborhood of a point
$p=(x;z)\in\mathcal{C},$ , the homogeneous coordinates $[y^{0}:\ldots:y^{n}]$
can be chosen so that $z=[0:\ldots:0:1]$ and the cone
$\widehat{\mathcal{C}_{x}}$ is tangent to the hyperplane $\\{y_{0}=0\\}$ along
$\hat{z}$. The affine coordinates $(y^{0},\ldots,y^{n-1})$ obtained from
setting $y^{n}=1,$ are used to derive the explicit expressions of the
invariants. Using these preferred coordinates in a sufficiently small
neighborhood of $p,$ it is assumed that $\mathcal{C}$ is given in terms of
homogeneous coordinates as a locus

(2.1) $L(x^{0},\ldots,x^{n};y^{0},\ldots,y^{n})=0$

where $L$ is homogeneous of degree $r$ in $y^{i}$ and, in terms of affine
coordinates, as a graph

(2.2) $y^{0}=F(x^{0},\ldots,x^{n};y^{1},\ldots,y^{n-1}).$

######  Example 2.2.

A cone structure of codimension one on $M^{n+1}$ whose fibers are hyperplanes
given by $\textrm{Ker}\,\omega$ for some $\omega\in T^{*}M$, and are
_maximally non-integrable_ , i.e.,
$\omega{\,{\wedge}\;}(\mathrm{d}\omega)^{k-1}\neq 0$, is called a contact
structure if $n=2k-1$, and a quasi-contact structure (or even contact
structure) if $n=2k$. These can be considered as “degenerate” examples of
codimension one cone structures and have no local invariants. In these cases,
the function $F$ in (2.2) is of the form $F(x;y)=c_{a}(x)y^{a}+c_{n}(x),$ with
$\omega=\mathrm{d}x^{0}+c_{a}(x)\mathrm{d}x^{a}+c_{n}(x)\mathrm{d}x^{n}.$

######  Example 2.3.

A pseudo-conformal structure, i.e., a conformal class of a pseudo-Riemannian
metric $g_{ij}$ of signature $(p+1,q+1),\ p,q\geq 0$, can be introduced by its
field of null cones, i.e., the set of quadrics

$\mathcal{C}_{x}:=\left\\{[y]\in\mathbb{P}T_{x}\ \vline\
g_{ij}y^{i}y^{j}=0\right\\},$

As a result, a pseudo-conformal structure is an example of a cone structure of
codimension one, which, in contrast to the previous example, is “non-
degenerate” (see section 2.3.3). In terms of the local form (2.2), a
representative $g_{ij}$ of the conformal class, at a point
$s=(x;y)\in\mathcal{C},$ can be put in the form
$2y^{0}y^{n}-\varepsilon_{ab}y^{a}y^{b},$ where

(2.3) $[\varepsilon_{ab}]=\mathbb{I}_{p,q}:=\begin{pmatrix}\mathbb{I}_{p}&0\\\
0&-\mathbb{I}_{q}\end{pmatrix},$

and therefore $F(s)=\varepsilon_{ab}y^{a}y^{b}.$

In Section 2.3.3, when a coframe
$(\omega^{0},\ldots,\omega^{n},\theta^{1},\ldots,\theta^{n-1})$ over
$\mathcal{C}$ is introduced, the corresponding coframe derivatives are denoted
by

$\mathrm{d}G=G_{;i}\omega^{i}+G_{;{\underline{a}}}\theta^{a}.$

###  2.2. The flat model

When setting up the equivalence problem for a given geometric structure,
finding the flat model can be very helpful for identifying the structure
bundle. Naturally, one expects the flat model for causal structures to
coincide with the flat model for conformal geometry. In causal geometry, the
flat model is obtained from the natural pseudo-conformal structure on a
quadric. Consider the quadric $Q_{p+1,q+1}^{n+1}\subset\mathbb{P}^{n+2},$
where $p+q=n-1,$ given by ${}^{t}\\!vSv=0,$ where
$v={}^{t}\\!(y,x^{0},x^{1},\cdots,x^{n},z)\in\mathbb{R}^{n+3}$ and

$S:=\begin{pmatrix}0&0&0&0&1\\\ 0&0&0&-1&0\\\ 0&0&\mathbb{I}_{p,q}&0&0\\\
0&-1&0&0&0\\\ 1&0&0&0&0\\\ \end{pmatrix}$

has signature $(p+2,q+2).$ It is equipped with a pseudo-conformal structure of
signature $(p+1,q+1)$, expressed as $2y^{0}y^{n}-\varepsilon_{ab}y^{a}y^{b}$
where $y^{i}$’s denote the fiber coordinates of $TQ_{p+1,q+1}^{n+1}$. The null
cone bundle of this pseudo-conformal class is denoted by
$\mathcal{C}\subset\mathbb{P}TQ_{p+1,q+1}^{n+1}.$ Consider the _flag manifold_

(2.4)
$\mathbb{N}\mathbb{F}^{n+1}_{0,1}:=\\{(V_{1},V_{2})\in\mathrm{Gr}^{0}_{1}(n+3)\times\mathrm{Gr}^{0}_{2}(n+3):V_{1}\subset
V_{2}\\}$

where $\mathrm{Gr}^{0}_{k}(m)$ denotes the space of $k$-dimensional subspaces
of $\mathbb{R}^{m}$ which are null with respect to the inner product $S,$
e.g., $\mathrm{Gr}_{1}^{0}(n+3)\cong Q_{p+1,q+1}^{n+1}.$ Taking a basis
$\\{e_{1},\cdots,e_{m}\\}$ for $\mathbb{R}^{m},$ the transitivity of the
action of $G_{p,q}:=\mathrm{O}(p+2,q+2)$ on $\mathrm{Gr}^{0}_{k}(m)$ results
in isomorphisms $\mathrm{Gr}^{0}_{k}(m)\cong G_{p,q}/P_{k}$ where $P_{k}$ is
the parabolic subgroup isomorphic to the stabilizer of the subspace spanned by
$\\{e_{1},\cdots,e_{k}\\}.$

Let $m=n+3,$ $n\geq 4$ and define the parabolic subgroup
$P_{k_{1},\cdots,k_{i}}:=P_{k_{1}}\cap\cdots\cap P_{k_{i}}$ of $G.$
Consequently, if follows that

(2.5) $\displaystyle\mathbb{N}\mathbb{F}^{4}_{0,1}$ $\displaystyle\cong
G_{p,q}/P_{1,2,3},\ p+q=2,$ $\displaystyle\mathbb{N}\mathbb{F}^{n+1}_{0,1}$
$\displaystyle\cong G_{p,q}/P_{1,2},\ p+q=n-1\geq 3$

The relevant fibrations are shown in the diagram below.

(2.6) $\begin{diagram}$

Note that $\mathbb{N}\mathbb{F}^{n+1}_{0,1}$ is isomorphic to $\mathcal{C}.$
The Maurer-Cartan form on $G_{p,q}$ determines a canonical Cartan connection
over $\mathcal{C}$ which, as will be shown, is associated to the flat causal
structure. Moreover, the fibration of $G_{p,q}$ over
$\mathrm{Gr}^{0}_{2}(n+3)$ gives the flat model for Lie contact structures
studied in [SY89, Miy91]. As will be seen, these structures arise on the space
of characteristic curves of certain classes of causal geometries.

The Maurer-Cartan form of $G_{p,q}$ can be written as

(2.7)
$\varphi=\begin{pmatrix}\phi_{0}+\phi_{n}&-\pi_{n}&-\pi_{b}&-\pi_{0}&0\\\
\omega^{n}&\phi_{0}-\phi_{n}&-\gamma_{b}&0&-\pi_{0}\\\
\omega^{a}&-\theta^{a}&-\phi^{a}{}_{b}&-\gamma^{a}&\pi^{a}\\\
\omega^{0}&0&-\theta_{b}&-\phi_{0}+\phi_{n}&-\pi_{n}\\\
0&\omega^{0}&-\omega_{b}&\omega^{n}&-\phi_{0}-\phi_{n}\end{pmatrix},$

with $\mathrm{d}\varphi+\varphi{\,{\wedge}\;}\varphi=0.$ The symbol
$\varepsilon_{ab}$ is used to lower and raise indices. The set of semi-basic
1-forms with respect to the projections $\upsilon,\upsilon\circ\pi$ and
$\upsilon\circ\tau$ are respectively

$\begin{aligned}
\mathcal{I}_{1}&=\\{\omega^{0},\cdots,\omega^{n},\theta^{1},\cdots,\theta^{n-1}\\},\\\
\mathcal{I}_{2}&=\\{\omega^{0},\cdots,\omega^{n}\\},\\\
\mathcal{I}_{3}&=\\{\omega^{0},\cdots,\omega^{n-1},\theta^{1},\cdots,\theta^{n-1}\\}.\end{aligned}.$

The tangent spaces $T_{p}\mathcal{C}$ are equipped with a filtration defined
as follows. At $p\in\mathcal{C},$ the line over $p,$ denote by $\hat{p}\subset
T_{\pi(p)}Q_{p+1,q+1}^{n+1},$ is null. Let $\omega^{0}$ denote a
representative of the dual projective class of $\hat{p}$ with respect to the
conformally defined inner product $S.$ For instance, if $p=[e_{n}],$ then take
$\omega^{0}=e_{n}^{\flat}.$ Define
$K^{2}:=\operatorname{Ann}(\pi^{*}\omega^{0})\subset T\mathcal{C}.$ Using the
tautological bundle

$K^{1}:=\\{v\in T_{p}\mathcal{C}:\left(\pi_{*}(p)\right)(v)\in\hat{p}\subset
T_{\pi(p)}Q_{p+1,q+1}^{n+1}\\},$

the filtration $K^{1}\subset K^{2}\subset T\mathcal{C}$ is obtained. At
$p\in\mathcal{C}$, let $K_{-1}(p):=K^{1}(p),$ $K_{-2}:=K^{2}(p)/K^{1}(p)$ and
$K_{-3}:=K^{3}(p)/K^{2}(p).$ The graded vector space

(2.8) $\mathfrak{m}(p)=K_{-1}(p)\oplus K_{-2}(p)\oplus K_{-3}(p)$

can be associated to the filtration of $T_{p}\mathcal{C}$. Note that at each
$p\in\mathcal{C},$ $\mathfrak{m}(p)\cong\mathfrak{m}.$ The Lie bracket on
$T\mathcal{C}$ induces a bracket on $\mathfrak{m}$ which is tensorial and
makes $\mathfrak{m}$ a graded nilpotent Lie algebra (GNLA), i.e.,
$[K_{i},K_{j}]=K_{i+j}$ with $K_{i}=0$ for all $i\geq 4$ and $K_{-1}$ the
generating component. Let $G_{0}$ denote the subgroup of automorphisms of
$T\mathcal{C}$, i.e., $\mathrm{O}(p+2,q+2),$ which preserves the grading of
$\mathfrak{m}$ via the adjoint action. Let $K_{0}$ denote the Lie algebra of
$G_{0}.$ Then, the Lie algebra of automorphisms of $\mathcal{C},$ i.e.,
$\mathfrak{g}=\mathfrak{o}(p+2,q+2),$ admits a $\mathbb{Z}$-grading defined by

$\mathfrak{g}_{k}:=\\{g\in\mathfrak{g}:g\cdot K_{i}\subset K_{i+k}\mathrm{\
for\ }0\geq i\geq-3\\}.$

This is a $|3|$-grading, i.e.,
$\mathfrak{g}=\oplus_{i=-3}^{3}\mathfrak{g}_{i}$ such that
$[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j}$ with
$\mathfrak{g}_{i}=0$ for $|i|>3,$ and $\mathfrak{g}_{-1}$ generates
$\mathfrak{g}_{-}:=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-3}.$

Note that $K^{1}=E\oplus\mathrm{Ver}$ where $\mathrm{Ver}$ is the vertical
distribution with respect to the fibration $\pi$ and
$E=\pi_{*}(\mathfrak{g}_{-1}).$ Moreover, $T\mathcal{C}$ is equipped with a
conformally defined quadratic form obtained by lifting the canonical one
defined on $Q_{p+1,q+1}^{n+1},$

$g=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b},$

which is degenerate on $\mathrm{Ver}.$ The group of automorphisms of
$T\mathcal{C}$, which preserves the grading of $\mathfrak{m}$, also preserves
the conformal class of the lifted quadratic form $g$ on $T\mathcal{C}.$ The
Lie algebra $\mathfrak{g}_{0}\subset\mathfrak{g}$ which corresponding to the
group of automorphisms of $\mathfrak{m}$ that preserves the grading, is
isomorphic to $\mathfrak{co}(p,q)\oplus\mathbb{R}.$

Using the Maurer-Cartan form $\varphi,$ the subspaces
$\mathfrak{g}_{-3},\mathfrak{g}_{-2},$ and $\mathfrak{g}_{-1}$ are dual to the
Pfaffian systems $\mathcal{I}_{-3}:=\\{\omega^{0}\\}$,
$\mathcal{I}_{-2}:=\\{\omega^{a}\\}_{a=1}^{n-1},$ and
$\mathcal{I}_{-1}:=\\{\omega^{n},\theta^{a}\\}_{a=1}^{n-1}.$

###  2.3. The first order structure equations

In this section the first order structure equations for causal geometries are
derived via the method of equivalence. The procedure involves successive
coframe adaptations, as a result of which various important aspects of causal
geometries are unraveled, e.g., the quasi-contact structure of $\mathcal{C},$
the projective invariants of the fibers $\mathcal{C}_{x},$ and the Wsf
curvature. Also, it is noted that a causal structure can be equivalently
formulated as a codimension one sub-bundle of $\mathbb{P}T^{*}M,$ which is
called a dual formulation of causal geometries. Such formulation is
traditionally preferred in the context of geometric control theory [AZ02].

The derivation of the first order structure equations is similar to that of
Finsler structures as presented in [Bry02].

####  2.3.1. The projective Hilbert form

Given a cone structure of codimension one $(\mathcal{C},M),$ the subset
$\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$ is a projective hypersurface, for all
$x\in M$. At a point $y\in\mathcal{C}_{x}$, the affine tangent space
$\widehat{T}_{y}\mathcal{C}_{x}$ is defined as the hyperplane
$T_{z}\widehat{\mathcal{C}}_{x}\subset T_{x}M$ tangent to the cone
$\widehat{\mathcal{C}}_{x}\subset T_{x}M,$ containing $z\in\hat{y}$. Note that
this subspace is independent of the choice of $z.$ It is important here to
distinguish between the affine tangent space
$\widehat{T}_{y}\mathcal{C}_{x}\subset T_{x}M$ and the vertical tangent space
$\mathrm{V}T_{(x;y)}\mathcal{C}$ corresponding to the fibration
$\pi\circ\iota:\mathcal{C}\rightarrow M.$

The 1-form $\omega^{0}\in T^{*}\mathcal{C},$ at $(x;y)\in\mathcal{C}$ is
defined as a multiple of $(\pi\circ\iota)^{*}(\alpha)$ where
$\alpha:=\operatorname{Ann}(\widehat{T}_{y}\mathcal{C}_{x})\in T_{x}^{*}M.$ By
definition, $\omega^{0}$ is defined up to a multiplication by a nowhere
vanishing smooth function on $\mathcal{C}$ and is therefore semi-basic with
respect to the fibration $\pi:\mathcal{C}\rightarrow M.$ In this article
$\omega^{0}$ is referred to as the projective Hilbert form.

####  2.3.2. The adapted flag

At each point $(x;y)\in\mathcal{C},$ there is a natural basis
$(\textstyle{\frac{\partial}{\partial\omega^{0}}},\dots,\textstyle{\frac{\partial}{\partial\omega^{n}}})$
for $T_{x}M$ adapted to the flag

$\hat{y}\subset\widehat{T}_{y}\mathcal{C}_{x}\subset T_{x}M,$

where $\\{\textstyle{\frac{\partial}{\partial\omega^{n}}}\\}$ and
$\\{\textstyle{\frac{\partial}{\partial\omega^{1}}},\ldots,\textstyle{\frac{\partial}{\partial\omega^{n}}}\\},$
respectively, span $\hat{y}$ and $\widehat{T}_{y}\mathcal{C}_{x}.$

In terms of the graph (2.2), the local expressions are

(2.9) $\displaystyle\textstyle\frac{\partial}{\partial\omega^{n}}$
$\displaystyle=\textstyle F\frac{\partial}{\partial
x^{0}}+y^{a}\frac{\partial}{\partial x^{a}}+\frac{\partial}{\partial x^{n}},$
$\displaystyle\textstyle\frac{\partial}{\partial\omega^{a}}$
$\displaystyle=\textstyle\partial_{\underline{a}}F\frac{\partial}{\partial
x^{0}}+\frac{\partial}{\partial x^{a}}$
$\displaystyle\textstyle\frac{\partial}{\partial\omega^{0}}$
$\displaystyle=\textstyle\frac{\partial}{\partial x^{0}}.$

The dual coframe $(\omega^{0},\ldots,\omega^{n})$ can be written as

(2.10) $\displaystyle\omega^{n}$ $\displaystyle=\mathrm{d}x^{n},$
$\displaystyle\omega^{a}$ $\displaystyle=\mathrm{d}x^{a}-y^{a}\mathrm{d}x^{n}$
$\displaystyle\omega^{0}$
$\displaystyle=\mathrm{d}x^{0}-\partial_{{\underline{a}}}F\mathrm{d}x^{a}+(y^{a}\partial_{\underline{a}}F-F)\mathrm{d}x^{n}.$

The pull-back of these 1-forms to $\mathcal{C}$ spans the semi-basic 1-forms
with respect to the fibration $\pi:\mathcal{C}\rightarrow M.$ At each point
$p\in\mathcal{C},$ given by $p=(x^{i};y^{i}),$ these 1-forms are adapted to
the flag

(2.11) $\mathrm{V}T_{p}\mathcal{C}\subset K_{p}\subset H_{p}\subset
T_{p}\mathcal{C},$

where $\mathrm{V}T_{p}M=\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n}\\}$ is
the vertical tangent bundle,
$K_{p}=\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n-1}\\}$ is the fiber of the
tautological vector bundle given by

$K_{p}=\\{v\in T_{p}\mathcal{C}\ \vline\ \pi_{*}(p)(v)\in\hat{y}\subset
T_{\pi(p)}M\\},$

where $p=(x;y)$, the line $\hat{y}$ corresponds to the projective class of $y$
and $H_{p}=\textrm{Ker}\,\\{\omega^{0}\\}$ is the quasi-contact distribution
(see Section 2.3.5) To clarify the notation, if $p=(x;y)$ then the fiber
$\mathrm{V}T_{p}\mathcal{C}$ is also denoted by $T_{y}\mathcal{C}_{x}$ and the
quasi-contact distribution can be expressed as the splitting
$T_{y}\mathcal{C}_{x}\oplus\widehat{T}_{y}\mathcal{C}_{x}$ which is not
unique.

####  2.3.3. Causal structures and the first coframe adaptation

Recall that a projective hypersurface $\mathcal{V}\subset\mathbb{P}^{n}$ is
called _tangentially non-degenerate_ if its projective second fundamental form
has maximal rank (see [Sas99, AG93]). Now the stage is set to define causal
structures.

######  Definition 2.4.

A causal structure of signature $(p+1,q+1)$ on $M^{n+1}$ is a cone structure
of codimension one, $(M,\mathcal{C}),$ where the fibers $\mathcal{C}_{x}$ are
tangentially non-degenerate projective hypersurfaces and their projective
second fundamental form has signature $(p,q)$ everywhere.

In section 2.3.6 it is shown that the tangential non-degeneracy assumption of
the fibers $\mathcal{C}_{x}$ implies that the signature of $h^{ab}$ at any
point of $\mathcal{C}$ determines its the signature of $h^{ab}$ everywhere.

Let $\mathrm{d}_{V}$ denote the vertical exterior derivative, i.e., the
exterior differentiation with respect to fiber variables $y^{i}$. Using the
expressions (2.10), it follows that

(2.12)
$\mathrm{d}_{V}\omega^{0}\equiv-\theta^{0}{}_{a}{\,{\wedge}\;}\omega^{a},\quad\textrm{mod}\quad
I\\{\omega^{0}\\},$

where
$\theta^{0}{}_{a}=\partial_{\underline{a}}\partial_{\underline{b}}F\mathrm{d}y^{b}.$
As a result of (2.12) one obtains

$\mathrm{d}\omega^{0}=-2\theta^{0}{}_{0}{\,{\wedge}\;}\omega^{0}-\theta^{0}{}_{a}{\,{\wedge}\;}\omega^{a}+T_{0i}\omega^{0}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}T_{ab}\omega^{a}{\,{\wedge}\;}\omega^{b}+T_{an}\omega^{a}{\,{\wedge}\;}\omega^{n}.$

By setting $\phi_{0}:=\theta^{0}{}_{0}-T_{0i}\omega^{i},$
$\theta_{a}:=\theta^{0}{}_{a}-\textstyle{\frac{1}{2}}T_{ab}\omega^{b}-T_{an}\omega^{n},$
one arrives at

(2.13)
$\mathrm{d}\omega^{0}=-2\phi_{0}{\,{\wedge}\;}\omega^{0}-\theta_{a}{\,{\wedge}\;}\omega^{a},$

which implies that $\omega^{0}$ has rank $2n-1$, i.e.,
$\omega^{0}{\,{\wedge}\;}(\mathrm{d}\omega^{0})^{n-1}\neq 0.$

The 1-forms $\theta_{a}$ define vertical 1-forms with respect to the fibration
$\pi:\mathcal{C}\rightarrow M.$ The 1-forms $\theta_{a}$ are linearly
independent everywhere when the fibers $\mathcal{C}_{x}$ are tangentially non-
degenerate at every point, i.e.,
$\det(\partial_{\underline{a}}\partial_{\underline{b}}F)\neq 0.$

Since the vertical 1-forms $\theta_{1},\ldots,\theta_{n-1}$ are linearly
independent, at each point $(x;y)\in\mathcal{C},$ one obtains a coframe
$\mathcal{I}_{\mathsf{tot}}:=\\{\omega^{0},\ldots,\omega^{n},\theta_{1},\ldots,\theta_{n-1}\\}$
for $T_{(x;z)}\mathcal{C}$ adapted to the flag

(2.14) $0\subset\mathrm{V}T_{p}\mathcal{C}\subset K_{p}\subset H_{p}\subset
T_{p}\mathcal{C},$

defined previously in (2.11). Such a coframe is called 1-adapted. For a
quadric, this filtration coincides with the one introduced in (2.8). In
analogy with the flat model, the following algebraic ideals are defined:

(2.15) $\displaystyle\mathcal{I}_{\mathsf{tot}}$
$\displaystyle=I\\{\omega^{0},\cdots,\omega^{n},\theta^{1},\cdots,\theta^{n-1}\\},$
$\displaystyle\mathcal{I}_{\mathsf{bas}}$
$\displaystyle=I\\{\omega^{0},\cdots,\omega^{n}\\},$
$\displaystyle\mathcal{I}_{\mathsf{char}}$
$\displaystyle=I\\{\omega^{0},\cdots,\omega^{n-1},\theta^{1},\cdots,\theta^{n-1}\\}.$

Using the expressions (2.10), equation (2.13) gives

(2.16)
$\theta_{a}=\partial_{\underline{a}}\partial_{\underline{b}}F\mathrm{d}y^{b}+\partial_{0}\partial_{\underline{a}}F\mathrm{d}x^{0}+\partial_{b}\partial_{\underline{a}}F\mathrm{d}x^{b}+(\partial_{\underline{a}}\partial_{n}F-\partial_{a}F-\partial_{0}F\partial_{\underline{a}}F)\mathrm{d}x^{n}$

From the expression above it follows that the linear independence of
$\theta_{a}$’s modulo $\omega^{i}$’s is equivalent to the non-degeneracy of
the vertical Hessian $\partial_{\underline{a}}\partial_{\underline{b}}F$.

The fact that such a coframing is adapted to the flag (2.14) and satisfies
(2.13) implies that different choices of 1-adapted coframes are related by

$\displaystyle\omega^{0}$
$\displaystyle\longmapsto\mathbf{a}^{2}\omega^{0},\qquad$
$\displaystyle\omega^{a}$
$\displaystyle\longmapsto\mathbf{E}_{a}\omega^{a}+\mathbf{A}^{a}{}_{b}\omega^{b},$
$\displaystyle\omega^{n}$
$\displaystyle\longmapsto\mathbf{e}\,\omega^{0}+\mathbf{B}_{a}\omega^{a}+\mathbf{b}^{2}\omega^{n},\qquad$
$\displaystyle\theta_{a}$
$\displaystyle\longmapsto\mathbf{C}_{a}\omega^{0}+\mathbf{S}_{ab}\left(\mathbf{A}^{-1}\right)^{b}{}_{c}\omega^{c}+\mathbf{a}^{2}\left(\mathbf{A}^{-1}\right)^{b}{}_{a}\theta_{b}.$

In the language of $G$-structures, the structure group of the bundle of
1-adapted coframes, $G_{1},$ is of the form

(2.17) $\left\\{\begin{pmatrix}\mathbf{a}^{2}&0&0&0\\\
\mathbf{E}&\mathbf{A}&0&0\\\ \mathbf{e}&\mathbf{B}&\mathbf{b}^{2}&0\\\
\mathbf{C}&{}^{t}\\!\mathbf{A}^{-1}\mathbf{S}&0&\mathbf{a}^{2}\,{}^{t}\\!\mathbf{A}^{-1}\end{pmatrix}\
\ \ \vline\ \begin{aligned}
&\mathbf{A}\in\operatorname{GL}_{n-1}(\mathbb{R}),\quad\mathbf{a},\mathbf{b}{}&{}\in{}\mathbb{R}\backslash\\{0\\}&,\\\
&{}^{t}\mathbf{B},\mathbf{C},\mathbf{E}\in\mathbb{R}^{n-1},\
&\mathbf{e}\in{}\mathbb{R}&,\\\ \quad&\mathbf{S}\in M_{n-1}(\mathbb{R}),\
&\mathbf{S}={}{}^{t}\\!\mathbf{S}&\end{aligned}\right\\}$

######  Remark 2.5.

The reason for using $\mathbf{a}^{2}$ and $\mathbf{b}^{2}$ in $G_{1}$ is as
follows. Let $L(x^{i};y^{i})$ be a Lagrangian function for a causal structure,
i.e., the vanishing set of $L$ defines $\mathcal{C}$. As is conventional in
the case of pseudo-conformal structures, it is desirable to restrict to the
set of causal transformations preserving the sign of $L$ in a neighborhood of
the null cones, i.e., restricting to the set of Lagrangians of the form
$S^{2}\,L$ for some nowhere vanishing homogeneous function
$S:TM\rightarrow\mathbb{R}.$ Recall that the projective Hilbert form in terms
of $L$ is given by $\omega^{0}=\partial_{\underline{i}}L\mathrm{d}x^{i}$.
Replacing $L$ by $S^{2}L$ and restricting to the zero locus of $L,$ the
projective Hilbert form transforms to
$S^{2}(L_{\underline{i}}\mathrm{d}x^{i})=S^{2}\omega^{0}.$ This observation
justifies the use of $\mathbf{a}^{2}$ in $G_{1}.$ Moreover, recall that a
vector bundle of rank 1 is called orientable if it has a non-vanishing global
section. In the case of contact manifolds, a contact distribution is called
_co-orientable_ or _transversally orientable_ if it has a globally defined
contact form [LM87]. Similarly, a quasi-contact structure is called co-
orientable if it has a globally defined quasi-contact form $\omega^{0}$. If
the quasi-contact distribution on $\mathcal{C}$ is given a co-orientation, it
is preserved under the action of $G_{1}.$ Similarly, if the characteristic
field of a quasi-contact structure (see Section 2.3.5) is given an
orientation, then having $\mathbf{b}^{2}$ in (2.17) implies that the structure
group $G_{1}$ preserves the orientation of the characteristic field. Moreover,
the conformal scaling of the quadratic form $g$ defined in 2.3.11 by the
action of $G_{1}$ is given by $\mathbf{a}^{2}\mathbf{b}^{2}$. As a result, it
is possible to define space-like and time-like horizontal vector fields $v$ in
$T\mathcal{C}$ via the signature of $g(v,v).$

To implement Cartan’s method of equivalence, one lifts the 1-adapted coframe
to the $G_{1}$-bundle in the following way (see [IL03, Gar89, Olv95] for the
background). Recall that the set of all coframes on $\mathcal{C},$ defines a
$\mathrm{GL}(2n)$-principal bundle
$\tilde{\varsigma}:\tilde{\mathcal{P}}\rightarrow\mathcal{C},$ with the right
action

$R_{g}(\vartheta_{p})=g^{-1}\cdot\vartheta_{p},$

where $\vartheta_{p}\in\varsigma^{-1}(p)$ is a coframe at $p\in\mathcal{C},$
$g\in\mathrm{GL}(2n),$ and the action on the right hand side is the ordinary
matrix multiplication.

Among the set of all coframes on $\mathcal{C}$, let $\vartheta$ represent
1-adapted coframes ${}^{t}(\omega^{0},\cdots,\theta_{n-1}),$ which are well-
defined up to the action of $G_{1}.$ As a result, the 1-adapted coframes are
sections of a $G_{1}$-bundle
$\varsigma_{1}:\mathcal{P}_{1}\rightarrow\mathcal{C}.$ On
$\mathcal{P}_{1}\cong\mathcal{C}\times G_{1}$ one can define a canonical set
of 1-forms by setting

$\underline{\vartheta}(p,g)={}^{t}(\underline{\omega}^{0},\cdots,\underline{\theta}_{n-1}):=g^{-1}\cdot\vartheta_{p}.$

The exterior derivatives of the 1-forms in $\underline{\vartheta}$ are given
by

(2.18)
$\mathrm{d}\underline{\vartheta}=\mathrm{d}g^{-1}{\,{\wedge}\;}\vartheta+g^{-1}\cdot\mathrm{d}\vartheta=-g^{-1}\cdot\mathrm{d}g\cdot{\,{\wedge}\;}g^{-1}\vartheta+g^{-1}\cdot\mathrm{d}\vartheta=-\alpha{\,{\wedge}\;}\underline{\vartheta}+T,$

where $\alpha(p,g)=g^{-1}\cdot\mathrm{d}g$ is $\mathfrak{g}_{1}$-valued 1-form
where $\mathfrak{g}_{1}$ denotes the Lie algebra of $G_{1}.$ The 2-forms
$T(p,g)$ can be expressed in terms of
$\underline{\omega}^{i}{\,{\wedge}\;}\underline{\omega}^{j}$,
$\underline{\theta}_{a}{\,{\wedge}\;}\underline{\omega}^{i}$,$\underline{\theta}_{a}{\,{\wedge}\;}\underline{\theta}_{b}.$
The terms constituting $T$ are called the torsion terms. They are semi-basic
with respect to the projection $\varsigma_{1}.$ Via $\alpha,$ one can identify
the vertical tangent spaces of the fibers of
$\varsigma_{1}:\mathcal{P}_{1}\rightarrow\mathcal{C}$ with $\mathfrak{g}_{1}.$
The $\mathfrak{g}_{1}$-valued 1-form $\alpha$ is called a _pseudo-connection_
on $\mathcal{P}_{1}$ which is not yet uniquely defined. Recall that the
difference between a _connection_ and a pseudo-connection on $G_{1}$-bundle
lies in the fact that a connection form $\tilde{\alpha}$ satisfies the
equivariance condition
$R^{*}_{g}\tilde{\alpha}=\mathrm{Ad}(g^{-1})\tilde{\alpha},$ where $g\in
G_{1},$ while a pseudo-connection in general does not.

A choice of pseudo-connection in (2.18) gives rise to structure equations for
the 1-forms $\underline{\vartheta}.$ For 1-adapted coframes, the pseudo-
connection forms are of the form,

(2.19) $\begin{pmatrix}2\phi_{0}&0&0&0\\\
\gamma^{a}{}_{0}&\psi^{a}{}_{b}&0&0\\\
\gamma^{n}{}_{0}&\gamma^{n}{}_{a}&2\phi_{n}&0\\\
\pi_{a}&\pi_{ab}&0&2\phi_{0}\,\delta^{a}_{b}-\psi^{a}{}_{b}\end{pmatrix}.$

where $\pi_{ab}=\pi_{ba}.$

It follows that

(2.20a) $\displaystyle\mathrm{d}\underline{\omega}^{0}{}={}$
$\displaystyle-2\phi_{0}{\,{\wedge}\;}\underline{\omega}^{0}-\underline{\theta}_{a}{\,{\wedge}\;}\underline{\omega}^{a},$
(2.20b) $\displaystyle\mathrm{d}\underline{\omega}^{a}{}={}$
$\displaystyle-\gamma^{a}{}_{0}{\,{\wedge}\;}\underline{\omega}^{0}-\psi^{a}{}_{b}{\,{\wedge}\;}\underline{\omega}^{b}-h^{ab}\,\underline{\theta}_{b}{\,{\wedge}\;}\underline{\omega}^{{n}},$
(2.20c) $\displaystyle\mathrm{d}\underline{\omega}^{{n}}{}={}$
$\displaystyle-\gamma^{n}{}_{0}{\,{\wedge}\;}\underline{\omega}^{0}-\gamma^{n}{}_{a}{\,{\wedge}\;}\underline{\omega}^{a}-2\phi_{{n}}{\,{\wedge}\;}\underline{\omega}^{{n}},$
(2.20d) $\displaystyle\mathrm{d}\underline{\theta}_{a}{}={}$
$\displaystyle-\pi_{a}{\,{\wedge}\;}\underline{\omega}^{0}-\pi_{ab}{\,{\wedge}\;}\underline{\omega}^{b}+\psi^{b}{}_{a}{\,{\wedge}\;}\underline{\theta}_{b}-2\phi_{0}{\,{\wedge}\;}\underline{\theta}_{a}+\sigma_{a}{\,{\wedge}\;}\underline{\omega}^{n}+\sigma^{b}{}_{a}{\,{\wedge}\;}\underline{\theta}_{b},$

where the 1-forms $\sigma_{a},\sigma^{b}{}_{a},$ are the torsion terms and
vanish modulo $\mathcal{I}_{\mathsf{tot}}.$ Their components are determined in
Section 2.3.9.

The equation of $\mathrm{d}\omega^{0}$ follows directly from (2.13). Moreover,
the Pfaffian system $\mathcal{I}_{\mathsf{bas}},$ is completely integrable
with the fibers $\mathcal{C}_{x}$ being its integral manifolds. It follows
that $\mathrm{d}\omega^{a}=\tau^{a}{}_{i}{\,{\wedge}\;}\omega^{i},$ and
$\mathrm{d}\omega^{n}=\tau^{n}{}_{i}{\,{\wedge}\;}\omega^{i},$ where
$\tau^{a}{}_{i},\tau^{n}{}_{i}\equiv 0$ modulo $\mathcal{I}_{\mathsf{tot}}$.
Using the fact that the action of $G_{1}$ preserves
$\mathcal{I}_{\mathsf{bas}},$ and that two pseudo-connection differ by a semi-
basic $\mathfrak{g}_{1}$-valued 1-form, all the 1-forms
$\tau^{a}_{i},\tau^{n}{}_{i}$ can be absorbed into the pseudo-connection
forms. The only torsion term left is
$h^{ab}\theta_{b}{\,{\wedge}\;}\omega^{n},$ for some function $h^{ab},$ which
is discussed in Section 2.3.6. From now on, since structure equations are
always written for the lifted coframe $\underline{\vartheta}$, the underline
in $\underline{\omega}^{i}$ and $\underline{\theta}_{a}$’s will be dropped.

######  Remark 2.6.

The oriented projectivized tangent bundle $\mathbb{P}_{+}TM$ is defined so
that at $x\in M$ the fiber $\mathbb{P}_{+}T_{x}M$ is isomorphic to the space
of rays in $T_{x}M$, or, equivalently, the set of oriented real lines in
$T_{x}M$ passing through the origin. As a result, $\mathbb{P}_{+}T_{x}M$ is
the nontrivial double cover of $\mathbb{P}T_{x}M.$

The above definition of cone structures, and more specifically causal
structures, can be changed so that $\mathcal{C}$ is a sub-bundle of
$\mathbb{P}_{+}TM.$ In the case of causal structures immersed in
$\mathbb{P}_{+}TM$, the assumption that the fibers
$\mathcal{C}_{x}=\iota^{-1}(x)$ be connected is not desirable. This is due to
the fact that $\mathcal{C}_{x}$ should be thought of as the light cones which
are comprised of the future and past light cones whose projectivizations are
disjoint. Thus, one expects
$\mathcal{C}_{x}=\mathcal{C}^{+}_{x}\sqcup\mathcal{C}^{-}_{x}.$ This way, in
analogy with Finsler geometry, a cone structure is called _reversible_ if
$\widehat{\mathcal{C}}=-\widehat{\mathcal{C}}.$ Otherwise, it is called _non-
reversible_. It should be noted that among oriented causal structures, non-
reversible ones can be of interest in certain problems due to possible
differences in their future and past light cones. This is similar to the case
of non-reversible Finsler structures containing the important class of Randers
spaces [BRS04]. In [KP67], Kronheimer and Penrose use the term _self-dual_ for
a reversible causal structure. This is due to the fact that they define the
dual of a causal structure to be the causal structure obtained from replacing
all the causal relations between the points of the space by their inverses,
i.e., interchanging the words “future” and “past”.

####  2.3.4. A dual formulation of causal structures

In the context of geometric control theory, geometric structures arising from
distributions are typically formulated in the cotangent bundle [AZ02].
Firstly, the definition of a Legendre transformation is recalled.

######  Definition 2.7.

Given a sufficiently smooth real-valued function $L$, usually referred to as a
Lagrangian, on $TM$, the associated Legendre transformation
$\textrm{Leg}:TM\rightarrow T^{*}M,$ is defined by

$\textrm{Leg}=\hat{\pi}\circ\mathrm{d}_{V}L,$

where $\mathrm{d}_{V}L$ defines a section of the product bundle $TM\times
T^{*}M,$ with the projection map $\hat{\pi}:TM\times T^{*}M\rightarrow
T^{*}M.$ In local coordinates, $\textrm{Leg}(x^{i};y^{i})=(x^{i};p_{i})$ where
$p_{i}=\frac{\partial L}{\partial y^{i}},$ and $(p_{0},\dots,p_{n})$ are local
coordinates for the fibers of $T^{*}M$.

The Legendre transformation is a local diffeomorphism provided that
$\textrm{Leg}_{*}=\frac{\partial^{2}L}{\partial y^{i}\partial y^{j}}$ is non-
degenerate everywhere. The inverse Legendre transformation $\textrm{Leg}^{-1}$
can be obtained by the same procedure using the Hamiltonian function
$Q:=\mathcal{A}\circ\textrm{Leg}^{-1}$ where
$\mathcal{A}:=\mathrm{d}L(\underline{A})-L$ with $\underline{A}$ being the
Liouville vector field, i.e., $\mathcal{A}=y^{i}\frac{\partial L}{\partial
y^{i}}-L$ in local coordinates. When the degree of homogeneity of $L$ is
$r\neq 1,$ Euler’s theorem gives $Q=(r-1)L\circ\textrm{Leg}^{-1}$.

Changing the defining function to $L^{\prime}(x,y)=S(x,y)L(x,y),$ the function
Leg remains invariant over $\mathcal{C},$ hence the vanishing set of $Q$ in
$T^{*}M$ remains invariant. Consequently, one can equivalently study the dual
causal structure
$\mathcal{C}^{*}:=\textrm{Leg}(\mathcal{C})\subset\mathbb{P}T^{*}M$ whose
fibers $\mathcal{C}^{*}_{x}\subset\mathbb{P}T^{*}_{x}M$ are the projective
dual of the projective hypersurfaces $\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$.

Via the Legendre transformation the projective Hilbert form
$\omega^{0}=\partial_{\underline{i}}L\,\mathrm{d}x^{i}$ is the pull-back of
the canonical contact form $\lambda=p_{i}\mathrm{d}x^{i}$ restricted to
$\mathcal{C}^{*}\subset\mathbb{P}T^{*}M$ i.e.,

$\textrm{Leg}^{*}(p_{i}\mathrm{d}x^{i})=\frac{\partial L}{\partial
y^{i}}(x,y)\mathrm{d}x^{i}.$

Consequently, the coframe adaptation can be carried out using $\lambda$ for
the dual cone structure. Hence, this dual formulation can be taken as the
starting point for studying causal structures.

####  2.3.5. Characteristic curves and a quasi-Legendrian foliation

A causal structure comes with a distinguished set of curves which can be
defined using the quasi-contact nature of $\mathcal{C}.$ Recall from Example
2.2 that a quasi-contact structure (or even contact structure) on a
$2n$-dimensional manifolds was defined locally in terms of a 1-form of rank
$2n-1.$ As a result, there is a unique direction along which $\omega^{0}$ and
$\mathrm{d}\omega^{0}$ are degenerate i.e., $[\mathbf{v}]\subset
T\mathbb{P}TM$ such that $\mathbf{v}\im\omega^{0}=0$ and
$\mathbf{v}\im\mathrm{d}\omega^{0}=0,$ which is called the _characteristic
field_ of the quasi-contact structure and its integral curves are called
_characteristic curves_ (c.f. [Hsu92]). A section of the characteristic field
is called a _characteristic vector field_.

In local coordinates using the defining function $F$ in (2.2), one obtains
that the vector field

$\textstyle F\frac{\partial}{\partial x^{0}}+y^{a}\frac{\partial}{\partial
x^{a}}+\frac{\partial}{\partial x^{n}}+A^{a}\frac{\partial}{\partial y^{a}}$

where

$A^{b}\partial_{\underline{a}}\partial_{\underline{b}}F=-F\partial_{0}\partial_{\underline{a}}F-y^{b}\partial_{b}\partial_{\underline{a}}F+\partial_{a}F+\partial_{0}F\partial_{\underline{a}}F-\partial_{\underline{a}}\partial_{n}F$

represents a characteristic vector field on $\mathcal{C}.$

######  Definition 2.8.

Let $M$ be a $2n$-dimensional manifold with a quasi-contact 1-form
$\omega\in\Gamma(T^{*}M).$ A submanifold $N\subset M$ is called _quasi-
Legendrian_ if $T_{x}N\in\textrm{Ker}\,\omega$ and
$T_{x}N\im\mathrm{d}\omega=0$ for all $x\in N,$ with the property that
$T_{x}N$ is transversal to the characteristic field of $\omega.$

The bundle $\mathcal{C}$ is $2n$-dimensional and $\omega^{0}\in
T^{*}\mathcal{C}$ has maximal rank $2n-1,$ therefore, $\mathcal{C}$ has a
quasi-contact structure. The fibers $\mathcal{C}_{x}$ form a quasi-Legendrian
foliation of $\mathcal{C}$ since the vertical tangent spaces
$T_{y}\mathcal{C}_{x}$ are annihilated by
$\mathcal{I}_{\mathsf{bas}}=\\{\omega^{0},\dots,\omega^{n}\\}.$

Moreover, using (2.13), the characteristic field of $\omega^{0}$ is the kernel
of the Pfaffian system
$\mathcal{I}_{\mathsf{char}}=\\{\omega^{0},\dots,\omega^{n-1},\theta_{1},\cdots,\theta_{n-1}\\},$
which justifies the subscript $\mathsf{char}.$

######  Remark 2.9.

Recall that the geodesics of a Finsler structure are defined as the projection
of integral curves of the Reeb vector field associated to the contact
structure on the indicatrix bundle induced by the Hilbert form [Bry02]. Here,
the projection of the characteristic curves gives an analogue of null
geodesics in pseudo-conformal structures.

####  2.3.6. Projective second fundamental form and the second coframe
adaptation

The next coframe adaptation involves the projective fundamental form of the
fibers $\mathcal{C}_{x}.$

Let $\omega^{a}\equiv-h^{ab}\theta_{b}{\,{\wedge}\;}\omega^{n}$ modulo
$I\\{\omega^{0},\cdots,\omega^{n-1}\\}.$ Using the identity
$\mathrm{d}^{2}\omega^{0}=0,$ it follows that

$h^{ab}\theta_{a}{\,{\wedge}\;}\theta_{b}{\,{\wedge}\;}\omega^{n}=0,$

i.e., $h^{ab}=h^{ba}.$ Using the local expressions of $\omega^{a}$ in (2.10),
one obtains that

$\mathrm{d}_{V}\omega^{a}\equiv-\mathrm{d}y^{a}{\,{\wedge}\;}\omega^{n}\quad\textrm{mod}\quad
I\\{\omega^{0},\cdots,\omega^{n-1}\\}.$

Recall from (2.12) that $\mathrm{d}y^{a}=h^{ab}\theta^{0}{}_{b}$ where
$(h^{ab})=(\partial_{\underline{a}}\partial_{\underline{b}}F)^{-1}.$

Using the structure equations (2.20), the infinitesimal action of the
structure group $G_{1}$ on $h^{ab}$ is obtained from the identity
$\mathrm{d}^{2}\omega^{a}=0,$ and is given by

(2.21)
$\mathrm{d}h^{ab}+h^{ac}\psi^{b}{}_{c}+h^{bc}\psi^{a}{}_{c}-2h^{ab}\left(\phi_{0}+\phi_{n}\right)\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

The relation above implies that the functions $h^{ab}(p,g_{0})$ and
$\tilde{h}^{ab}(p,g_{1})$ where $g_{1}=gg_{0},$ with $g\in G_{1}$ represented
by the matrix (2.17), are related by

(2.22)
$\tilde{h}^{ab}={\mathbf{a}^{2}\mathbf{b}^{2}}(\mathbf{A})^{-1}{}^{a}{}_{c}(\mathbf{A})^{-1}{}^{b}{}_{d}h^{cd}.$

It follows that the bilinear form $\phi_{2}=h^{ab}\theta_{a}\circ\theta_{b}$
when pulled-back to the fibers $\mathcal{C}_{x}$ coincides with its projective
second fundamental form.

######  Remark 2.10.

The action of the structure group on $h^{ab}$ can also be obtained directly
from the structure equations (2.20). However, in implementing Cartan’s method
of equivalence finding the infinitesimal group action by differentiating the
structure equations is more straightforward and computationally easier. This
is why from now on the focus will be primarily on the infinitesimal action of
the structure group on the torsion terms.

Note that by definition 2.4, tangential non-degeneracy of the fibers
$\mathcal{C}_{x}$ implies that $h^{ab}$ is non-degenerate everywhere in
$\mathcal{C}.$ As a result, the signature of $h^{ab},$ as a well-defined
tensor over $\mathcal{C},$ remains the same everywhere.

Assuming that $h^{ab}$ has signature $(p,q)$, the relation (2.22) implies that
$\mathbf{A}^{a}{}_{b}$ can be chosen so that $h^{ab}$ is normalized to
$\varepsilon^{ab}.$ By restricting the $G_{1}$-bundle to the coframes for
which $h^{ab}=\varepsilon^{ab},$ and using the relation (2.21), it follows
that

(2.23) $\varepsilon^{ac}\phi^{b}{}_{c}+\varepsilon^{bc}\phi^{a}{}_{c}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}},\quad\mathrm{where}\quad\phi^{a}{}_{b}=\psi^{a}{}_{b}-\left(\phi_{0}+\phi_{n}\right)\delta^{a}{}_{b}.$

As a result, for such coframes one can replace $\psi^{a}{}_{b}$ by
$\phi^{a}{}_{b}+\left(\phi_{0}+\phi_{n}\right)\delta^{a}{}_{b}$ where the
matrix-valued 1-forms $\phi^{a}{}_{b}$ take values in $\mathfrak{o}(p,q)$
modulo $\mathcal{I}_{\mathsf{tot}}$

Coframings with normalized $h^{ab}$ are 2-adapted, and they are local sections
of a $G_{2}$-bundle over $\mathcal{C}$ where

(2.24) $G_{2}=\left\\{\begin{pmatrix}\mathbf{a}^{2}&0&0&0\\\
\mathbf{E}&\mathbf{a}\mathbf{b}\mathbf{A}&0&0\\\
\mathbf{e}&\mathbf{B}&\mathbf{b}^{2}&0\\\
\mathbf{C}&\mathbf{A}\mathbf{S}&0&\frac{\mathbf{a}}{\mathbf{b}}\mathbf{A}\end{pmatrix}\
\ \ \vline\ \begin{aligned}
&\mathbf{A}\in\mathrm{O}(p,q),\quad\mathbf{a},\mathbf{b}{}&{}\in{}\mathbb{R}\backslash\\{0\\}&,\\\
&{}^{t}\mathbf{B},\mathbf{C},\mathbf{E}\in\mathbb{R}^{n-1},\
&\mathbf{e}\in{}\mathbb{R}&,\\\ \quad&\mathbf{S}\in M_{n-1}(\mathbb{R}),\
&\mathbf{S}={}{}^{t}\\!\mathbf{S}&\end{aligned}\right\\}.$

######  Remark 2.11.

By a result of Griffiths and Harris [GH79], if the projective second
fundamental form of a projective hypersurface
$\mathcal{C}_{x}\subset\mathbb{P}T_{x}M$ has degeneracy of rank $k,$ then its
projective dual $\mathcal{C}^{*}_{x}\subset\mathbb{P}T^{*}_{x}M$ has
codimension $k+1.$ In other words, let $(M,\mathcal{C})$ be a cone structure
$(M,\mathcal{C})$ of codimension one, whose fibers have projective second
fundamental form with degeneracy of rank $k.$ Then, its dual cone structure
$(M,\mathcal{C}^{*})$ has fibers of dimension $n-k-1$. Therefore, it is more
natural to study cone structure of codimension one whose fibers have
degenerate second fundamental form via their dual formulation since
$\dim\mathcal{C}=2n,$ and $\dim\mathcal{C}^{*}=2n-k.$

####  2.3.7. The third coframe adaptation

The 1-forms $\phi^{a}{}_{b}$ are $\mathfrak{o}(p,q)$-valued, i.e.,
$\phi_{ab}+\phi_{ba}\equiv 0$ modulo $\mathcal{I}_{\mathsf{tot}},$ where
$\phi_{ab}=\varepsilon_{ac}\phi^{c}{}_{b}.$ From (2.23) it follows that

(2.25)
$\psi_{ab}=\phi_{ba}+(\phi_{0}+\phi_{n})\delta_{ab}+E_{aib}\omega^{i}+F_{a}{}^{c}{}_{b}\theta_{c},$

where $E_{aib}=E_{bia}$ and $F_{a}{}^{c}{}_{b}=F_{b}{}^{c}{}_{a}.$ Hence, with
respect to 2-adapted coframes, the structure equations are

(2.26a) $\displaystyle\mathrm{d}\omega^{0}={}$
$\displaystyle-2\phi_{0}{\,{\wedge}\;}\omega^{0}-\theta_{a}{\,{\wedge}\;}\omega^{a},$
(2.26b) $\displaystyle\mathrm{d}\omega^{a}={}$
$\displaystyle-\gamma^{a}{}_{0}{\,{\wedge}\;}\omega^{0}-\phi^{a}{}_{b}{\,{\wedge}\;}\omega^{b}-\left(\phi_{0}+\phi_{n}\right){\,{\wedge}\;}\omega^{a}-\varepsilon^{ab}\,\theta_{b}{\,{\wedge}\;}\omega^{{n}},$
$\displaystyle-E^{a}{}_{ic}\,\omega^{i}{\,{\wedge}\;}\omega^{c}-F^{ab}{}_{c}\,\theta_{b}{\,{\wedge}\;}\omega^{c},$
(2.26c) $\displaystyle\mathrm{d}\omega^{{n}}={}$
$\displaystyle-\gamma^{n}{}_{0}{\,{\wedge}\;}\omega^{0}-\gamma^{n}{}_{a}{\,{\wedge}\;}\omega^{a}-2\phi_{{n}}{\,{\wedge}\;}\omega^{{n}},$
(2.26d) $\displaystyle\mathrm{d}\theta_{a}={}$
$\displaystyle-\pi_{a}{\,{\wedge}\;}\omega^{0}-\pi_{ab}{\,{\wedge}\;}\omega^{b}+\phi^{b}{}_{a}{\,{\wedge}\;}\theta_{b}-\left(\phi_{0}-\phi_{n}\right){\,{\wedge}\;}\theta_{a}$
$\displaystyle+\sigma_{a}{\,{\wedge}\;}\omega^{n}+\sigma^{b}{}_{a}{\,{\wedge}\;}\theta_{b},$

where $\varepsilon^{ab}$ is used to raise the indices in $E_{aib}$ and
$F_{a}{}^{c}{}_{b}.$ By replacing

$\displaystyle\phi^{a}{}_{b}$
$\displaystyle\longmapsto\phi^{a}{}_{b}+\frac{1}{2}\left(E^{a}{}_{bc}\,\omega^{c}-\varepsilon^{ae}E^{d}{}_{ec}\varepsilon_{da}\,\omega^{c}+\varepsilon_{cd}E^{c}{}_{ab}\,\omega^{d}\right)$
$\displaystyle\phi_{n}$
$\displaystyle\longmapsto\phi_{n}+E^{b}{}_{nb}\,\omega^{{n}},\qquad\gamma^{a}{}_{0}\longmapsto\gamma^{a}{}_{0}+E^{a}{}_{0b}\,\omega^{b},$

it can be assumed that $E^{a}{}_{bc}$ and $E^{a}{}_{0c}$ vanish, and
$E^{a}{}_{nb}$ is trace-free.

Differentiating $\mathrm{d}\omega^{a},$ and collecting the terms of the form
$\tau^{a}{}_{b}{\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\omega^{n}$ for some
1-form satisfying $\tau_{ab}=\tau_{ba}$ and $\tau^{a}{}_{a}\equiv 0$ modulo
$\mathcal{I}_{\mathsf{tot}}$, one obtains

$\left[\left(\mathrm{d}E_{anb}+2E_{anb}\phi_{n}+2E_{cn(a}\phi^{c}{}_{b)}+\pi_{ab}\right){\,{\wedge}\;}\omega^{n}-\mathrm{d}\left(\phi_{0}+\phi_{n}\right)\varepsilon_{ab}\right]{\,{\wedge}\;}\omega^{b}=0,$

where $E_{anb}=\varepsilon_{ac}E^{c}{}_{nb}.$ Contracting the above equations
with $\varepsilon^{ab}$, and using the properties
$E_{anb}=E_{bna},E^{a}{}_{na}=0,$ and $\phi_{ab}=-\phi_{ba},$ it follows that
$\mathrm{d}(\phi_{0}+\phi_{n})\equiv\textstyle\frac{1}{n-1}\pi_{ab}\varepsilon^{ab}{\,{\wedge}\;}\omega^{n}$
modulo $\mathcal{I}_{\mathsf{char}}$. As a result, the infinitesimal action of
the structure group on $E^{a}{}_{nb}$ is given by

(2.27)
$\mathrm{d}E_{anb}+2E_{anb}\phi_{n}+2E_{cn(a}\phi^{c}{}_{b)}+\left(\pi_{ab}-\textstyle\frac{1}{n-1}\pi_{cd}\varepsilon^{cd}\varepsilon_{ab}\right)\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Similarly to Section 2.3.6, it is possible to restrict to coframes with
vanishing $E^{a}{}_{nb},$ and consequently, by relation (2.27), the 1-forms
$\pi_{ab}$ are pure trace modulo $\mathcal{I}_{\mathsf{tot}}$, i.e.,
$\pi_{ab}=\pi_{n}\varepsilon_{ab},$ modulo $\mathcal{I}_{\mathsf{tot}},$ for
some 1-form $\pi_{n}.$ Such coframes are called 3-adapted.

The 3-adapted coframes are local sections of a $G_{3}$-bundle, where $G_{3}$
consists of matrices

(2.28) $\left\\{\begin{pmatrix}\mathbf{a}^{2}&0&0&0\\\
\mathbf{E}&\mathbf{a}\mathbf{b}\mathbf{A}&0&0\\\
\mathbf{e}&\mathbf{B}&\mathbf{b}^{2}&0\\\
\mathbf{C}&\mathbf{c}{}^{t}\\!\mathbf{A}\varepsilon&0&\frac{\mathbf{a}}{\mathbf{b}}{}^{t}\\!\mathbf{A}\end{pmatrix}\
\ \ \vline\ \begin{aligned}
&\mathbf{A}\in\mathrm{O}(p,q),\quad\mathbf{a},\mathbf{b}{}&{}\in{}\mathbb{R}\backslash\\{0\\}&,\\\
&{}^{t}\mathbf{B},\mathbf{C},\mathbf{E}\in\mathbb{R}^{n-1},\
&\mathbf{c},\mathbf{e}\in{}\mathbb{R}&\end{aligned}\right\\}.$

####  2.3.8. Symmetries of $F^{abc}$

In order to show $F^{abc}$ is completely symmetric, one first differentiates
equation (2.26b) to obtain

$0\equiv\varepsilon^{ab}\left(\mathrm{d}\theta_{b}-\phi^{c}{}_{b}{\,{\wedge}\;}\theta_{c}+\left(\phi_{0}-\phi_{n}\right)\theta_{b}+F^{cd}{}_{b}\,\theta_{d}{\,{\wedge}\;}\theta_{c}\right){\,{\wedge}\;}\omega^{n},\quad\textrm{mod}\quad\omega^{0},\ldots,\omega^{n-1}.$

Differentiating equation (2.26a) yields,

$0=\left(\mathrm{d}\theta_{c}-\phi^{a}{}_{c}{\,{\wedge}\;}\theta_{a}+\left(\phi_{0}-\phi_{n}\right)\theta_{c}+F^{ab}{}_{c}\,\theta_{a}{\,{\wedge}\;}\theta_{b}\right){\,{\wedge}\;}\omega^{c},\quad\textrm{mod}\quad
I\\{\omega^{0},\omega^{n}\\}.$

Comparing the two equations above, one obtains,
$F^{ab}{}_{c}\,\theta_{a}{\,{\wedge}\;}\theta_{b}=0$, i.e.,
$F^{ab}{}_{c}=F^{ba}{}_{c}.$ Recall, in (2.25) the quantities $F^{ab}{}_{c}$
satisfy $F^{abc}=F^{cba},$ where $F^{abc}=\varepsilon^{cd}F^{ab}{}_{d}.$ As a
result, $F^{abc}$ is totally symmetric.

####  2.3.9. The fourth coframe adaptation, Fubini cubic form and the Wsf
curvature

The next coframe adaptation reveals the essential invariants of a causal
structure, which are the Fubini cubic forms of the fibers $\mathcal{C}_{x}$
and the Weyl shadow flag curvature (also referred to as the Wsf curvature).

Differentiating (2.26b) and collecting terms of the form
$\tau^{ab}{}_{c}{\,{\wedge}\;}\theta_{b}{\,{\wedge}\;}\omega^{c}$ satisfying
$\tau_{a}{}^{b}{}_{c}=\tau_{c}{}^{b}{}_{a}$, one has

(2.29)
$\begin{split}0\equiv{}&\left[\left(\mathrm{d}F^{abc}+F^{abd}\phi^{c}{}_{d}+F^{adc}\phi^{b}{}_{d}+F^{dbc}\phi^{a}{}_{d}-F^{abc}(\phi_{0}-\phi_{n})\right.\right.\\\
&\left.\left.-(\gamma^{n}{}_{d}\varepsilon^{dc}\varepsilon^{ab}-\gamma^{a}{}_{0}\varepsilon^{bc)})\right){\,{\wedge}\;}\theta_{b}-\mathrm{d}(\phi_{0}+\phi_{n})\varepsilon^{ac}\right]{\,{\wedge}\;}\omega_{c}\quad\textrm{mod}\quad\omega^{0},\omega^{n}\end{split}$

Contacting by $\varepsilon_{ab}$, and defining

$F^{a}:=\varepsilon_{bc}F^{abc},$

one arrives at

$\mathrm{d}(\phi_{0}+\phi_{n})=\textstyle\frac{1}{n-1}(\gamma^{b}{}_{0}-\gamma^{n}{}_{d}\varepsilon^{db}+\mathrm{d}F^{b}){\,{\wedge}\;}\theta_{b}.$

Putting this back in (2.29), and contracting with $\varepsilon_{bc}$ gives

$\mathrm{d}F^{a}\equiv-F^{b}\phi^{a}{}_{b}+F^{a}(\phi_{0}-\phi_{n})-\textstyle\frac{n(n-2)}{n-1}(\gamma^{a}{}_{0}-\gamma^{n}{}_{b}\varepsilon^{ba})\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

As before, the infinitesimal action above makes it possible to translate
$F^{a}$ to zero. Therefore, the class of 4-adapted coframes can be defined by
the apolarity relation

(2.30) $F^{a}=0.$

The resulting reduction in the structure group is realized via the relation
$\gamma^{a}{}_{0}\equiv\varepsilon^{ab}\gamma^{n}{}_{b}$ modulo
$\mathcal{I}_{\mathsf{tot}}.$

It can be shown that the cubic form

$F_{3}=F^{abc}\theta_{a}\circ\theta_{b}\circ\theta_{c},$

when pulled-back to the fiber $\mathcal{C}_{x},$ coincides with Fubini cubic
form of $\mathcal{C}_{x}.$ This becomes evident in the derivation of the local
expression of $F^{abc}$ in (2.65).

It is noted that because the second fundamental form is normalized, for any
choice of 4-adapted coframe, the Fubini cubic form is well-defined tensor on
$\mathcal{C}.$

The structure group $G_{4}$ is reduced to the group of matrices of the form

(2.31) $\left\\{\begin{pmatrix}\mathbf{a}^{2}&0&0&0\\\
\mathbf{E}&\mathbf{a}\mathbf{b}\mathbf{A}&0&0\\\
\mathbf{e}&\mathbf{B}&\mathbf{b}^{2}&0\\\
\mathbf{C}&\mathbf{c}{}^{t}\\!\mathbf{A}\varepsilon&0&\frac{\mathbf{a}}{\mathbf{b}}{}^{t}\\!\mathbf{A}\end{pmatrix}\
\ \ \vline\ \begin{aligned}
&\mathbf{A}\in\mathrm{O}(p,q),\quad\mathbf{a},\mathbf{b}{}&{}\in{}\mathbb{R}\backslash\\{0\\}&,\\\
&{}^{t}\mathbf{B},\mathbf{C}\in\mathbb{R}^{n-1},\
&\mathbf{c},\mathbf{e}\in{}\mathbb{R}&\\\
&\mathbf{E}=\frac{a}{b}\mathbf{A}\,{}^{t}\\!\mathbf{B}\end{aligned}\right\\},$

where ${}^{t}\mathbf{B}$ denotes the transpose of $\mathbf{B}$ with respect to
$\varepsilon$.

By setting
$\gamma^{a}{}_{0}-\varepsilon^{ab}\gamma^{n}{}_{b}=K^{a}{}_{i0}\,\omega^{i}+K^{ab}\theta_{b},$
the structure equations after the fourth coframe adaptation are of the form

(2.32a) $\displaystyle\mathrm{d}\omega^{0}={}$
$\displaystyle-2\phi_{0}{\,{\wedge}\;}\omega^{0}-\theta_{a}{\,{\wedge}\;}\omega^{a},$
(2.32b) $\displaystyle\mathrm{d}\omega^{a}={}$
$\displaystyle-\varepsilon^{ab}\,\gamma_{b}{\,{\wedge}\;}\omega^{0}-\phi^{a}{}_{b}{\,{\wedge}\;}\omega^{b}-\left(\phi_{0}+\phi_{n}\right){\,{\wedge}\;}\omega^{a}-\varepsilon^{ab}\,\theta_{b}{\,{\wedge}\;}\omega^{n},$
$\displaystyle-K^{a}{}_{i0}\,\omega^{i}{\,{\wedge}\;}\omega^{0}-K^{ab}\,\theta_{b}{\,{\wedge}\;}\omega^{0}-F^{ab}{}_{c}\,\theta_{b}{\,{\wedge}\;}\omega^{c},$
(2.32c) $\displaystyle\mathrm{d}\omega^{n}={}$
$\displaystyle-\gamma^{n}{}_{0}{\,{\wedge}\;}\omega^{0}-\gamma_{a}{\,{\wedge}\;}\omega^{a}-2\phi_{n}{\,{\wedge}\;}\omega^{n},$
(2.32d) $\displaystyle\mathrm{d}\theta_{a}={}$
$\displaystyle-\pi_{a}{\,{\wedge}\;}\omega^{0}-\varepsilon_{ab}\,\pi_{n}{\,{\wedge}\;}\omega^{b}+\phi^{b}{}_{a}{\,{\wedge}\;}\theta_{b}-\left(\phi_{0}-\phi_{n}\right){\,{\wedge}\;}\theta_{a}$
$\displaystyle+A^{b}{}_{a}\theta_{b}{\,{\wedge}\;}\omega^{n}+W_{anbn}\omega^{b}{\,{\wedge}\;}\omega^{n}+B^{bc}{}_{a}\theta_{b}{\,{\wedge}\;}\theta_{c}$
$\displaystyle+E_{an}{}^{c}{}_{b}\theta_{c}{\,{\wedge}\;}\omega^{b}+W_{anbc}\omega^{b}{\,{\wedge}\;}\omega^{c}$

where $\gamma^{n}{}_{a}$ is being replaced by $\gamma_{a}.$

Differentiating (2.32a) yields

$\displaystyle A^{a}{}_{b}{}=0,\quad W_{anbn}{}=W_{bnan},\quad
E_{anbc}{}=E_{ancb},$ $\displaystyle K^{ab}=K^{ba},\quad B^{ab}{}_{c}=0,\quad
W_{[a|n|bc]}{}=0$

By making the following changes

$\displaystyle\gamma_{a}$
$\displaystyle\longmapsto\gamma_{a}-K_{cn0}\,\omega^{n}-K_{(ab)0}\omega^{a},\qquad\phi_{n}\longmapsto\phi_{n}-K_{cn0}\,\omega^{c},$
$\displaystyle\phi^{a}_{b}$
$\displaystyle\longmapsto\phi^{a}_{b}+\textstyle{\frac{1}{2}}\left(T^{a}{}_{bc}\,\omega^{c}-\varepsilon^{ae}T^{d}{}_{ec}\varepsilon_{da}\,\omega^{c}+\varepsilon_{cd}T^{c}{}_{ab}\,\omega^{d}\right)+\textstyle{\frac{1}{2}}\left(K^{a}{}_{b}-\varepsilon^{ac}K^{d}{}_{c}\varepsilon_{eb}\,\right)\omega^{0},$

where $T^{a}{}_{bc}=\delta^{a}{}_{b}K_{cn0}-\delta^{a}{}_{c}K_{bn0},$ and
$K_{an0}=\varepsilon_{ab}K^{b}{}_{n0},$ it can be assumed $K^{a}{}_{i0}$
vanishes

By replacing

$\pi_{n}\longmapsto\pi_{n}+\textstyle\frac{1}{n-1}\varepsilon^{ac}E_{anbc}\,\theta^{b}-\textstyle{\frac{1}{n-1}}\varepsilon^{cd}W_{cndn}\,\omega^{n},$

the traces of $W_{anbn}$ and $E^{a}{}_{nbc}$ are absorbed in $\pi_{n},$ and
therefore, $W_{anbn}$ and $E_{ancb}$ can be chosen to be trace-free in
$(a,b).$

The symmetric trace-free tensor $W_{anbn}$ is called the Weyl shadow flag
curvature (the Wsf curvature), which, as will be shown, is the second
essential invariant of causal structures.

By differentiating (2.32b), and collecting 3-forms
$A_{ab}\,\theta^{a}{\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\omega^{n}$ such that
$A_{ab}=A_{ba},$ and $A^{a}{}_{a}=0,$ it follows that

$E_{anbc}=-f_{abc},$

where $f_{abc}=F_{abc;n}$, i.e., the derivative of $F_{abc}$ along the
characteristic curves.

####  2.3.10. The fifth coframe adaptation

There is one more coframe adaptation that can be imposed on any causal
structure, which reduces the structure group by one dimension. First, note
that the derivative of (2.32b) gives

(2.33)
$\displaystyle\left(\left(\mathrm{d}K^{ab}+2K^{c(a}\phi^{b)}{}_{c}-2K^{ab}\left(\phi_{0}-\phi_{n}\right)-\varepsilon^{ab}\gamma^{n}{}_{0}-F^{abc}\gamma_{c}\right){\,{\wedge}\;}\theta_{b}+\mathrm{d}\gamma^{a}\right){\,{\wedge}\;}\omega^{0}\equiv
0$

modulo $I\\{\omega^{1},\cdots,\omega^{n}\\}.$ To find the infinitesimal group
action on $K^{ab}$ via the relation above one needs to find the 2-form
components $\nu^{ab}{\,{\wedge}\;}\theta_{b}$ of $\mathrm{d}\gamma^{a}$ where
$\nu^{ab}\nequiv 0$ modulo $\mathcal{I}_{\mathsf{tot}}.$ Differentiating
(2.32c), it follows that

$\left(\mathrm{d}\gamma_{a}+F^{bc}{}_{a}\gamma_{b}{\,{\wedge}\;}\theta_{c}+\Lambda_{a}\right){\,{\wedge}\;}\omega^{a}\equiv
0$

where $\Lambda_{a}$ is not of the form $\nu^{b}{}_{a}{\,{\wedge}\;}\theta_{b}$
with $\nu^{b}{}_{a}\nequiv 0$ modulo $\mathcal{I}_{\mathsf{tot}}.$ Inserting
the above expression of $\mathrm{d}\gamma_{a}$ into (2.33), one sees that the
infinitesimal action of the structure group $G_{5}$ on $K^{ab}$ is given by

$\mathrm{d}K^{ab}+2K^{c(a}\phi^{b)}{}_{c}-2K^{ab}\left(\phi_{0}+\phi_{n}\right)+\varepsilon^{ab}\gamma^{n}{}_{0}-2F^{abc}\gamma_{c}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}$

Note that $K^{ab}$ is not a tensor as its infinitesimal transformation rule in
terms of the structure group involves $F^{ab}{}_{c}.$

Using the apolarity relation (2.30), and the fact that $K_{ab}$ and
$\phi_{ab}$ are symmetric and skew-symmetric respectively, the infinitesimal
action of the structure group $G_{4}$ on $K:=\varepsilon_{ab}K^{ab}$ is found
to be

$\mathrm{d}K+K\left(\phi_{0}-\phi_{n}\right)+(n-1)\gamma^{n}{}_{0}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Now, the 5-adapted coframes can be defined by the property that $K=0.$ Hence,
one obtains

$\gamma^{0}{}_{n}=M_{ni}\,\omega^{i}+L^{a}\,\theta_{a}.$

By replacing

$\gamma_{a}\longmapsto\gamma_{a}-M_{na}\,\omega^{0}-\varepsilon_{ab}M_{nn}\,\omega^{b},\qquad\phi_{n}\longmapsto\phi_{n}-\textstyle{\frac{1}{2}}M_{nn}\,\omega^{0},$

it can be assumed that $M_{ni}=0.$

The 5-adapted coframes are local sections of a $G_{5}$-bundle where $G_{5}$ is
the matrix group defined by

(2.34) $\left\\{\begin{pmatrix}\mathbf{a}^{2}&0&0&0\\\
\mathbf{E}&\mathbf{a}\mathbf{b}\mathbf{A}&0&0\\\
\mathbf{e}&\mathbf{B}&\mathbf{b}^{2}&0\\\
\mathbf{C}&\mathbf{c}\,{}^{t}\\!\mathbf{A}\varepsilon&0&\frac{\mathbf{a}}{\mathbf{b}}\,{}^{t}\\!\mathbf{A}\end{pmatrix}\
\vline\ \begin{aligned} &\mathbf{a},\mathbf{b}\in\mathbb{R}\backslash\\{0\\},\
\mathbf{c}\in\mathbb{R},\\\ &\mathbf{A}\in\mathrm{O}(p,q),\
\mathbf{B},\mathbf{C}\in\mathbb{R}^{n-1}\\\
&\mathbf{E}=\frac{\mathbf{a}}{\mathbf{b}}\mathbf{A}\,{}^{t}\mathbf{B},\
\mathbf{e}=\frac{1}{2\mathbf{b}^{2}}\mathbf{B}\,{}^{t}\mathbf{B}\end{aligned}\right\\}.$

####  2.3.11. Invariant conformal quadratic form

According to (2.34), the structure group $G_{5}$ preserves the conformal class
of the quadratic form

(2.35)
$g:=2\,\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\,\omega^{a}\circ\omega^{b},$

which is semi-basic with respect to the fibration $\pi:\mathcal{C}\rightarrow
M.$ As will be shown in Section 2.3.14, the vanishing of the Fubini form
implies that the null cones of the causal structure coincide with those of
$g.$ The $\mathfrak{o}(p,q)$-valued pseudo-connection associated to the
structure group $G_{5}$ acting on $g,$ is given by

${\begin{pmatrix}\phi_{n}-\phi_{0}&\theta&0\\\
{}^{t}\\!\gamma&\phi&{}^{t}\\!\theta\\\
0&\gamma&\phi_{0}-\phi_{n}\end{pmatrix},}$

where ${}^{t}\\!\gamma$ and ${}^{t}\\!\theta$ are column vectors of 1-forms
$\gamma^{a}:=\varepsilon^{ab}\gamma_{b}$ and
$\theta^{a}:=\varepsilon^{ab}\theta_{b},$ respectively. From now on,
$\varepsilon_{ab}$ will be used to raise and lower the indices $a,b,c.$

####  2.3.12.  The first order structure equations

Having carried out five coframe adaptations, the first structure equations for
causal structures can now be expressed as follows.

The pseudo-connection form for the first order structure group is

(2.36) $\begin{pmatrix}2\phi_{0}&0&0&0\\\
\gamma^{a}&\phi^{a}{}_{b}+\left(\phi_{0}+\phi_{n}\right)\delta^{a}{}_{b}&0&0\\\
0&\gamma_{b}&2\phi_{{n}}&0\\\
\pi_{a}&\varepsilon_{ab}\,\pi_{n}&0&-\phi^{a}{}_{b}+\left(\phi_{0}-\phi_{{n}}\right)\delta^{a}{}_{b}\end{pmatrix}.$

Consequently, the structure equations can be written as

(2.37a) $\displaystyle\mathrm{d}\omega^{0}{}={}$
$\displaystyle-2\phi_{0}{\,{\wedge}\;}\omega^{0}-{}\theta_{a}{\,{\wedge}\;}\omega^{a},$
(2.37b) $\displaystyle\mathrm{d}\omega^{a}{}={}$
$\displaystyle-\gamma^{a}{\,{\wedge}\;}\omega^{0}-\phi^{a}{}_{b}{\,{\wedge}\;}\omega^{b}-\left(\phi_{0}+\phi_{{n}}\right){}{\,{\wedge}\;}\omega^{a}-\theta^{a}{\,{\wedge}\;}\omega^{{n}}$
$\displaystyle-K^{a}{}_{b}\,\theta^{b}{\,{\wedge}\;}\omega^{0}-F^{a}{}_{bc}\,\theta^{b}{\,{\wedge}\;}\omega^{c},$
(2.37c) $\displaystyle\mathrm{d}\omega^{{n}}{}={}$
$\displaystyle-\gamma_{a}{\,{\wedge}\;}\omega^{a}-2\phi_{{n}}{\,{\wedge}\;}\omega^{{n}}-L^{a}\,\theta_{a}{\,{\wedge}\;}\omega^{0}$
(2.37d) $\displaystyle\mathrm{d}\theta_{a}{}={}$
$\displaystyle-\pi_{a}{\,{\wedge}\;}\omega^{0}{}\,-\pi_{{n}}{\,{\wedge}\;}\omega_{a}+\phi^{b}{}_{a}{\,{\wedge}\;}\theta_{b}-\left(\phi_{0}-\phi_{{n}}\right){\,{\wedge}\;}\theta_{a}$
$\displaystyle-{}f_{abc}\,\theta^{b}{\,{\wedge}\;}\omega^{c}+W_{anbn}\,\omega^{b}{\,{\wedge}\;}\omega^{{n}}+\textstyle{\frac{1}{2}}W_{anbc}\,\omega^{b}{\,{\wedge}\;}\omega^{c},$

with the following symmetries

(2.38) $\begin{gathered}F_{abc}=F_{(abc)},\quad F^{a}{}_{ab}=0,\quad
K^{ab}=K^{ba},\quad K^{a}{}_{a}=0,\\\ W_{anbn}=W_{bnan},\quad
W^{a}{}_{nan}=0,\quad\quad W_{anbc}=-W_{0acb},\quad
W_{[a|n|bc]}=0.\end{gathered}$

The infinitesimal group action on the torsion coefficients is equivalent to
saying that the following 1-forms are semi-basic with respect to the fibration
$\pi:\mathcal{C}\rightarrow M.$

(2.39a) $\displaystyle\mathrm{D}F_{abc}$
$\displaystyle:=\mathrm{d}F_{abc}-F_{abd}\,\phi^{d}{}_{c}-F_{adc}\,\phi^{d}{}_{b}-F_{dbc}\,\phi^{d}{}_{a}-F_{abc}\,(\phi_{0}-\phi_{n}),$
(2.39b) $\displaystyle\mathrm{D}K_{ab}$
$\displaystyle:=\mathrm{d}K_{ab}-K_{ac}\,\phi^{c}{}_{b}-K_{bc}\,\phi^{c}{}_{a}-2K_{ab}\,\left(\phi_{0}-\phi_{n}\right)-2F_{abc}\,\gamma^{c},$
(2.39c) $\displaystyle\mathrm{D}L_{a}$
$\displaystyle:=\mathrm{d}L_{a}-L_{b}\,\phi^{b}{}_{a}-3L^{a}\,(\phi_{0}-\phi_{n})+K^{ab}\,\gamma_{b},$
(2.39d) $\displaystyle\mathrm{D}W_{anbn}$
$\displaystyle:=\mathrm{d}W_{anbn}-W_{ancn}\,\phi^{c}{}_{b}-W_{bncn}\,\phi^{c}{}_{a}-4W_{anbn}\,\phi_{n},$
(2.39e) $\displaystyle\mathrm{D}W_{anbc}$
$\displaystyle:=\mathrm{d}W_{anbc}-W_{dnbc}\,\phi^{d}{}_{a}-W_{andc}\,\phi^{d}{}_{b}-W_{anbd}\,\phi^{d}{}_{c},$
$\displaystyle\quad-
W_{anbc}\,\left(\phi_{0}-3\phi_{n}\right)+\left(W_{anbn}\,\varepsilon_{cd}+W_{ancn}\,\varepsilon_{bd}\right)\gamma^{d}.$

By the infinitesimal actions (2.39), it follows that no more reduction of the
structure group is possible, unless certain non-vanishing conditions on the
torsion coefficients are assumed. Such special cases are considered in
subsequent sections.

####  2.3.13. An $\\{e\\}$-structure

Using the first order structure equations, an $\\{e\\}$-structure can be
associated to causal structures. The necessary computations are carried out in
the Appendix.

######  Theorem 2.12.

To a causal structure $(M^{n+1},\mathcal{C}^{2n}),$ $n\geq 4,$ one can
associate an $\\{e\\}$-structure on the principal bundle
$P_{1,2}\rightarrow\mathcal{P}\rightarrow\mathcal{C}$ of dimension
$\frac{(n+3)(n+2)}{2}$ where $P_{1,2}\subset\mathrm{O}(p+2,q+2)$ is the
parabolic subgroup defined in (2.5). If $n=3,$ then $P_{1,2}$ is replaced by
$P_{1,2,3}.$ The essential invariants of a causal structure are the Fubini
cubic form of the fibers and the Wsf curvature. The vanishing of the essential
invariants implies that the $\\{e\\}$-structure coincides with the Maurer-
Cartan forms of $\mathfrak{(}p+2,q+2)$. The structure equations can be written
as

(2.40)
$\mathrm{d}\phi+\phi{\,{\wedge}\;}\phi=\begin{pmatrix}-\Phi_{0}-\Phi_{n}&-\Pi_{n}&-\Pi_{b}&-\Pi_{0}&0\\\
\Omega^{n}&\Phi_{n}-\Phi_{0}&\Gamma_{b}&0&\Pi_{0}\\\
\Omega^{a}&\Theta^{a}&\Phi^{a}{}_{b}&\Gamma^{a}&-\Pi^{a}\\\
\Omega^{0}&0&\Theta_{b}&\Phi_{0}-\Phi_{n}&\Pi_{n}\\\
0&-\Omega^{0}&\Omega_{b}&-\Omega^{n}&+\Phi_{0}+\Phi_{n}\end{pmatrix},$

where $\phi$ is defined in (2.7) and the right hand side of (2.40) is
expressed in (A.17).

The proof of the above theorem will be continued in the Appendix and involves
lots of tedious calculations. Of course, it is desirable to find the right
change of basis so that the $\\{e\\}$-structure defines a Cartan connection.
This issue is discussed in Section 2.4.3. It is noted the number of torsion
elements on the right hand side of the first order structure equations is
minimal due to the special absorption process that was undertaken here. When
obtaining the $\\{e\\}$-structure in the Appendix, to simplify computations,
one term will be added to Equation (2.37d) (see Equation A.1.4).

An immediate corollary of Theorem 2.12 is the following.

######  Corollary 2.13.

The maximal dimension of the symmetry algebra of causal structures on a
$(n+1)$-dimensional manifolds is $\frac{(n+3)(n+2)}{2}$ which occurs only in
the flat model.

###### Proof.

As explained in Section 2.2 the flat model has the maximal symmetries. Now,
suppose there is non-flat model that is maximally symmetric. Then, either the
Fubini form or the Wsf have a non-zero entry. However, by the infinitesimal
group actions (2.39), the structure group acts by scaling on both of the
essential invariants. As a result, the symmetry group of the causal structure
has to be a proper sub-group of $\mathrm{O}(p+1,q+1)$ obtained after
normalizing the value of that non-zero entry of the essential invariants.
Thus, it cannot be maximally symmetric. ∎

The corollary above is clear from the parabolic point of view, since any
parabolic geometry $(G,P)$ has maximal symmetry dimension equal to $\dim G$
which can only be reached in the flat case [ČS09].

####  2.3.14. The case of conformal structures

Using the first order structure equations, an observation is made regarding
(pseudo-)conformal structures viewed as causal structures.

######  Proposition 2.14.

A causal structure $(M,\mathcal{C})$ of signature $(p+1,q+1)$ with vanishing
Fubini cubic form induces a pseudo-conformal structure of the same signature
on $M.$ The resulting pseudo-conformal structure is flat if the initial causal
structure has vanishing Wsf curvature, and hence flat. Conversely, any pseudo-
conformal structure defines a causal structure of the same signature on its
bundle of null cones. The resulting causal structure is flat if the initial
pseudo-conformal structure is flat.

###### Proof.

To show the first part, note that if the Fubini form vanishes then, by the
Bianchi identities (A.19), the quantities $K_{ab}$ and $L_{a}$ vanish as well.
The first order structure equations (2.36) can be used to show that the Lie
derivatives of the quadratic form $g,$ defined in Section 2.3.11, along
vertical vector fields
$\frac{\partial}{\partial\theta_{1}},\cdots,\frac{\partial}{\partial\theta_{n-1}}$
are given by

(2.41) $\mathcal{L}_{\frac{\partial}{\partial\theta_{a}}}g=\lambda g,$

for some non-vanishing function $\lambda$ on $\mathcal{C}.$ The vector fields
$\frac{\partial}{\partial\omega^{i}},$ and
$\frac{\partial}{\partial\theta_{a}}$ are defined as dual to the coframe
$\omega^{0},\cdots,\theta_{n-1}.$ The vertical vector fields are integrable
with the fibers $\mathcal{C}_{x}$ as their integral manifolds, and $M$ as the
quotient space. Consequently, the conformal class of the quadratic form $g$
descends to $M$ and because it is non-degenerate and of maximal rank on $M,$ a
pseudo-conformal structure on $M$ is obtained with the same signature as that
of $g.$ If the causal structure is flat, i.e., the $\\{e\\}$-structure is
given by the Maurer-Cartan forms of $\mathrm{O}(p+2,q+2),$ then the structure
bundle fibers over $M$ as well, implying that the resulting pseudo-conformal
structure has to be flat too.

The converse part is shown similarly. Given a pseudo-conformal structure of
signature $(p+1,q+1)$, choose a coframe $(\omega^{0},\cdots,\omega^{n})$ so
that a representative of the pseudo-conformal structure takes the form
$g=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b}.$ On
its null cone bundle $\pi:\mathcal{C}\rightarrow M,$ the metric $g$ pulls back
to define a quadratic form on $\mathcal{C}$ satisfying (2.41). The 1-forms
$\omega^{i}$ define a set of semi-basic 1-forms on $\mathcal{C}$. Using
conformal transformations, $\omega^{i}$’s can be chosen to become adapted to
the flag (2.11). This can be seen as follows. The affine tangent space
$\widehat{T}_{y}\mathcal{C}_{x}\subset T_{x}M$ is a null hyperplane and by a
conformal transformation $\omega^{0}$ can be taken to be its annihilator.
Moreover, the dual of $\omega^{0}$ with respect to the metric $g$ lies on
$\hat{y}.$ More explicitly, let $y\in\hat{y}\subset\widehat{\mathcal{C}}.$
Because, $g_{ij}y^{i}y^{j}=0,$ the 1-form $\omega^{0}$ as the annihilator of
$\widehat{T}_{y}\mathcal{C}_{x}$ can be expressed as
$\omega^{0}=g_{ij}y^{j}\mathrm{d}x^{i}.$ Now, it is apparent that the dual of
$\omega^{0}$ lies in $\hat{y}.$ Since the metric is expressed as
$g=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b},$ it
follows that $\omega^{a}(\hat{y})=0,$ and $\omega^{n}(\hat{y})\neq 0.$ Hence,
$\omega^{i}$’s are adapted to the flag (2.11).

Since $g$ satisfies (2.41), the resulting causal structure have vanishing
Fubini form. If the conformal structure is flat, i.e., isomorphic to the flat
model, then as discussed in Section 2.2, the corresponding causal structure is
flat as well. Because the construction is reversible, the result follows from
the first part of the proposition. ∎

The following corollary follows immediately.

######  Corollary 2.15.

If a pseudo-conformal structure lifts to a causal structure with vanishing Wsf
curvature, then the resulting causal structure is flat.

###### Proof.

By Proposition 2.14, a pseudo-conformal structure defines a causal structure
with vanishing Fubini cubic form. If the resulting causal structure has
vanishing Wsf curvature, then all of its essential invariants are zero and is
therefore flat. By Proposition 2.14, one obtains that the initial pseudo-
conformal structure has be flat as well. ∎

######  Remark 2.16.

As was mentioned previously, the discussion above goes through in the case of
conformal structures of positive definite signature. The extra step would be
to complexify the inner product and the tangent bundle and work with complex
causal structures endowed with a reality condition. See [LeB83] for a
treatment of complex conformal structures.

###  2.4. Causal structures as parabolic geometries

This section provides the necessary definitions and theorems needed to
characterize causal structures as parabolic geometries. A main reference on
parabolic geometries is [ČS09], but, for the purposes of this section the
articles [KT14, ANN15, Yam93, Zel09] suffice.

The main step is the characterization of causal structures by a bracket
generating distribution with certain properties, described in Section 2.4.2.
As a result, causal structures on an $(n+1)$-dimensional manifold can be
viewed as regular normal infinitesimal flag structures of type
$(B_{n},P_{1,2})$ or $(D_{n},P_{1,2})$ if $n\geq 4,$ or $(D_{3},P_{1,2,3})$ if
$n=3$. By a result of Tanaka-Morimoto-Čap-Schichl, such structures can be
canonically identified as regular normal parabolic geometries of the same
type.

####  2.4.1. Parabolic geometries and Tanaka Theory

A parabolic geometry is a Cartan geometry $(\mathcal{P},\omega)$ of type
$(G,P)$ where $G$ is a real or complex semi-simple Lie group and $P\subset G$
is a parabolic subgroup. Recall that a Cartan geometry $(\mathcal{P},\omega)$
of type $(G,H)$ is a principle bundle
$H\longrightarrow\mathcal{P}\longrightarrow M,$ endowed with a Cartan
connection $\omega.$ The 1-form $\omega$ takes value in $\mathfrak{g}$, the
Lie algebra of $G,$ and satisfies the following conditions.

  1. (1)

$\omega_{u}:T_{u}\mathcal{P}\longrightarrow\mathfrak{g}$ is an isomorphism of
vector space for all $u\in\mathcal{P}.$

  2. (2)

$V^{\dagger}\im\omega=V$ for all $V\in\mathfrak{h}$ where $V^{\dagger}$ is the
fundamental field of $V.$

  3. (3)

$(R_{h})^{*}\omega=\mathrm{Ad}(h^{-1})\omega$ for all $h\in H.$

A canonical way of identifying a geometric structure as a parabolic geometry
is by characterizing it in terms of certain _bracket generating distributions_
possibly equipped with a tensor field, e.g., conformal structures or semi-
Riemannian structures. Recall the following definitions. To any distribution
$D\subset TN$, one can associate a Pfaffian system

$D^{\perp}:=\\{\omega\in T^{*}\mathcal{C}\ \vline\ \omega(u)=0\ \forall\ u\in
D\\}.$

Given a distribution $D\subset T\mathcal{C},$ its space of sections is denoted
by $\Gamma(D)$ and its _derived distribution_ is defined by $D^{2}=D+[D,D].$
Here, $D+[D,D]$ means a distribution whose sections are obtained from
$\Gamma(D)+[\Gamma(D),\Gamma(D)].$ Define $D^{1}=D.$ The _ $k$-th weak derived
distribution_ of $D$ is defined by

(2.42) $D^{k+1}=D^{k}+[D^{k},D].$

One obtains that

(2.43) $(D^{k+1})^{\perp}=\\{\omega\in(D^{k})^{\perp}\ \vline\
\mathrm{d}\omega\equiv 0\quad\textrm{mod}\quad(D^{k})^{\perp}\\}.$

A distribution $D\subset TN$ is called bracket generating if there exists
$k>1$ such that $D^{k}=TN.$ The small growth vector of a bracket generating
distribution $D$ is $(d_{1},\cdots,d_{k})$ where $d_{k}=\dim(D^{k}).$

On a manifold $M,$ a bracket generating distribution, $D,$ of depth $\mu$
induces a filtration of $T_{x}M$ for all $x\in M,$ by

$D(x)=:D^{1}(x)\subset D^{2}(x)\subset\cdots\subset D^{\mu}(x)=T_{x}M.$

Let $\mathfrak{m}(x)$ denote the grading of $T_{x}M$ that corresponds to the
filtration above, i.e.,

$\mathfrak{m}(x)=\bigoplus_{i=-1}^{-\mu}\mathfrak{g}_{i}(x),\qquad\mathfrak{g}_{i}(x):=D^{-i}(x)/D^{-i-1}(x).$

At each $x\in M,$ the Lie bracket induces a bracket on $\mathfrak{m}(x)$
called the Levi bracket which is tensorial and turns $\mathfrak{m}(x)$ into a
graded nilpotent Lie algebra (GNLA). The Lie algebra $\mathfrak{m}(x)$ is
called the _symbol algebra_ of $D(x)$. A distribution $D$ of _constant type_
has the property that $\mathfrak{m}(x)\cong\mathfrak{m}$ for some GNLA
$\mathfrak{m}$ for all $x\in M.$

An interesting class of GNLA arises from pairs $(G,P)$ where $G$ is a real or
complex semi-simple group and $P\subset G$ is a parabolic subgroup in the
following way. A $|\mu|$-grading for the Lie algebra of $G,$ denoted by
$\mathfrak{g},$ is a vector space decomposition

$\mathfrak{g}=\mathfrak{g}_{\mu}\oplus\cdots\oplus\mathfrak{g}_{\mu},$

satisfying (i)$[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j};$
(ii) $[\mathfrak{g}_{i},\mathfrak{g}_{-1}]=\mathfrak{g}_{i-1}$ for all
$i\leq-1,$ i.e., the negative part of the grading,

$\mathfrak{g}_{-}:=\mathfrak{g}_{-\mu}\oplus\cdots\oplus\mathfrak{g}_{-1},$

is generated by $\mathfrak{g}_{-1};$ and (iii) $\mathfrak{g}_{\pm\mu}\neq 0.$

The subalgebra $\mathfrak{g}_{-}$ is GNLA and

$\mathfrak{p}:=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{\mu}$

is always a parabolic subalgebra of $\mathfrak{g}.$ Its corresponding
subgroup, $P\subset G,$ is a subgroup that lies between the normalizer
$N_{G}(\mathfrak{p})$ and its identity component. The subgroup $G_{0}\subset
P$ defined as

$G_{0}=\\{g\in P\ \vline\
\mathrm{Ad}(\mathfrak{g}_{i})\subset\mathfrak{g}_{i}\ \mathrm{for\ all\
}i=-\mu,\cdots,\mu\\}$

is called the _Levi subgroup_ corresponding to $P.$

For such a choice of $(G,P),$ there is a canonical construction of a bracket
generating distribution whose symbol algebra is $\mathfrak{g}_{-}.$ Let
$N_{\mathfrak{g}}$ be the simply connection Lie group with Lie algebra
$\mathfrak{g}_{-}.$ The left-invariant vector fields obtained from
$\mathfrak{g}_{-1}$ induces a distribution of constant type on
$N_{\mathfrak{g}}$ whose symbol algebra is isomorphic to $\mathfrak{g}_{-}.$

For a semi-simple Lie algebra $\mathfrak{g}$ every derivation is inner.
Therefore given a $|\mu|$-grading of $\mathfrak{g}$ it admits a unique
_grading element_ i.e., $e\in\mathfrak{g}_{0}$ such that $[e,g]=jg$ for all
$g\in\mathfrak{g}_{j}$ and $-\mu\leq j\leq\mu.$

Consequently, the space of $q$-forms on $\mathfrak{g}_{-}$ with values in
$\mathfrak{g}$, denoted by $\Lambda^{q}(\mathfrak{g}_{-},\mathfrak{g}),$
admits a decomposition
$\oplus_{p}\Lambda^{p,q}(\mathfrak{g}_{-},\mathfrak{g})$ where
$\Lambda^{p,q}(\mathfrak{g}_{-},\mathfrak{g})$ is the set of $q$-forms
$\omega,$ with weight $p,$ i.e., $\mathcal{L}_{e}\omega=p\omega.$161616In
[Yam93], the space $\Lambda^{p,q}$ is referred to as $C^{p-q+1,q}.$ The
decomposition can be written as

$\Lambda^{p,q}(\mathfrak{g}_{-},\mathfrak{g})=\bigoplus_{k}\mathfrak{g}_{k+p}\otimes\wedge^{q}_{k}(\mathfrak{g}_{-}^{*}),\quad\mathrm{where\quad}\wedge^{q}_{k}(\mathfrak{g}_{-}^{*}):=\sum_{\begin{subarray}{l}r_{1}+\cdots+r_{q}=k\\\
r_{1},...,r_{q}<0\end{subarray}}\mathfrak{g}^{*}_{r_{1}}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\mathfrak{g}^{*}_{r_{q}}.$

According to a theorem of Tanaka [Yam93], if
$H^{1,p}(\mathfrak{g}_{-},\mathfrak{g})=0,$ for all $p\geq 0$, then via a
process called the _Tanaka prolongation_ the Lie algebra of infinitesimal
symmetries of a distribution with symbol algebra $\mathfrak{g}_{-}$ is
isomorphic to $\mathfrak{g},$ i.e., $\mathfrak{inf}(D)=\mathfrak{g}.$

The above theorem can be applied to _regular infinitesimal flag structures_ of
type $(G,P),$ which is defined as follows. Consider a manifold $M$ endowed
with a bracket generating distribution of constant type whose symbol algebra
is $\mathfrak{m}.$ The graded frame bundle is a right principal bundle
$\mathrm{Aut}_{gr}(\mathfrak{m})\rightarrow\mathcal{F}_{gr}(M)\rightarrow M,$
where $\mathrm{Aut}_{gr}(\mathfrak{m})$ is the group of automorphisms of
$\mathfrak{m}$ that preserve the grading.171717In case $M$ is endowed with
additional structure, e.g., conformal structures or semi-Riemannian
structures, this principal bundle can be reduced further. If the symbol
algebra of the distribution is isomorphic to $\mathfrak{g}_{-}$ and
$\mathrm{Aut}_{gr}(\mathfrak{m})$ is isomorphic181818If there exists
additional structure on the distribution then
$\mathrm{Aut}_{gr}(\mathfrak{m})$ reduces to $G_{0}.$ to
$\mathrm{Ad}(G_{0})\subset\mathrm{Aut}_{gr}(\mathfrak{g}_{-}),$ where $G_{0}$
is the Levi subgroup of $(G,P)$ and $\mathrm{Ad}$ is the adjoint action, then
one says that the distribution induces a regular infinitesimal flag structure
of type $(G,P).$

By a result of Tanaka [Tan79], Morimoto [Mor93] and Čap-Schichl [ČS00], there
is an equivalence of categories between _regular, normal_ parabolic geometries
and regular infinitesimal flag structures via Tanaka prolongation. Regularity
and normality of a parabolic geometry $(G,P)$ is defined in terms of its
curvature function of its Cartan connection $(\mathcal{P},\omega)$ as follows.

Therefore, at each point $s\in\mathcal{P},$ the curvature
$\Omega=\mathrm{d}\omega+[\omega,\omega]$ of the Cartan connection can be used
to define the curvature function
$K:\mathcal{P}\rightarrow\wedge^{2}\mathfrak{g}_{-}^{*}\otimes\mathfrak{g}$ as

$K(s)(X,Y)=\Omega(\omega^{-1}(X)(s),\omega^{-1}(Y)(s)),\quad
X,Y\in\mathfrak{g}_{-},$

where the space $\mathfrak{g}_{-}^{*}$ can be identified with
$\mathfrak{g}_{+}:=\oplus_{i>0}\mathfrak{g}_{i}$ via the Killing form. The
Spencer cohomology groups
$H^{p,q}\subset\mathfrak{g}\otimes\wedge^{q}(\mathfrak{g}_{-}^{*}),$ are
defined using the Kostant operators
$\partial:\Lambda^{p,q}\rightarrow\Lambda^{p-1,q+1},$ and
$\partial^{*}:\Lambda^{p,q}\rightarrow\Lambda^{p+1,q-1},$ given by

$\displaystyle\partial c(X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q+1})$
$\displaystyle=\sum_{i}(-1)^{i+1}[X_{i},c(X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\hat{X}_{i}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q+1})]$
$\displaystyle+\sum_{i<j}$
$\displaystyle(-1)^{i+j}c([X_{i},X_{j}]{\,{\wedge}\;}X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\hat{X}_{i}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\hat{X}_{j}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q+1})$
$\displaystyle\partial^{*}c(X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q-1})$
$\displaystyle=\sum_{i}[e^{*}_{i},c(e_{i}{\,{\wedge}\;}X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q-1})]$
$\displaystyle+\frac{1}{2}\sum_{i,j}$
$\displaystyle(-1)^{j+1}c([e^{*}_{i},X_{j}]_{-}{\,{\wedge}\;}e_{i}{\,{\wedge}\;}X_{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\hat{X}_{j}{\,{\wedge}\;}\cdots{\,{\wedge}\;}X_{q-1}),$

where $c\in\Lambda^{p,q}(\mathfrak{g})$ and $[e^{*}_{i},X_{j}]_{-}$ denotes
the $\mathfrak{g}_{-}$-component of $[e^{*}_{i},X_{j}],$ with respect to the
decomposition $\mathfrak{g}=\mathfrak{g}_{-}\oplus\mathfrak{p}.$ The operators
above result in the orthogonal decomposition

(2.44) $\Lambda^{p,q}=H^{p,q}+\square\Lambda^{p,q},$

where the operator $\square:=\partial^{*}\partial+\partial\partial^{*}$ is
referred to as the Kostant Laplacian and $\square H^{p,q}=0.$

The $\Lambda^{p,2}$-component of the curvature function $K$ is denoted by
$K^{(p)}$. Consequently, $K$ can be decomposed as

(2.45) $K=\sum_{i=-\mu+2}^{3\mu}K^{(i)},$

where
$K^{(p)}:\mathfrak{g}_{i}\times\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{i+j+p}$
is the component of $K$ with homogeneity $p.$ Additionally, one can decompose
the curvature function as

$K=\sum_{i=-\mu}^{\mu}K_{p},$

where $K_{p}$ takes value in $\mathfrak{g}_{p}.$ The
$\mathfrak{g}_{-}$-component of $K$, $\sum_{i=-\mu}^{-1}K_{i},$ is denoted by
$K_{-}.$ The components of the curvature lying in $H^{p,2}$ are called
_harmonic curvature_ which constitute the set of essential invariants.

A Cartan connection is called torsion-free if $K_{-}=0,$ normal if
$\partial^{*}K=0,$ and regular if it is normal and $K^{(i)}=0,$ for all $i\leq
0.$

####  2.4.2. Characterizing causal structures by a bracket generating
distribution

The fact that $[\varepsilon^{ab}]$ is non-degenerate allows one to give a
characterization of causal structures in terms of a bracket generating
distribution with a certain small growth vector obtained as follows.

Define the distributions $\Delta$ and $D$ such that
$(\Delta)^{\perp}=\mathcal{I}_{\mathsf{bas}}$ and
$D^{\perp}=\\{\omega^{0},\cdots,\omega^{n-1}\\}.$ The rank of $D$ is $n$ and
$\Delta$ is a completely integrable sub-distribution of $D$ of corank $1$
whose integral manifolds are given by the fibers $\mathcal{C}_{x}$. Let
$\mathbf{v}$ be a characteristic vector field. It follows that $\mathbf{v}\in
D$ and $\mathbf{v}\notin\Delta.$

Knowing $\omega^{a}\equiv-\varepsilon^{ab}\theta_{b}{\,{\wedge}\;}\omega^{n},$
modulo $I\\{\omega^{0},\cdots,\omega^{n-1}\\}$ and that $\Delta$ is completely
integrable implies that $(D^{2})^{\perp}=\\{\omega^{0}\\}.$ Similarly, the
equation $\mathrm{d}\omega^{0}\equiv-\theta_{a}{\,{\wedge}\;}\omega^{a},$
modulo $I\\{\omega^{0}\\},$ implies $(D^{3})^{\perp}=\\{0\\},$ i.e.
$D^{3}=T\mathcal{C}.$ As a result, the small growth vector of $D$ is
$(n,2n-1,2n).$

Let $\\{f_{1},\cdots,f_{n-1}\\}$ be a dual basis for $\Delta$ and $e_{n}$ be a
basis for $D/\Delta.$ Since the growth vector is $(n,2n-1,2n),$ it follows
that $\operatorname{rank}(D^{2}/D)=n-1.$ Because $\Delta$ is Frobenius, one
obtains that the set of vectors $e_{a}=[e_{n},f_{a}]$ are linearly independent
and span $D^{2}/D.$ This relation among $e_{a},e_{n}$ and $f_{a}$ is
equivalent to the structure equation
$\mathrm{d}\omega^{a}\equiv-\varepsilon^{ab}\theta_{b}{\,{\wedge}\;}\omega^{n}$
modulo $I\\{\omega^{0},\cdots,\omega^{n-1}\\}.$

Furthermore, the equation
$\mathrm{d}\omega^{0}\equiv\theta_{a}{\,{\wedge}\;}\omega^{a}$ modulo
$I\\{\omega^{0}\\},$ implies that there is a non-degenerate pairing between
$e_{a}$’s and $f_{a}$’s, i.e., $[e_{\mathbf{a}},f_{\mathbf{a}}]=e_{0}$ (no
summation involved ) for all $1\leq\mathbf{a}\leq n-1,$ where $e_{0}$ spans
$D^{3}/D^{2}.$

One can investigate the action induced by the structure group $G_{5}$ in
(2.34) on the _symbol algebra_ of $D$ as follows. Each tangent space
$T_{x}\mathcal{C}$ admits a filtration $D(x)\subset D^{2}(x)\subset D^{3}(x)$
which results in the grading
$\mathfrak{m}(x)=\mathfrak{g}_{-1}(x)\oplus\mathfrak{g}_{-2}(x)\oplus\mathfrak{g}_{-3}(x)$
where $\mathfrak{g}_{i}(x)=D^{-i}/D^{-i-1}$ with $D^{0}=\\{0\\}.$ Using the
Lie bracket, $\mathfrak{m}(x)$ becomes a GNLA which would be the symbol
algebra of $D.$ Since $\mathfrak{g}_{-1}$ is given by the kernel of
$\mathcal{I}_{1}:=I\\{\omega^{0},\cdots,\omega^{n-1}\\},$ the action of the
structure group $G_{5}$ on
$\mathcal{I}_{\mathsf{tot}}=I\\{\omega^{0},\cdots,\omega^{n},\cdots,\theta^{n-1}\\},$
modulo $\mathcal{I}_{1},$ is given by scaling on $\omega^{n}$ and by conformal
transformations on $\theta^{a},$ i.e. ,

$\omega^{n}\rightarrow\mathbf{b}^{2}\omega^{n},\qquad\theta^{a}\rightarrow\frac{\mathbf{a}}{\mathbf{b}}\mathbf{A}^{a}{}_{b}\theta^{b}.$

This action is isomorphic to the action of the _Levi subgroup_ of
$\mathrm{O}(p+2,q+2)$ corresponding to the parabolic subgroup $P_{1,2}$ if
$n\geq 3$ or $P_{1,2,3}$ if $n=3,$ on the coframe bundle. In other words, in
causal structures the group of automorphisms of the coframe bundle which
preserves the grading of $T_{x}\mathcal{C}$ is reduced to $G_{0}.$

####  2.4.3. Normal Cartan connection

Using the flat model 2.2, it follows that the symbol algebra arising from the
bracket generating distribution $D,$ defined in Section 2.4.2 as
$\\{\omega^{0},\cdots,\omega^{n-1}\\}^{\perp},$ is isomorphic to the negative
part of the $|3|$-grading for $\mathfrak{g}=\mathfrak{o}(p+2,q+2),$ i.e.,

$\mathfrak{m}\cong\mathfrak{g}_{-}=\mathfrak{g}_{-3}\oplus\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}.$

The parabolic subalgebra that corresponds to this $|3|$-grading is denoted as
$\mathfrak{p}_{1,2}$ when $n\geq 4$ and $\mathfrak{p}_{1,2,3}$ when $n=3.$ The
corresponding parabolic subgroup is given in Section 2.2, i.e., the parabolic
subgroup of $\mathrm{O}(p+2,q+2)$ that is isomorphic to the stabilizer of the
subspace spanned by $\\{e_{1},e_{2}\\}$ where $\\{e_{1},\cdots,e_{n+3}\\}$ is
some basis for $\mathbb{R}^{n+3}$ and $p+q=n-1.$ Alternatively,
$\mathfrak{p}_{1,2}$ can be described by crossing the first two nodes in the
Dynkin diagrams $B_{n}$ and $D_{n}$ when $n\geq 4$ and all three nodes when
$n=3.$

Furthermore, the distribution $D$ induces a regular infinitesimal flag
structure. This is due to the fact that the condition
$H^{p,1}(\mathfrak{g}_{-},\mathfrak{g})=0$ is satisfied for all $p\geq 0.$
More directly, by the discussion in Section 2.4.2 and the first order
structure equations, one obtains that $\mathrm{Aut}_{gr}(\mathfrak{g}_{-})$ is
reduced to the Levi subgroup. Thus, by Tanaka prolongation one can view causal
structures as regular, normal parabolic geometries of type $(B_{n},P_{1,2})$
or $(D_{n},P_{1,2})$ if $n\geq 4$, or $(D_{3},P_{1,2,3}).$ In Section 2.5.1 it
is discussed that when $n\geq 4,$ causal structures are canonically isomorphic
to regular, normal parabolic geometries of type $(B_{n},P_{1,2})$ or
$(D_{n},P_{1,2})$. When $n=3,$ causal structures are _“subgeometries”_ of
regular, normal parabolic geometries of type $(D_{3},P_{1,2,3})$, i.e., the
integrability of the fibers $\mathcal{C}_{x},$ implies the vanishing of one of
the harmonic curvature components of such parabolic geometries.

Knowing the existence of a normal Cartan connection for causal structures, the
normality conditions of Tanaka are straightforward to obtain and can be used
to recursively define the correct change of basis which transforms the
$\\{e\\}$-structure to a Cartan connection (see [Mak16]). Since finding the
explicit expressions for the change of basis is very tedious and the result is
not particularly illuminating, finding the normal Cartan connection is not
carried out here.

###  2.5. Essential invariants and the Raychaudhuri equation

In this section the essential invariants whose vanishing imply the flatness of
the causal structure are introduced. Their geometric interpretation is given.
A variational problem is naturally associated to a causal geometry which
allows one to define the notion of null Jacobi fields in the causal setting.
Finally, the section ends with a discussion on associated Weyl connections of
a causal structure and a derivation of the Raychaudhuri equation for causal
structures. In this section there are references to the Appendix, particularly
to the $\\{e\\}$-structure obtained in Section A.1.4.

####  2.5.1. The harmonic curvature

According to the $\\{e\\}$-structure obtained in Section A.1.4, obstructions
to the flatness of a causal structure, as listed in Sections A.2 and A.3, are
generated by $F_{abc}$ and $W_{anbn}.$ For instance, differentiating (2.37d)
gives

(2.46) $\displaystyle W_{anbc}$
$\displaystyle=\textstyle{\frac{2}{3}}\left(W_{an[b|n;|{\underline{c}}]}+F_{a[c}{}^{d}W_{b]ndn}\right)$
$\displaystyle+\textstyle\frac{4}{3(n+2)}\left(\varepsilon_{a[c|}W^{d}{}_{n|b]n;{\underline{d}}}+\varepsilon_{a[b}F^{de}{}_{c]}W_{dnen}\right)$

In Section A.2, the quantities $W_{ijkl}$ are expressed as the coframe
derivatives of $W_{anbn}$ with respect to
$\frac{\partial}{\partial\theta^{a}}.$ As a result, in the case $F_{abc}=0,$
the vanishing of $W_{nabn}$ implies flatness of the causal structure.

For an understanding of $W_{anbn},$ consider the 2-plane given by
$\lambda{\,{\wedge}\;}\mu$ where
$\lambda=\lambda^{i}\mathbf{e}_{i},\mu=\mu^{i}\mathbf{e}_{i}\in TM$ for some
basis $\\{\mathbf{e}_{0},\cdots,\mathbf{e}_{n}\\}.$ The _sectional conformal
curvature_ of a pseudo-conformal structure evaluated at
$\lambda{\,{\wedge}\;}\mu$ is defined by

$W(\lambda{\,{\wedge}\;}\mu)=W_{ijkl}\lambda^{i}\mu^{j}\lambda^{k}\mu^{l}.$

As a result, at the point $(x,[y])\in\mathcal{C},$ the invariant $W_{nabn}$
can be interpreted as the part of the conformal sectional curvature acting on
2-planes
$\xi^{a}=\frac{\partial}{\partial\omega^{n}}{\,{\wedge}\;}\frac{\partial}{\partial\omega^{a}}.$
According to the construction of the coframes in (2.11), $\xi$ is a subset of
$\widehat{T}_{y}\mathcal{C}_{x}$ and contains $\hat{y}$ where
$[\hat{y}]=y\in\mathcal{C}_{x}.$ It follows that
$\xi^{a}=\mu^{a}{\,{\wedge}\;}y^{i}\partial_{i},$ where $\mu^{a}$ is an
element of the _shadow space_

(2.47) $S_{(x,[y])}:=\widehat{T}_{y}\mathcal{C}_{x}/\hat{y}.$

As a result, $W_{anbn}$ can be thought of as the restriction of the bilinear
form $W_{ijkl}y^{j}y^{l}$ to the shadow space. This restriction is well-
defined up to scale. Infinitesimally, this scaling action is realized by the
1-form $\phi_{n}$ as is apparent from (2.39d).

Finally, it can be argued that the Fubini cubic form and the Wsf curvature
coincide with the harmonic curvatures of the parabolic geometries that
correspond to causal structures. By the discussion in Section 2.4.3, causal
structures on an $(n+1)$-dimensional manifold correspond to parabolic
geometries of type $(B_{n},P_{1,2})$ and $(D_{n},P_{1,2})$ if $n\geq 4,$ and
$(D_{3},P_{1,2,3}).$ As discussed in Section 2.4.1, a grading is induced on
the cohomology classes of parabolic geometries. Moreover, the set of essential
invariants for a parabolic geometry is given by the harmonic curvature. It
turns out that for parabolic geometries of type $(B_{n},P_{1,2})$ and
$(D_{n},P_{1,2})$ if $n\geq 4,$ the harmonic curvature has two irreducible
components. Using the grading (2.45) of the curvature, one component has
homogeneity one whose $\mathfrak{g}_{0}$-irreducible representation is given
as a totally symmetric and trace-free cubic tensor. The other irreducible
component has homogeneity two whose $\mathfrak{g}_{0}$-irreducible
representation is given as a symmetric and trace-free bilinear form (see
[KT14, ANN15, Yam93]). The structure equations of causal geometries imply that
the essential invariant $F_{abc}$ corresponds to a cohomology class of
homogeneity one and the Wsf curvature corresponds to a cohomology class of
homogeneity two. Also, from the infinitesimal group actions (2.39a) and
(2.39d) one obtains that the Fubini cubic form and the Wsf curvature are
closed under the action of the Levi subgroup. Moreover, one can impose the
Tanaka normality conditions on the pseudo-connection form of the
$\\{e\\}$-structure to derive a normal Cartan connection (see [Mak16]). It
turns out that the necessary change of coframe does not affect the Fubini
cubic form and the Wsf curvature, i.e., the essential invariants of the
$\\{e\\}$-structure coincide with the harmonic curvature.

The situation is different in four dimensions. Four dimensional causal
structures are a special class of parabolic geometries of type
$(D_{3},P_{1,2,3}).$ The harmonic curvature of these parabolic geometries is
five dimensional, three of which have homogeneity one and two have homogeneity
two. Consider a four dimensional causal structure of split signature. The
Fubini cubic form of projective surfaces and the Wsf curvature are comprised
of two entries (see Section 4). By Equation (4.8) all these four entries are
irreducible under the action of the Levi subgroup. As a result, four
dimensional causal structures are special cases of parabolic geometries of
type $(D_{3},P_{1,2,3})$ in which one of the harmonic curvatures of
homogeneity one is zero.

####  2.5.2. The ssf curvature

The action of the structure group $G_{5}$ defined in (2.34) on the bilinear
form (2.35) is a scaling action $g\rightarrow\mathbf{a}^{2}\mathbf{b}^{2}g.$
Fix a representative $g$ and consider the sub-bundle
$\mathcal{P}_{1}=\\{p\in\mathcal{P}\ \vline\
\mathbf{a}^{2}\mathbf{b}^{2}=1\\},$ where $\mathcal{P}$ denotes the structure
bundle associated to the derived $\\{e\\}$-structure (see Section A.1.4). It
follows that over $\mathcal{P}_{1}$ one has

(2.48)
$0=\mathrm{d}\left(\mathbf{a}^{2}\mathbf{b}^{2}\right)=2(\phi_{0}+\phi_{n})+s_{i}\omega^{i}+s_{\underline{a}}\theta^{a},$

for some functions $s_{i}$ and $s_{\underline{a}}.$ Differentiating this
equation again, it follows that

(2.49)
$\mathrm{d}s_{i}\equiv\pi_{i}+\delta^{0}_{i}\left(s_{a}\gamma^{a}+s_{a}\pi^{a}\right)+\delta^{a}_{i}\left(s_{b}\phi^{b}{}_{a}+s_{n}\gamma_{a}+s_{\underline{a}}\pi_{n}\right)\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Consequently, $s_{i}$ can be translated to zero by the action of the structure
group. It follows that

(2.50) $\pi_{i}=p_{ij}\omega^{j}+p_{i{\underline{a}}}\theta^{a}.$

Substituting (2.50) and $\phi_{n}=-\phi_{0}+s_{\underline{a}}\theta^{a}$ into
the structure equations, equation (2.37d) becomes

(2.51)
$\mathrm{d}\theta^{a}=\phi^{a}{}_{b}{\,{\wedge}\;}\theta^{b}-2\phi_{0}{\,{\wedge}\;}\theta^{a}+\textstyle{\frac{1}{2}}R^{a}{}_{nij}\omega^{i}{\,{\wedge}\;}\omega^{j}+E^{a}{}_{nbi}\theta^{b}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C^{a}{}_{nbc}\theta^{b}{\,{\wedge}\;}\theta^{c}.$

where

(2.52)
$\begin{gathered}R^{a}{}_{nbn}=W^{a}{}_{nbn}+p_{nn}\delta^{a}{}_{b},\quad
R^{a}{}_{nbc}=W^{a}{}_{nbc}+\delta^{a}{}_{c}p_{nb}+\delta^{a}{}_{b}p_{nc},\\\
R^{a}{}_{n0b}=-p^{a}{}_{b}+\delta^{a}{}_{b}p_{n0},\quad
R^{a}{}_{n0n}=p^{a}{}_{n}\end{gathered}$

In conformal geometry, the symmetric part of $p_{ij}$ is referred to as the
_Schouten tensor_. The quantities $R^{a}{}_{nbn}$ and $p_{nn}$ are called the
_shadow flag curvature_ and the _scalar shadow flag curvature_ (or the ssf
curvature) of the causal structure respectively.

####  2.5.3. Null Jacobi fields and the tidal force

As was mentioned in section 2.3.5, the characteristic curves of causal
structures is intrinsically defined by the quasi-contact structure of
$\mathcal{C}.$ Nevertheless, one can associate a _variational problem_ to
causal geometries via the Griffiths formalism [Gri83] (see also [Hsu92]). As a
result, the notion of null Jacobi fields naturally arises. The corresponding
Jacobi equations result in another interpretation of the Wsf curvature. The
fact that Jacobi equations of null Jacobi fields involve the Wsf curvature has
to do with the second Lie derivative of $g$ in (2.35), which is referred to as
the _tidal force_ [O’N83].

Recall that for the Pfaffian system
$\mathcal{I}_{\mathsf{char}}=\\{\omega^{0},\dots,\omega^{n-1},\theta^{1},\dots,\theta^{n-1}\\},$
after fixing an interval $I=(a,b),$ one can define the space of smooth
immersions of its integral curves as

$\widehat{\mathscr{U}}(I)=\\{\gamma:I\rightarrow\mathcal{C}\ \vline\
\gamma^{*}(\mathcal{I}_{\mathsf{tot}})=0\\}.$

The space $\widehat{\mathscr{U}}(I)$ can be given a Whitney
$C^{\infty}$-topology. After making a choice of orientation for the
characteristic curves, the space of the characteristic curves of
$\mathcal{C}$, $\mathscr{U}(I),$ is defined as
$\widehat{\mathscr{U}}(I)/\mathrm{Diff}(I)$, i.e., the space obtained from
$\widehat{\mathscr{U}}(I)$ by identifying the characteristic curves that are
reparametrizations of each other.

Consider the functional $\Phi:\mathscr{U}\rightarrow\mathbb{R},$ which at a
point $[\gamma]\in\mathscr{U}(I)$ is defined by

$\Phi([\gamma])=\int_{I}\gamma^{*}\omega^{0}.$

The triple $(\mathcal{C},I,\omega^{0})$ is referred to as a _variational
problem_ on $\mathcal{C},$ and is intrinsic to any causal structure. Since
$\omega^{0}(\dot{\gamma})=0$ everywhere, the above definition of $\Phi$ is
independent of the choice of representative $\gamma\in[\gamma].$ This is in
contrast with Finsler geometry where the geodesics are defined as the extremal
curves of the arc-length functional which is always positive and are equipped
with a specific parametrization.

Intuitively, the infinitesimal variations of a characteristic curve $\gamma$
can be identified with the tangent space $T_{\gamma}\mathscr{U}(I)$ which is