the space of smooth variational vector fields along $\gamma.$ To give a more
precise definition consider a parametrization for $I=(a,b)$ and a map
$\Gamma:(-\epsilon,\epsilon)\rightarrow\mathscr{U}(I)$. As a result of such
parametrization, the curves given by $\Gamma(s)$ can be parametrized. Using
this parametrization, one can define
$\widehat{\Gamma}:(-\epsilon,\epsilon)\times I\rightarrow\mathcal{C},$ as
$\widehat{\Gamma}(s,t)=\left(\Gamma(s)\right)(t).$ The map $\Gamma$ is called
a compact variation of the curve $\gamma$ if the corresponding map
$\widehat{\Gamma}$ is smooth, $\widehat{\Gamma}(0,t)=\gamma(t),$
$\widehat{\Gamma}^{*}(s_{0},t)(\mathcal{I}_{\mathsf{char}})=0,$ for any fixed
$s_{0},$ and $\widehat{\Gamma}$ is compactly supported in $I$, i.e., for all
values of $s,$ the curves $\widehat{\Gamma}(s,t)$ and $\widehat{\Gamma}(0,t)$
coincide outside of a compact subset of $I.$ Then, $T_{\gamma}\mathscr{U}(I)$
is expressed as

$T_{\gamma}\mathscr{U}(I):=\left\\{\frac{\partial\widehat{\Gamma}}{\partial
s}(0,t)\ \vline\ \Gamma:(-\epsilon,\epsilon)\rightarrow\mathscr{U}(I)\mathrm{\
is\ a\ compact\ variation\ of\ }\gamma\right\\}.$

The _variational equations_ of a characteristic curve can be thought of as the
first order approximation of $T_{\gamma}\mathscr{U}(I).$ A practical
derivation of the variational equations can be given in terms of the Pfaffian
system $\mathcal{I}_{\mathsf{char}}.$

Let $\eta\in\mathcal{I}_{\mathsf{char}},$ and $\Gamma$ a variation of the
characteristic curve $\gamma.$ By the definition of $\widehat{\Gamma}(s,t),$
one has $\widehat{\Gamma}^{*}(s_{0},t)\eta=0,$ and therefore
$\mathcal{L}_{\frac{\partial}{\partial s}}\widehat{\Gamma}^{*}(s,t)\eta\equiv
0$ modulo $I\\{\mathrm{d}s\\}.$ Setting $s=0,$ the latter expression implies
that

$\gamma^{*}(\hat{J}\im\mathrm{d}\eta+\mathrm{d}(\hat{J}\im\eta))=0,$

where $\hat{J}(t)=\frac{\partial}{\partial s}\widehat{\Gamma}(0,t)$ is the
infinitesimal variations associated to $\Gamma(s).$

In the expression of the variational equations one can drop the pull-back
$\gamma^{*}$ and write them as

$\hat{J}\im\mathrm{d}\eta+\mathrm{d}(\hat{J}\im\eta)\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{char}}.$

The tangential component of the vector field $\hat{J}(t)$ along $\gamma(t)$
has no effect on the variational equations. Therefore, $\hat{J}$ along the
curve $\gamma(t)$ can be expressed as

$\hat{J}(t)=V^{a}(t)\frac{\partial}{\partial\omega^{a}}+V^{{\underline{a}}}(t)\frac{\partial}{\partial\theta^{a}}.$

The vector field $J=\pi_{*}\hat{J}$ is called a _null Jacobi field_ along the
null geodesics $\pi(\gamma(t))$.

The variational equations for $\hat{J}(t)$ can be written as

(2.53a)
$\displaystyle\hat{J}\im\mathrm{d}\omega^{0}+\mathrm{d}(\hat{J}\im\omega^{0})=0,$
(2.53b)
$\displaystyle\hat{J}\im\mathrm{d}\omega^{a}+\mathrm{d}(\hat{J}\im\omega^{a})=0,\quad\quad\textrm{mod}\quad
I\\{\omega^{0},\cdots,\omega^{n},\theta_{1}\cdots,\theta_{n-1}\\}.$ (2.53c)
$\displaystyle\hat{J}\im\mathrm{d}\theta_{a}+\mathrm{d}(\hat{J}\im\theta_{a})=0,$

Equation (2.53a) gives no new information. The last two equations give

(2.54)
$\displaystyle\mathrm{d}V^{a}+V^{b}\phi^{a}{}_{b}-V^{\underline{a}}\omega^{n}\equiv
0,$
$\displaystyle\mathrm{d}V^{\underline{a}}+V^{\underline{b}}\phi^{a}{}_{b}+W^{a}{}_{nbn}V^{b}\omega^{n}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{char}}.$

Using the pseudo-connection from Section 2.3.12, the covariant derivative
along characteristic curves of vector fields lying within the shadow space
(2.47) is given by

$D_{\mathbf{v}}\hat{J}\equiv\mathbf{v}\im\left(\mathrm{d}V^{a}\frac{\partial}{\partial\omega^{a}}+V^{b}\phi^{a}{}_{b}\frac{\partial}{\partial\omega^{a}}\right),\quad\textrm{mod}\quad\frac{\partial}{\partial\theta^{1}},\cdots,\frac{\partial}{\partial\theta^{n-1}},$

where $\mathbf{v}$ is the characteristic vector field satisfying
$\omega^{n}(\mathbf{v})=1.$

Using equations (2.54), it follows that

(2.55) $\displaystyle D_{\mathbf{v}}D_{\mathbf{v}}\hat{J}$
$\displaystyle\equiv-W^{a}{}_{nbn}V^{b}\frac{\partial}{\partial\omega^{a}},\quad\textrm{mod}\quad\frac{\partial}{\partial\theta^{1}},\cdots,\frac{\partial}{\partial\theta^{n-1}}$
$\displaystyle\equiv-\overset{\mathrm{sf}}{W}\hat{J},\quad\textrm{mod}\quad\frac{\partial}{\partial\theta^{1}},\cdots,\frac{\partial}{\partial\theta^{n-1}}$

where $\overset{\mathrm{sf}}{W}$ is viewed as an automorphism of the shadow
space. Recall from Section 2.5.1 that $W^{a}{}_{nbn}$ is given by
$\omega^{a}(W(\frac{\partial}{\partial\omega^{b}},\mathbf{v})\mathbf{v}.$
Hence, defining

$J^{\prime}=\pi_{*}(\mathrm{D}_{\mathbf{v}}\hat{J}),$

the null Jacobi equation can be expressed as

(2.56) $J^{\prime\prime}+W(J,\mathbf{v})\mathbf{v}=0,$

where $J=\pi_{*}\hat{J}$ and $W(J,\mathbf{v})\mathbf{v}$ denotes
$\pi_{*}\left(W(\hat{J},\mathbf{v})\mathbf{v}\right).$ For convenience, in
Section 2.5.4 Equation (2.56) is expressed as

(2.57) $J^{\prime\prime}+\overset{\mathrm{sf}}{W}J=0.$

In some references [O’N83] (cf. [HS11]) the automorphism of the shadow space
$W^{a}{}_{nba}:S_{(x,[y])}\rightarrow S_{(x,[y])}$ is called the trace-free
part of the _tidal force_ in analogy with the terminology from General
Relativity. Equation (2.55) shows that the Wsf curvature plays a rule
analogous to that of the conformal sectional curvature in the case of
variational theory of null geodesics.

The relation between the tidal force and the trace-free part of the Wsf
curvature can be realized differently via the second Lie derivative the
bilinear form $g$ in (2.35) along the characteristic curves (cf. [HS11,
Bry02]). The structure equations (2.37) imply

$\mathcal{L}^{2}_{\mathbf{v}}g=4\omega^{0}\circ\pi_{n}-2\varepsilon_{ab}\theta^{a}\circ\theta^{b}-2W_{anbn}\omega^{a}\circ\omega^{b}.$

Therefore, for any two vector fields $X_{1},X_{2}\in\textrm{Ker}\,\omega^{0},$
$X_{\epsilon}=X_{\epsilon}^{a}\frac{\partial}{\partial\omega^{a}}+X_{\epsilon}^{\underline{a}}\frac{\partial}{\partial\theta^{a}}$
where $\epsilon=1,2,$ it follows that

$W_{anbn}X_{1}^{a}X_{2}^{b}=-\textstyle{\frac{1}{2}}\mathcal{L}^{2}_{\mathbf{v}}g(X_{1},X_{2})-\varepsilon_{ab}X_{1}^{\underline{a}}X_{2}^{\underline{b}}.$

####  2.5.4. The Raychaudhuri equation

In this section, a causal analogue of the Raychaudhuri equation along null
geodesics is derived. The approach that is taken here is presented in [BEE96,
Chapter 12].

Taking a section of the structure bundle,
$s:\mathcal{C}\rightarrow\mathcal{P},$ define the null Jacobi fields
$\hat{J}_{1}(t),\cdots,\hat{J}_{n-1}(t)$ along the characteristic curve
$\gamma(t)$, $t\in[a,b]$ by the initial conditions

$\hat{J}_{a}(t_{0})=0,\quad\mathrm{D}_{\mathbf{v}(t_{0})}\hat{J}_{a}(t_{0})=\frac{\partial}{\partial(s^{*}\omega^{a})}$

where $t_{0}\in[a,b]$ and $\mathbf{v}(t)$ is tangent vector to $\gamma(t).$
Define the $(1,1)$ tensor field called the _null Jacobi tensor_ as

$A(t)=[J_{1}(t),\cdots,J_{n-1}(t)],$

which is a $(n-1)\times(n-1)$ matrix whose $b$th column is the null Jacobi
field $J_{b}(t)=\pi_{*}(\hat{J}_{b}(t)).$

The optical invariants along null geodesics are defined as follows.

######  Definition 2.17.

Let $A$ be a null Jacobi tensor along a null geodesic and define
$B=A^{\prime}A^{-1},$ where $A$ is invertible. The _expansion_ , $\theta$,
_vorticity tensor,_ $\omega$ and _shear tensor_ , $\sigma,$ are defined as

$\theta=B^{a}{}_{a},\quad\omega^{a}{}_{b}=\textstyle{\frac{1}{2}}\left(B^{a}{}_{b}-\left(B^{*}\right)^{a}{}_{b}\right),\quad\sigma=\textstyle{\frac{1}{2}}\left(B^{a}{}_{b}+\left(B^{*}\right)^{a}{}_{b}\right)-\frac{\theta}{n-1}\delta^{a}{}_{b},$

where $B^{*}$ denotes the adjoint matrix with respect to the bilinear form
$g.$

Following the discussion in Section 2.5.3, after taking a section of the
structure bundle one obtains that a null Jacobi field satisfies the equation

$J^{\prime\prime}+\overset{\mathrm{sf}}{R}J=0,$

rather than Equation (2.57), where $\overset{\mathrm{sf}}{R}$ represents the
matrix form of the shadow flag curvature
$R^{a}{}_{nbn}=W^{a}{}_{nbn}+p_{nn}\delta^{a}{}_{b}$ defined in (2.52).

It follows that

$A^{\prime\prime}+\overset{\mathrm{sf}}{R}A=0.$

Moreover, one has

$B^{\prime}=(A^{\prime}A^{-1})^{\prime}=A^{\prime\prime}A^{-1}-A^{\prime}A^{-1}A^{\prime}A^{-1}=-\overset{\mathrm{sf}}{R}-B^{2}.$

Since $B=\omega+\sigma+\frac{\theta}{n-1}\mathrm{Id},$ one obtains

$\theta^{\prime}=\mathrm{tr}(B^{\prime})=-\mathrm{tr}(\overset{\mathrm{sf}}{R})-\mathrm{tr}(B^{2})=-\mathrm{tr}(\overset{\mathrm{sf}}{R})-\mathrm{tr}\left[\left(\omega+\sigma+\frac{\theta}{n-1}\mathrm{Id}\right)^{2}\right]$

Using the fact that
$\mathrm{tr}(\omega)=\mathrm{tr}(\sigma)=\mathrm{tr}(\omega\sigma)=0$ and
$\mathrm{tr}(\overset{\mathrm{sf}}{R})=(n-1)p_{nn}$ one arrives at

(2.58)
$\theta^{\prime}=-(n-1)p_{nn}-\mathrm{tr}(\omega^{2})-\mathrm{tr}(\sigma^{2})-\frac{\theta^{2}}{n-1},$

which is the causal analogue of the Raychaudhuri equation along null
geodesics.

###  2.6. Examples

In this section three examples of causal structures are given. The first one
is obtained from the product of two Finsler structures and the second and
third ones are given in terms of a defining function. These examples are used
in Section 4.

####  2.6.1. Product of Finsler structures

A Finsler manifold is denoted by $(N,\Sigma),$ where $N$ is $n$-dimensional
and $\Sigma\subset TN$ is of codimension one whose fibers $\Sigma_{p}\subset
T_{p}N$ are strictly convex hypersurfaces. The fiber bundle $\Sigma$ is often
referred to as the indicatrix bundle over which the Finsler metric has unit
value. It is known [Car34, Bry02] that $\Sigma$ is equipped with unique
coframing
$\eta=(\eta^{0},\eta^{1},\cdots,\eta^{n-1},\zeta^{1},\cdots,\zeta^{n-1})$
satisfying

(2.59) $\displaystyle\mathrm{d}\eta^{0}$
$\displaystyle=-\zeta_{a}{\,{\wedge}\;}\eta^{a},$
$\displaystyle\mathrm{d}\eta^{a}$
$\displaystyle=\phantom{-}\zeta^{a}{\,{\wedge}\;}\eta^{0}-\psi^{a}{}_{b}{\,{\wedge}\;}\eta^{b}-I^{ab}{}_{c}\zeta_{b}{\,{\wedge}\;}\eta^{c},$
$\displaystyle\mathrm{d}\zeta_{a}$
$\displaystyle=\phantom{-}\psi^{b}{}_{a}{\,{\wedge}\;}\zeta_{b}+R_{0a0b}\eta^{0}{\,{\wedge}\;}\eta^{b}+\textstyle{\frac{1}{2}}R_{0abc}\eta^{b}{\,{\wedge}\;}\eta^{c}-J_{a}{}^{b}{}_{c}\zeta_{b}{\,{\wedge}\;}\eta^{c}$

for some invariants $I_{abc},J_{abc},$ $R_{0a0b}$ and $R_{0abc}.$ The 1-form
$\eta^{0}$ is called the Hilbert form of the Finsler structure which, at the
point $v\in\Sigma_{p},$ is uniquely characterized by the conditions

$\eta^{0}(H_{v}\Sigma_{p})=0,\quad\eta^{0}(v)=1,$

where $H_{v}\Sigma_{p}\subset T_{p}N$ is the tangent hyperplane to
$\Sigma_{p}$ at the point $v$. As a result of the structure equation,
$\eta^{0}$ induces a contact structure on $\Sigma$ whose associated Reeb
vector field defines the geodesic flow of the Finsler structure on $\Sigma$.
Moreover, the quadratic form
$\omega^{0}\circ\omega^{0}+\delta_{ab}\omega^{b}\circ\omega^{c}$ is well-
defined for the Finsler structure. It turns out that the quantity
$I_{abc}=\delta_{ad}\delta_{be}I^{de}{}_{c}$ can be used to define the cubic
form $I_{abc}\zeta^{a}\circ\zeta^{b}\circ\zeta^{c}$, which, when restricted to
a fiber $\Sigma_{p},$ coincides with the centro-affine cubic form of the
hypersurface $\Sigma_{p}\subset T_{p}N.$ The quantities $J_{abc}$ measure the
rate of change of $I_{abc}$ along the geodesics, i.e.,
$J_{abc}=\frac{\partial}{\partial\eta^{0}}I_{abc}$. The quantity $R_{0a0b}$ is
symmetric using which the flag curvature of the Finsler structure can be
represented by

$R_{0a0b}\eta^{a}\circ\eta^{b}.$

Let $(N,\Sigma)$ and $(\tilde{N},\tilde{\Sigma})$ be two Finsler manifolds.
Given a strictly positive function $f$ on their product space, at a point
$p\in M=N\times{}_{f}\tilde{N},$ one can define a projective hypersurface
whose associated cone is given by

(2.60) $\widehat{\mathcal{C}}_{p}=\left\\{(v,\tilde{v})\in
T_{p}(N\times\tilde{N})\ \ \vline\ \ \lambda
fv\in\Sigma_{\pi(p)},\lambda\tilde{v}\in\tilde{\Sigma}_{\tilde{\pi}(p)}\mathrm{\
for\ some\ }\lambda\in\mathbb{R}\right\\}.$

As a result, $M$ is equipped with a causal structure of signature
$(n,\tilde{n})$ where $\tilde{n}=\dim\tilde{N}.$ Note that it is assumed that
the Finsler structures are reversible.

If $G$ and $\tilde{G}$ represent the Finsler metrics on $N$ and $\tilde{N},$
respectively, then the causal structure on $M$ is locally given as the zero
locus of $G-f\tilde{G}.$ If $f$ only depends on $N,$ then the above causal
structure corresponds to the warped product of the Finsler structures on $N$
and $\tilde{N}.$

More specifically, let $M=N\times{}_{f}\mathbb{R}$, where $N$ is a
3-dimensional Finsler manifold. On $N$ there is the unique coframing
$(\eta^{0},\eta^{1},\eta^{2},\zeta^{1},\zeta^{2})$ satisfying (2.59). If
$\eta^{3}$ represents a 1-form on $\mathbb{R},$ one has

$\mathrm{d}f=f_{,0}\eta^{0}+f_{,1}\eta^{1}+f_{,2}\eta^{2}+f_{,3}\eta^{3},$

where $f_{,i}=\frac{\partial}{\partial\eta^{i}}f$ are called the coframe
derivatives of $f.$ The projective Hilbert form is a multiple of

$\omega^{0}={\textstyle{\frac{1}{\sqrt{2}}}}(\eta^{0}-f\eta^{3}),$

where, by abuse of notation, the pull-backs are dropped and $\eta^{0}$ and
$\eta^{3}$ represent $\pi^{*}(\eta^{0})$ and $\tilde{\pi}^{*}(\eta^{3}).$

A choice of 3-adapted coframing for the causal structure on $M$ is given by

(2.61)
$\begin{gathered}\omega^{1}=\eta^{1},\quad\omega^{2}=\eta^{2},\quad\omega^{3}={\textstyle{\frac{1}{\sqrt{2}}}}(-\eta^{0}-f\eta^{3}),\\\
\theta_{1}={\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{1}-{\textstyle{\frac{1}{\sqrt{2}}}}f_{,1}\eta^{3},\quad\theta_{2}={\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{2}-{\textstyle{\frac{1}{\sqrt{2}}}}f_{,2}\eta^{3},\\\
\gamma_{1}=-{\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{1}+{\textstyle{\frac{1}{\sqrt{2}}}}f_{,1}\eta^{3},\quad\gamma_{2}=-{\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{2}+{\textstyle{\frac{1}{\sqrt{2}}}}f_{,2}\eta^{3},\\\
\phi_{0}=-\phi_{3}=-\textstyle{\frac{1}{2}}f_{,0}\eta^{3},\quad\phi=\pi_{3}=0.\end{gathered}$

The quantities $F^{ab}{}_{c}$ in the structure equations (2.26) coincide with
$I^{ab}{}_{c}$ in (2.59). By absorbing the trace of $F_{abc}$ and $K_{ab}$, a
5-adapted coframing $(\tilde{\omega}^{i},\tilde{\theta}_{a})$ is given by

$\tilde{\omega}^{0}=\omega^{0},\quad\tilde{\omega}^{a}=\omega^{a}+A^{a}\omega^{0},\quad\tilde{\omega}^{3}=\omega^{3}+B\omega^{0},\quad\tilde{\theta}_{a}=\theta_{a},$

where

$\begin{gathered}A^{1}=-\textstyle{\frac{1}{2}}(I_{111}+I_{122}),\quad
A^{2}=\textstyle{\frac{1}{2}}(I_{112}+I_{222}),\\\
B=\textstyle{\frac{1}{4}}\left((A^{1})^{2}+(A^{2})^{2}\right)-\textstyle{\frac{1}{2\sqrt{2}}}\left(A^{1}{}_{,\underline{1}}+A^{2}{}_{,\underline{2}}\right)-A^{1}F_{111}-A^{2}F_{222},\end{gathered}$

where $F_{111}=\textstyle{\frac{1}{4}}\left(I_{111}-3I_{122}\right),$
$F_{222}=\textstyle{\frac{1}{4}}\left(I_{222}-3I_{211}\right)$ are the
components of the Fubini cubic form of the causal structure and
$A^{a}{}_{,\underline{b}}$ denotes the coframe derivative
$\frac{\partial}{\partial\zeta^{b}}A^{a}$. A choice of connection forms for
the first order structure equations is given by

$\displaystyle\tilde{\gamma}^{1}$
$\displaystyle=\gamma^{1}+B_{,1}\omega^{0}-\left({\textstyle{\frac{1}{\sqrt{2}}}}B_{,0}+A^{1}{}_{,1}\right)\tilde{\omega}^{1}-\textstyle{\frac{1}{2}}\left(A^{1}{}_{,2}+A^{2}{}_{;1}\right)\tilde{\omega}^{2}+{\textstyle{\frac{1}{\sqrt{2}}}}A^{1}{}_{,0}\tilde{\omega}^{3}+B\theta_{1},$
$\displaystyle\tilde{\gamma}^{2}$
$\displaystyle=\gamma^{2}+B_{,2}\omega^{0}-\textstyle{\frac{1}{2}}\left(A^{1}{}_{,2}+A^{2}{}_{,1}\right)\tilde{\omega}^{1}-\left({\textstyle{\frac{1}{\sqrt{2}}}}B_{,0}+A^{2}{}_{,2}\right)\tilde{\omega}^{2}+{\textstyle{\frac{1}{\sqrt{2}}}}A^{2}{}_{,0}\tilde{\omega}^{3}+B\theta_{2},$
$\displaystyle\tilde{\phi}_{0}$
$\displaystyle=-\textstyle{\frac{1}{2}}A^{1}\theta_{1}-\textstyle{\frac{1}{2}}A^{2}\theta_{2},$
$\displaystyle\tilde{\phi}_{3}$
$\displaystyle=\textstyle{\frac{1}{2\sqrt{2}}}(-B_{,0}\omega^{0}+A^{1}{}_{,0}\tilde{\omega}^{1}+A^{2}{}_{,0}\tilde{\omega}^{2}),$
$\displaystyle\tilde{\phi}$
$\displaystyle=\textstyle{\frac{1}{2}}(A^{1}{}_{,2}-A^{2}{}_{,1})\omega^{0}+\textstyle{\frac{1}{2\sqrt{2}}}A^{2}{}_{,0}\tilde{\omega}^{1}-\textstyle{\frac{1}{2\sqrt{2}}}A^{1}{}_{,0}\tilde{\omega}^{2}-\textstyle{\frac{1}{2}}A^{2}\theta_{1}+\textstyle{\frac{1}{2}}A^{1}\theta_{2}$
$\displaystyle\tilde{\pi}_{3}$
$\displaystyle=-{\textstyle{\frac{1}{\sqrt{2}}}}A^{1}{}_{,0}\theta_{1}-{\textstyle{\frac{1}{\sqrt{2}}}}A^{2}{}_{,0}\theta_{2}-\textstyle{\frac{1}{2}}\left(R_{0101}+R_{0202}+(\ln
f)_{,11}+(\ln f)_{,22}\right)\tilde{\omega}^{3}.$

Consequently, the Wsf curvature is given by the trace-free part of

(2.62) $\textstyle{\frac{1}{2}}R_{0a0b}+\textstyle{\frac{1}{2}}(\ln f)_{,ab},$

where $(\ln
f)_{;ab}:=\frac{\partial}{\partial\eta^{b}}\frac{\partial}{\partial\eta^{a}}\ln
f.$ Moreover, the quantities $K_{ab}$ and $L_{a}$ are given by

$K_{12}=K_{21}=-\sqrt{2}A^{1}{}_{,\underline{2}}-F_{112}A^{1}-F_{122}A^{2},\quad
K_{11}=-K_{22}=-{\textstyle{\frac{1}{\sqrt{2}}}}(A^{1}{}_{,\underline{1}}-A^{2}{}_{,\underline{2}}),\quad
L_{a}=-\sqrt{2}B_{,\underline{a}}.$

To see the symmetric relation $K_{12}=K_{21}$ one needs to realize
$A^{1}{}_{,\underline{2}}=A^{2}{}_{,\underline{1}}.$ It is rather
straightforward to generalize this example to obtain a causal structure on
$M=N\times{}_{f}\mathbb{R}$ where $N$ is an $n$-dimensional Finsler manifold.
The corresponding Wsf is given by expression (2.62) and the Fubini cubic form
is obtained from the centro-affine cubic form of indicatrices of the Finsler
structure.

####  2.6.2. Cayley’s cubic scroll

Using the description (2.2), consider the causal structure on a 4-dimensional
manifold given locally as the graph of the function

$F(x^{i},y^{a})=\textstyle\frac{1}{3}c_{111}(y^{1})^{3}+c_{12}y^{1}y^{2}+c_{1}y^{1}+c_{2}y^{2}+c_{0},$

where the coefficients $c_{i\cdots k}$ are functions of $x^{i}$’s.

A choice of 1-adapted coframing is given by

(2.63) $\displaystyle\omega^{0}$
$\displaystyle=\mathrm{d}x^{0}-\textstyle\frac{\partial}{\partial
y^{a}}F\mathrm{d}x^{a}+(y^{a}\textstyle\frac{\partial}{\partial
y^{a}}F-F)\mathrm{d}x^{3},$ $\displaystyle\omega^{a}$
$\displaystyle=\mathrm{d}x^{a}-y^{a}\mathrm{d}x^{3},$
$\displaystyle\omega^{3}$ $\displaystyle=\mathrm{d}x^{3},$
$\displaystyle\theta_{a}$
$\displaystyle=\textstyle\frac{\partial^{2}}{\partial x^{i}\partial
y^{a}}F\mathrm{d}x^{i}-(\textstyle\frac{\partial}{\partial
x^{a}}F+\textstyle\frac{\partial}{\partial
x^{0}}F\textstyle\frac{\partial}{\partial
y^{a}}F)+\textstyle\frac{\partial^{2}}{\partial y^{a}\partial
y^{b}}F\mathrm{d}y^{b}.$

A 2-adapted coframing is given by

$\omega^{1}\longrightarrow\omega^{1}-\textstyle\frac{c_{111}}{c_{12}}y^{1}\omega^{2},\quad\omega^{3}\longrightarrow\textstyle\frac{1}{c_{12}}\omega^{3},\quad\theta_{1}\longrightarrow\theta_{1}-\textstyle\frac{c_{111}}{c_{12}}y^{1}\theta_{2},$

with respect to which the second fundamental form is normalized to
$h^{ab}=\left[\begin{smallmatrix}0&1\\\ 1&0\end{smallmatrix}\right].$ The
details of higher coframe adaptations won’t be discussed. This example is
discussed further in Section 4.1.

It should be noted that when the only non-zero coefficients are $c_{12}=1$ and
$c_{00}=(x^{1})^{2},$ then one obtains a local description of the split
signature pp-wave metric. Moreover, when $c_{0}=c_{1}=c_{2}=0$ one obtains a
local description of causal structures whose null cones are at each point
projectively equivalent to Cayley’s cubic scroll whose graph can be expressed
as $y^{0}=\textstyle{\frac{1}{3}}(y^{1})^{3}+y^{1}y^{2}$ in $\mathbb{P}^{3}.$
In Section 4.2, this example is used to characterize the submaximal causal
structure of split signature.

####  2.6.3. Locally isotrivially flat causal structures

This class of causal structures are the analogue of locally Minkowskian
Finsler structures.

######  Definition 2.18.

A causal structure $(\mathcal{C},M^{n+1})$ is called locally isotrivially flat
if it is locally equivalent to $(M\times\mathcal{V},M)$ where
$\mathcal{V}\subset\mathbb{P}^{n}$ is a projective hypersurface.

Alternatively, a locally isotrivially flat causal structure has the property
that locally it can be expressed as the graph of a function $F(x;y),$ as in
(2.2), which has no dependency on $x^{i}$’s.

The local expressions (2.10) and (2.16) of the 1-forms $\omega^{i}$ and
$\theta_{a}$ can be used to express the Fubini cubic form and Wsf curvature.
Firstly, note that
$\theta_{a}=\partial_{\underline{a}}\partial_{\underline{b}}F\mathrm{d}y^{b}.$
To express $F_{abc},$ consider the following transformation

(2.64)
$\displaystyle\tilde{\omega}^{0}=\mathbf{a}^{2}\omega^{0},\quad\tilde{\omega}^{a}=\mathbf{E}^{a}\omega^{0}+\omega^{a},\quad\tilde{\omega}^{n}=\mathbf{e}\omega^{0}+\omega^{n},\quad\tilde{\theta}_{a}=\mathbf{a}^{2}\theta_{a}.$

Differentiating $\tilde{\omega}^{0},$ it follows that

$\mathrm{d}\tilde{\omega}^{0}=-2\phi_{0}{\,{\wedge}\;}\tilde{\omega}^{0}-\tilde{\theta}_{a}{\,{\wedge}\;}\tilde{\omega}^{a}.$

where
$\phi_{0}=-\frac{\mathrm{d}\mathbf{a}}{\mathbf{a}}-\frac{\mathbf{E}^{a}}{2\mathbf{a}^{2}}\tilde{\theta}_{a}.$
Consequently, in Equation (2.20b) for 1-adapted coframes one obtains

$\gamma^{a}{}_{0}=\textstyle\frac{1}{\mathbf{a}^{2}}\mathrm{d}\mathbf{E}^{a}+\frac{1}{\mathbf{a}^{4}}\left(\mathbf{E}^{a}\mathbf{E}^{b}+H^{ab}\right)\theta_{b},\quad\psi^{a}{}_{b}=\frac{1}{\mathbf{a}^{2}}\mathbf{E}^{a}\theta_{b},\quad
h^{ab}=\frac{1}{\mathbf{a}^{2}}H^{ab},$

where $[H^{ab}]$ denotes the inverse matrix of the vertical Hessian
$[\partial_{\underline{a}}\partial_{\underline{b}}F].$

By composing Equations (2.21) and (2.25), it follows that the Fubini cubic
form $F^{abc}$ is defined via the equation

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

Transformations (2.64) imply that $\phi_{n}=0$. Using the above expressions
for $\phi_{0}$ and $\psi^{a}{}_{b}$, without normalizing $h^{ab}$ to
$\varepsilon^{ab},$ one can impose the apolarity condition $h^{ab}F_{abc}=0,$
as in (2.30), by setting

$\mathbf{a}=\left(\det(\partial_{\underline{a}}\partial_{\underline{b}}F)\right)^{\frac{1}{2(n+1)}},\quad\mathbf{E}^{d}=-\frac{\mathbf{a}^{2}}{n+1}H^{ab}H^{cd}\partial_{\underline{a}}\partial_{\underline{b}}\partial_{\underline{c}}F.$

Finally, using $\partial_{{\underline{a}}}\partial_{\underline{b}}F$ to lower
indices, it follows that

(2.65)
$F_{abc}=\mathbf{a}^{-2}\left(\partial_{\underline{a}}\partial_{\underline{b}}\partial_{{\underline{c}}}F-\frac{1}{n+1}\left(\partial_{{\underline{a}}}\partial_{\underline{b}}FF_{c}+\partial_{{\underline{b}}}\partial_{\underline{c}}FF_{a}+\partial_{{\underline{c}}}\partial_{\underline{a}}FF_{b}\right)\right),$

where
$F_{a}=H^{bc}\partial_{\underline{a}}\partial_{\underline{b}}\partial_{{\underline{c}}}F.$

The above computations go through for any causal structure. The expression
(2.65) (see also [Sas99, AG93]) shows that the Fubini cubic form is a third
order invariant. Using the Bianchi identities (A.19), it follows that the jet
order of the invariants $K^{ab}$ and $L^{a}$ over $\mathcal{C}$ are four and
five, respectively.

To find the Wsf curvature, note that without normalizing $h^{ab},$ the
quantities $E_{anb}$ in Equation (2.25) are defined via

$\mathrm{d}h^{ab}+h^{ac}\psi^{b}{}_{c}+h^{bc}\psi^{a}{}_{c}-2h^{ab}\left(\phi_{0}+\phi_{n}\right)=E^{a}{}_{n}{}^{b}\omega^{n},\quad\textrm{mod}\quad
I\\{\omega^{0},\cdots,\omega^{n-1},\theta_{1},\cdots,\theta_{n-1}\\},$

which by isotrivially flatness is zero. Hence, the reduction of $\pi_{ab}$ via
(2.27), gives
$\pi_{ab}-\frac{1}{n-1}\varepsilon^{cd}\pi_{cd}\varepsilon_{ab}=0.$ Combining
this with the fact that $\theta_{a}=0$ modulo
$I\\{\theta_{1},\cdots,\theta_{n-1}\\}$, it follows that $W^{a}{}_{nbn}=0,$
i.e., locally isotrivially flat causal structures have vanishing Wsf
curvature.

##  3\. Causal structures with vanishing Wsf curvature

In this section causal structures with vanishing Wsf curvature are studied.
For such classes, a compatibility condition is derived which can be used to
give a causal analogue of Landsberg spaces in Finsler geometry (see [BCS00]
for an account of various types of Finsler structures). Moreover, under
certain conditions, a rigidity theorem similar to that of Akbar-Zadeh’s [AZ88]
is obtained. Finally, the section ends with a discussion on Lie contact
structures on the locally defined space of characteristic curves of a causal
structure with vanishing Wsf curvature.

###  3.1. A compatibility condition

In this Section a class of causal structures which can be regarded as
analogues of Landsberg spaces in Finsler geometry is introduced. Their
similarity is the content of Proposition 3.1.

Since $\mathrm{D}F_{abc}$ in (2.39a) (2.39a) is semi-basic,

(3.1)
$\mathrm{d}F_{abc}-F_{abd}\,\phi^{d}{}_{c}-F_{adc}\,\phi^{d}{}_{b}-F_{dbc}\,\phi^{d}{}_{a}-F_{abc}\,(\phi_{0}-\phi_{n})=F_{abc;i}\omega^{i}+F_{abc;{\underline{d}}}\theta^{d}.$

After taking the Lie derivative of (3.1) along characteristic curves and using
the structure equations, one arrives at

$\mathrm{d}f_{abc}-f_{abd}\,\phi^{d}{}_{c}-f_{adc}\,\phi^{d}{}_{b}-f_{dbc}\,\phi^{d}{}_{a}-f_{abc}\,(\phi_{0}+\phi_{n})+F_{abc}\pi_{n}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Assume $f_{abc}=\lambda F_{abc},$ for some function $\lambda.$ Then, if
$F_{abc}\neq 0,$ the infinitesimal group action on $\lambda$ is given by

$\mathrm{d}\lambda-2\lambda\phi_{n}+\pi_{n}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

By translating $\lambda$ to zero, the 1-form $\pi_{n}$ reduces to

(3.2) $\pi_{n}=p_{ni}\omega^{i}+p_{n{\underline{a}}}\theta^{a}.$

for some coefficients $p_{ni}$ and $p_{n{\underline{a}}}.$ Replacing $\pi_{n}$
with the above expression in equation (2.37d), one finds the infinitesimal
action of the group $G_{5}$ on $p_{n{\underline{a}}}$ to be given by

$dp_{n{\underline{a}}}-p_{n{\underline{a}}}(\phi_{0}+\phi_{n})-p_{n{\underline{d}}}\phi^{d}{}_{a}+\pi_{a}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Translating $p_{n{\underline{a}}}$ to zero results in the reduction

(3.3) $\pi_{a}=p_{ai}\omega^{i}+p_{a{\underline{b}}}\theta^{b}.$

Consequently, equation (2.37d) changes to

(3.4)
$\mathrm{d}\theta_{a}=(\phi^{b}{}_{a}-(\phi_{0}-\phi_{n})\delta^{b}{}_{a}){\,{\wedge}\;}\theta_{b}+\textstyle{\frac{1}{2}}W_{anij}\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{anb0}\theta^{b}{\,{\wedge}\;}\omega^{0},$

where

$\displaystyle W_{anbn}=p_{nn},\quad
W_{anbc}=-p_{nb}\varepsilon_{ac}+p_{nc}\varepsilon_{ab},$ $\displaystyle
W_{an0n}=p_{a0},\quad W_{an0b}=p_{ab},\quad E_{anb0}=-p_{a{\underline{b}}}.$

Similarly, one finds that the normalization
$\varepsilon^{ab}p_{a{\underline{b}}}=0$ results in $\pi_{0}\equiv 0$ modulo
$\mathcal{I}_{\mathsf{tot}}.$ It follows that

(3.5) $\pi_{0}=p_{0i}\omega^{i}+p_{0{\underline{a}}}\theta^{a}.$

Assume furthermore that $p_{nn}\neq 0.$ Using the relation (2.39d), similar to
the derivation of (3.10), the infinitesimal group action on $p_{nn}$ is found
to be

(3.6) $\mathrm{d}p_{nn}-4p_{nn}\phi_{n}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Hence, the normalization $p_{nn}=\pm 1$ can be made, resulting in $\phi_{n}$
reduced to

(3.7) $\phi_{n}=r_{i}\omega^{i}+r_{\underline{a}}\theta^{a}.$

Furthermore, the infinitesimal group action of the reduced structure group on
$r_{\underline{a}}$ is given by

$\mathrm{d}r_{{\underline{a}}}+r_{{\underline{b}}}\phi^{b}{}_{a}-\gamma_{a}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

As a result, by translating $r_{{\underline{a}}}$ to zero, the 1-forms
$\gamma_{a}$ are reduced, i.e.,

(3.8) $\gamma_{a}=q_{ai}\omega^{i}+q_{a{\underline{b}}}\theta^{b}.$

After these reductions, the proposition below can be proved.

######  Proposition 3.1.

A causal structure with vanishing Wsf curvature satisfying $f_{abc}=\lambda
F_{abc}$ has vanishing ssf curvature.

###### Proof.

As was discussed, the condition $f_{abc}=\lambda F_{abc}$ results in the
reduction of the 1-forms $\pi_{i}$ according to equations (3.2), (3.3) and
(3.5). In the case $p_{nn}\neq 0,$ equation (3.6) it can be shown that
$\phi_{n}$ can be reduced to (3.7) by normalizing $p_{nn}$ to a non-zero value
$c$. Consequently, the 1-forms $\gamma_{a}$ can be reduced to (3.8).

Now that all the 1-forms $\pi_{i},\phi_{n}$ and $\gamma^{a}$ are reduced
equation (3.4) can be written as

$\mathrm{d}\theta_{a}=(\phi^{b}{}_{a}-\phi_{0}\delta^{b}{}_{a}){\,{\wedge}\;}\theta_{b}+c\,\omega^{a}{\,{\wedge}\;}\omega^{n}+(p_{ab}-p_{n0}\varepsilon_{ab})\omega^{0}{\,{\wedge}\;}\omega^{b}+p_{an}\omega^{0}{\,{\wedge}\;}\omega^{n}\\\
-(p_{a{\underline{b}}}+r_{0}\varepsilon_{ab})\theta^{b}{\,{\wedge}\;}\omega^{0}-p_{nb}\varepsilon_{ac}\omega^{b}{\,{\wedge}\;}\omega^{c}-r_{c}\varepsilon_{ab}\theta^{b}{\,{\wedge}\;}\omega^{c}-r_{n}\theta^{a}{\,{\wedge}\;}\omega^{n}$

where $c$ denotes the normalized value of $p_{nn}.$ It is noted that the
translations
$r_{\underline{a}},p_{n{\underline{a}}},\varepsilon^{ab}p_{a{\underline{b}}}$
to zero are used in the above equation. Similarly, express
$\mathrm{d}\omega^{a}$ and $\mathrm{d}\omega^{n}$ by replacing $\gamma^{a}$
and $\phi_{n}$ with their reduced forms in (2.37).

The reduced structure equations can be used to express the identity
$\mathrm{d}^{2}\theta_{a}=0.$ Collecting the coefficients of the 3-form
$\theta^{a}{\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\omega^{n},$ it follows that

(3.9)
$cF_{abc}+(r_{c;n}-r_{n;c}+\textstyle{\frac{1}{2}}p_{cn}-p_{nc}-r_{n}q_{cn}-r_{c}r_{n})\varepsilon_{ab}+p_{nb}\varepsilon_{ac}+p_{an}\varepsilon_{bc}+W_{abcn}=0,$

where $W_{abcn}=-W_{bacn}$ is a torsion term for $\Phi_{ab}$ as defined in
(A.3) and (A.11). According to (A.20), the value of $W_{abcn}$ is zero before
the reduction. After reducing $\pi_{a}$ and $\gamma_{a},$ it is found that

$W_{abcn}=p_{bn}\varepsilon_{ac}-p_{an}\varepsilon_{bc}.$

If $c\neq 0,$ then equation (3.9) and the fact that $F_{abc}$ is totally
symmetric and trace-free imply $F_{abc}=0$. This is a contradiction with the
assumptions $f_{abc}=\lambda F_{abc}$ and $F_{abc}$ being non-vanishing. As a
result, $c$ has to vanish. ∎

######  Remark 3.2.

The proposition above has an analogue in Finsler geometry. Recall that a
Finsler metric is called Landsberg if the derivative of its Cartan torsion
along geodesics is zero, i.e.,

$\dot{I}_{ijk}=0.$

It can be shown that if the flag curvature of a Landsberg metric with non-
vanishing Cartan torsion is constant, then it has to be zero [BCS00, Section
12.1]. By the proposition above, the condition $F_{abc;n}=\lambda F_{abc}$ can
be used to translate $F_{abc;n}$ to zero. As a result of this translation, if
the causal structure has vanishing Wsf curvature, then its ssf curvature has
to be zero. This observation suggests that causal structures satisfying
$f_{abc}=\lambda F_{abc}$ are causal analogues of Landsberg spaces.

###  3.2. A rigidity theorem

In this section, under some conditions, a theorem involving the global
vanishing of the Fubini form on a compact manifold is obtained.

As was discussed in Section 2.5.2, picking a representative of the conformal
class of the quadratic form (2.35) results in the reduction (2.48).
Consequently, translating $s_{i}$ to zero, using the relation (2.49), results
in further reductions (2.50).

The change in the structure equations after the reductions above can be
obtained from (A.17) and (A.3) where the 1-form $\phi_{n}$ is replaced by its
reduced form $-\phi_{0}+s_{\underline{a}}\theta^{a}$ and $\pi_{i}$’s are
replaced by their reduced form (2.50).

Assuming that the causal structure has vanishing Wsf curvature, i.e.,
$W_{anab}=0,$ the Bianchi identity (2.46) implies that $W_{anbc}=0.$ Using
$\mathrm{d}^{2}\theta_{a}=0,$ it is found that $p_{nn}$ is defined up to a
scale as the infinitesimal action of the reduced structure group on it is
given by

(3.10) $\mathrm{d}p_{nn}+4p_{nn}\phi_{0}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Hence, if it is non-zero, depending on its signature, it can be normalized to
$-1$ or $1,$ resulting in the reduction

(3.11) $\phi_{0}=t_{i}\omega^{i}+t_{\underline{a}}\theta^{a}.$

Using the reduced structure equations, the identity $\mathrm{d}\phi_{0},$ can
be used to give the infinitesimal action of the reduced structure group on
$t_{\underline{a}}$ as

$\mathrm{d}t_{{\underline{a}}}-t_{{\underline{b}}}\phi^{b}{}_{a}-\textstyle{\frac{1}{2}}\gamma_{a}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Therefore, by translating $t_{\underline{a}}$ to zero, the 1-forms
$\gamma_{a}$ are reduced to

$\gamma_{a}=q_{ai}\omega^{i}+q_{a{\underline{b}}}\theta^{b}.$

######  Remark 3.3.

Note that as a result of the conditions $F\neq 0$ and $p_{nn}\neq 0$ and the
subsequent reductions, the first structure equations take the form

(3.12a) $\displaystyle\mathrm{d}\omega^{0}{}={}$
$\displaystyle{}-\theta_{a}{\,{\wedge}\;}\omega^{a}+T^{0},$ (3.12b)
$\displaystyle\mathrm{d}\omega^{a}{}={}$
$\displaystyle-\phi^{a}{}_{b}{\,{\wedge}\;}\omega^{b}-\theta^{a}{\,{\wedge}\;}\omega^{{n}}+T^{a},$
(3.12c) $\displaystyle\mathrm{d}\omega^{{n}}{}={}$ $\displaystyle T^{n},$
(3.12d) $\displaystyle\mathrm{d}\theta^{a}{}={}$
$\displaystyle-\phi^{a}{}_{b}{\,{\wedge}\;}\theta^{b}+T^{{\underline{a}}}.$

where

$T^{\epsilon}=\textstyle{\frac{1}{2}}T^{\epsilon}{}_{ij}\omega^{i}{\,{\wedge}\;}\omega^{j}+T^{\epsilon}{}_{{\underline{a}}i}\theta^{a}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}T^{\epsilon}{}_{{\underline{a}}{\underline{b}}}\theta^{a}{\,{\wedge}\;}\theta^{b},$

for some coefficients
$T^{\epsilon}{}_{ij},T^{\epsilon}{}_{{\underline{a}}i},T^{\epsilon}{}_{{\underline{a}}{\underline{b}}},$
with $\epsilon$ representing $i$ and ${\underline{a}}.$ As was mentioned,
these are obtained from (A.17) and (A.3) after replacing the connection forms
by their reduced expressions.

The structure group is reduced to

(3.13) $\left\\{\begin{pmatrix}1&0&0&0\\\ 0&\mathbf{A}&0&0\\\ 0&0&1&0\\\
0&0&0&\mathbf{A}\end{pmatrix}\ \vline\ \mathbf{A}\in\mathrm{O}(p,q)\right\\}$

Therefore, the bilinear forms

(3.14)
$g=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b},\quad
h=2\omega^{0}\circ\omega^{n}-\varepsilon_{ab}\omega^{a}\circ\omega^{b}+\varepsilon_{ab}\theta^{a}\circ\theta^{b}$

are preserved by the reduced structure group.

In the case $p_{nn}<0,$ the normalization $p_{nn}=-1$ can be achieved which
results in the reductions that were explained above. Using
$\mathrm{d}^{2}\theta_{a}=0,$ using the structure equations after the
reductions, the vanishing of the coefficient of
$\theta^{a}{\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\omega^{n}$ implies the
equation

(3.15) $F_{abc;nn}-F_{abc}=0.$

If $M$ is compact, then all the components of $F_{abc}$ have to be bounded.

Recall from Section 2.3.5, that a vector field tangent to the characteristic
curves is annihilated by $\omega^{0},\omega^{a},\theta^{a}.$ Now, let
$s:\mathcal{C}\rightarrow\mathcal{P},$ be a section of the structure bundle,
inducing the coframe $(s^{*}\omega^{i},s^{*}\theta^{a}).$ The corresponding
frame is denoted by
$\frac{\partial}{\partial(s^{*}\omega^{i})},\frac{\partial}{\partial(s^{*}\theta^{a})}.$
The vector field $\frac{\partial}{\partial(s^{*}\omega^{n})}$ is a tangent
vector to the characteristic curves.

######  Definition 3.4.

A characteristic curve $\gamma,$ of a causal structure is called complete if
for each section $s:\mathcal{C}\rightarrow\mathcal{P},$ the parametrization of
$\gamma$ with respect to $\frac{\partial}{\partial(s^{*}\omega^{n})}$ can be
extended to $\mathbb{R}.$

Now the stage is set to state the following theorem.

######  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.

###### Proof.

Restrict a section of the structure bundle,
$s:\mathcal{C}\rightarrow\mathcal{P},$ to a characteristic curve and take a
representative of the conformal class of the quadratic form (2.35). After
carrying out the reductions, equation (3.15) gives a set of decoupled ODEs of
the form $f^{\prime\prime}-f=0$ for each component $F_{abc}$ of the Fubini
cubic form. Assuming the completeness of the characteristic curves, it follows
that the any non-zero solution of the ODE has to be unbounded. Since the
manifold is compact, the components $F_{abc}$ in the reduced structure
equations are bounded and, consequently, have to vanish. ∎

The above theorem has the following immediate 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.

###  3.3. Semi-integrable Lie contact structures

In this section it is shown how a causal structure with vanishing Wsf
curvature descends to a Lie contact structures on $\mathcal{N}.$ The
construction is analogous to Grossman’s notion of semi-integrable Segré
structures arising from torsion-free path geometries [Gro00].

Let $p\in\mathcal{C}$ be a generic point, that is a point admitting an open
neighborhood $U\subset\mathcal{C}$ foliated by characteristic curves. As a
result, the quotient map $\tau:U\rightarrow\mathcal{N}^{2n-1}$ defines the
space of characteristic curves in $U$.

Recall from Section 2.3.5 that a characteristic vector field $\mathbf{v}$
spans the unique degenerate direction of $\omega^{0}$ and
$\mathrm{d}\omega^{0},$ at every point i.e.,

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

It follows that $\mathcal{L}_{\mathbf{v}}\omega^{0}=0.$ Therefore, there
exists a 1-form $\tilde{\omega}^{0}\subset\Gamma(T^{*}\mathcal{N})$ of rank of
$2n-1$ such that $\tau^{*}\tilde{\omega}^{0}=\omega^{0}.$ As a result, the
quasi-contact structure on $\mathcal{C}$ defined by $\omega^{0}$ induces a
contact distribution on $\mathcal{N}.$ By abuse of notation, $\omega^{0}$ is
used to denote $\tilde{\omega}^{0}$ as the distinction is clear from the
context.

Because

$\mathbf{v}\im\omega^{a}=\mathbf{v}\im\theta^{a}=0,$

the 1-forms $\omega^{0},\cdots,\theta^{n-1}$ introduce a coframe on
$\mathcal{N},$ and the $\\{e\\}$-structure $\mathcal{F}$ fibers over
$\mathcal{N}.$ To understand the resulting structure on $\mathcal{N},$
consider the flat model. Recall from Section 2.2 that the flat causal geometry
corresponds to the null cone bundle of a hyperquadric
$Q_{p+1,q+1}\subset\mathbb{P}^{n+2},$ and its space of characteristic curves
is the space of isotropic 2-planes denoted by $G_{0}(2,n+4).$ The resulting
structure was first studied by Sato and Yamaguchi in [SY89] referred to as Lie
contact structures. As a parabolic geometry, this geometry is modeled on
$(B_{n},P_{2})$ or $(D_{n},P_{2}),$ for $n\geq 4$ or $(D_{3},P_{1,2})$ (see
Section 2.4.1).

In this section it is shown how a causal structure with vanishing Wsf
curvature descends to a Lie contact structures on $\mathcal{N}.$ As was
discussed in Section 2.3.13, no attempt is made to introduce a Cartan
connection for the Lie contact structure from the derived $\\{e\\}$-structure.

First, a definition of Lie contact structures in terms of a field of Segré
cones of type $(2,\mathsf{N})$ is given. Let $\mathbb{V}$ be a
$2\mathsf{N}$-dimensional vector space with a decomposition
$\mathbb{V}\cong\mathbb{R}^{2}\otimes\mathbb{R}^{\mathsf{N}}.$ An element
$p\in\mathbb{V}$ is said to be of rank one if and only if $p=a\otimes b.$ A
Segré cone of type $(2,\mathsf{N}),$ denoted by $S(2,\mathsf{N}),$ is defined
as the set of rank one elements of this decomposition. In terms of local
coordinates
$(z^{1}_{1},\cdots,z^{1}_{\mathsf{N}},z^{2}_{1},\cdots,z^{2}_{\mathsf{N}})$
adapted to this decomposition, the Segré cone is given by the points
satisfying the homogeneous quadratic equations

$z^{1}_{\mu}z^{2}_{\nu}-z^{2}_{\mu}z^{1}_{\nu}=0,$

where $1\leq\mu,\nu\leq\mathsf{N}.$ As a result, a Segré cone of type
$(2,\mathsf{N})$ has a double ruling. A $(\mathsf{N}-1)$-parameter ruling by
2-planes $E_{v}\cong\mathbb{R}^{2}\otimes v$ where
$v\in\mathbb{R}^{\mathsf{N}}.$ Such 2-planes are called $\alpha$-planes
[AG96]. Another ruling of $S(2,\mathsf{N})$ is given by the 1-parameter family
of $\mathsf{N}$-planes $u\otimes\mathbb{R}^{\mathsf{N}}$ where
$u\in\mathbb{R}^{2}.$ Such $\mathsf{N}$-planes are called
$\beta$-planes.191919The relation between these $\alpha$\- and $\beta$-planes
in the section and those defined in Section 4.1 has to do with the fact that
the null cones of a metric of split signature in four dimensions is a Segré
cone of type $(2,2).$

The proposition below is used to show how a Lie contact structure is induced
on $\mathcal{N}.$

######  Proposition 3.7.

Given a causal structure with adapted coframe $(\omega^{i},\theta_{a})$, the
span of the bilinear forms

(3.16) $\Upsilon^{ab}=\theta^{a}\circ\omega^{b}-\theta^{b}\circ\omega^{a}$

restricted to $H=\mathrm{Ker}\,\omega^{0}$ is invariant under the action of
the structure group $G_{5}$. Moreover, $\mathcal{C}$ has vanishing Wsf
curvature, if and only if the $\mathrm{Span}\\{\Upsilon^{ab}\\}$ is invariant
along characteristic curves.

###### Proof.

By the matrix form of the structure group $G_{5}$ in (2.34), it follows that
the span of $\Upsilon^{ab}$ modulo $I\\{\omega^{0}\\}$ is invariant under the
action of $G_{5}$.

Let $\mathbf{v}$ denote a characteristic vector field. To verify the second
claim it suffices to note that (2.37) implies

$\mathcal{L}_{\mathbf{v}}\Upsilon^{ab}=A^{ab}{}_{cd}\Upsilon^{cd}+(\mathbf{v}\im\omega^{n})(W^{b}{}_{ncn}\omega^{a}-W^{a}{}_{ncn}\omega^{b})\circ\omega^{c},$

where
$A^{ab}{}_{cd}=\mathbf{v}\im(2\phi_{0}\delta^{a}_{c}\delta^{b}_{d}+\delta^{b}_{d}\phi^{a}{}_{c}-\delta^{a}_{c}\phi^{b}{}_{d}).$
Thus, if the condition $W^{a}{}_{nbn}=0$ is satisfies, i.e., causal structure
has vanishing Wsf curvature, then the $\mathrm{span}\\{\Upsilon^{ab}\\}$ is
invariant along the characteristic curves. ∎

As a result of the proposition above, the set of tangent vectors
$\mathbf{u}\in H$ satisfying $\Upsilon^{ab}(\mathbf{u},\mathbf{u})=0$ is well-
defined and gives a Segré cone of type $(2,n-1)$ in the contact distribution
$H\subset T\mathcal{N}$.

######  Definition 3.8.

A manifold $N^{2\mathsf{N}+1}$ with a contact distribution $H$ is said to have
a Lie contact structure of signature $(p,q),$ $p+q=\mathsf{N}$ if there is a
smoothly varying family of Segré cones $S(2,\mathsf{N})\subset H_{x}$ such
that at every point $x\in N$, its $\beta$-planes are endowed with conformally
invariant inner product of signature $(p,q)$ and are Lagrangian with respect
to the symplectic 2-form that is induced by the contact 1-form on the contact
distribution.

A Lie contact structure is called _semi-integrable_ if there exists a
1-parameter family of foliations by Legendrian submanifolds with the property
that given a point $x\in N$ and a $\beta$-plane in $S(2,\mathsf{N})\subset
H_{x},$ there passes a unique member of the family through $x$, tangent to
that $\beta$-plane.

There are other equivalent definitions of a Lie contact structure discussed in
[Zad09].

######  Remark 3.9.

Semi-integrability condition gives rise to a foliation of the bundle
$\tilde{\rho}:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ where
$\mathcal{E}_{\beta}$ denotes the bundle of $\beta$-planes of the Segré cones.
Because $\beta$-planes are Lagrangian subspaces of the contact distribution,
$\mathcal{E}_{\beta}$ is a sub-bundle of $\mathrm{LG}(\mathcal{N})$ where
$\rho:\mathrm{LG}(\mathcal{N})\rightarrow\mathcal{N}$ is the bundle of
Lagrangian-Grassmannians of the contact distribution $H.$ Since, the bundle
map $\tilde{\rho}$ is the restriction of $\rho$ to the sub-bundle
$\mathcal{E}_{\beta},$ by abuse of notation, it is denoted by $\rho$ as well.

The theorem below gives a characterization of causal structures with vanishing
Wsf curvature in terms of semi-integrable Lie contact structures induced on
their space of characteristic curves.

######  Theorem 3.10.

Given a causal structure $(\mathcal{C},M)$ with vanishing Wsf curvature of
signature $(p+1,q+1)$ its space of characteristic curves admits a semi-
integrable Lie contact structure of signature $(p,q).$ Conversely, any semi-
integrable Lie contact structure of signature $(p,q)$ on a space $\mathcal{N}$
induces a causal structure of signature $(p+1,q+1)$ with vanishing Wsf
curvature on its space of ruling Legendrian submanifolds.

###### Proof.

To see how $\mathcal{N}$ is endowed with a Lie contact structure, firstly note
that by Proposition 3.7 the contact distribution carries a Segré structure of
type $(2,n-1).$ By the definition of $\Upsilon^{ab},$ it follows that the
$\beta$-planes, i.e., the 1-parameter ruling of the Segré cone, are given by

(3.17) $\textrm{Ker}\,\\{A\theta^{a}+B\omega^{a}\ \vline\ a=1,\dots,n-1\\},$

$[A:B]\in\mathbb{P}^{1}.$ Moreover, the inner product
$\varepsilon_{ab}\theta^{a}\circ\theta^{b}$ is of signature $(p,q)$ and its
conformal class is well-defined on each $\beta$-plane, i.e., it is conformally
invariant under the action of $G_{5}$ modulo
$\mathcal{I}_{\mathsf{bas}}=I\\{\omega^{0},\cdots,\omega^{n}\\}.$ Finally, to
show $\beta$-planes are Lagrangian consider a $\beta$-plane $E_{\beta_{1}}$
corresponding to $[A:B]\in\mathbb{P}^{1}$ in (3.17). The tangent vectors
$(v_{1},\cdots,v_{n-1})$ where
$v_{a}=B\frac{\partial}{\partial\theta^{a}}-A\frac{\partial}{\partial\omega^{a}}$
define a basis for $E_{\beta_{1}}.$ Because
$\mathrm{d}\omega^{0}\equiv\varepsilon_{ab}\theta^{a}{\,{\wedge}\;}\omega^{b},$
modulo $I\\{\omega^{0}\\},$ it follows that
$\mathrm{d}\omega^{0}(v_{a},v_{b})=0.$ Hence, $E_{\beta_{1}}$ is Lagrangian
with respect to the conformal symplectic form $\mathrm{d}\omega^{0}$.

As for the semi-integrability of these Lie contact structures, the following
lemma will be useful.

######  Lemma 3.11.

Given a causal structure with vanishing Wsf curvature, there is a canonical
bundle diffeomorphism $\varpi:\mathcal{C}\rightarrow\mathcal{E}_{\beta}$
between the fiber bundles $\tau:\mathcal{C}\rightarrow\mathcal{N}$ and
$\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N},$.

###### Proof.

Let $\gamma_{t}=(x(t);y(t))$ be a characteristic curve of $\mathcal{C}$ which
corresponds to the point $[\gamma]\in\mathcal{N}.$ Let $\mathcal{D}$ be the
completely integrable distribution of $T\mathcal{C}$ given by the vertical
tangent spaces
$T_{y}\mathcal{C}_{x}:=\textrm{Ker}\,\mathcal{I}_{\mathsf{bas}}$ with
$\mathcal{D}_{t}:=T_{y(t)}\mathcal{C}_{x(t)}.$ By the discussion above, it
follows that, at $t=t_{0},$ $\tau_{*}\mathcal{D}_{t_{0}}\subset H_{[\gamma]}$
is isotropic, $(n-1)$-dimensional and lies in the zero locus of
$\Upsilon^{ab},$ i.e. $\tau_{*}\mathcal{D}_{t_{0}}$ is a $\beta$-plane. By
defining

(3.18) $\varpi(\gamma_{t_{0}})=\tau_{*}\mathcal{D}_{t_{0}},$

one obtains the desired diffeomorphism. ∎

Continuing the proof of Theorem 3.10, note that the fibers $\mathcal{C}_{x},$
as the integral manifolds of $\mathcal{D},$ give a foliation of
$\mathcal{E}_{\beta}$ via the diffeomorphism $\varpi.$ Because of the
transversality of the fibers $\mathcal{C}_{x}$ and characteristic curves
$\gamma$ passing through $x,$ it follows that the subsets
$\tau(\mathcal{C}_{x})\subset\mathcal{N}$ are immersed as $(n-1)$-dimensional
submanifolds of $\mathcal{N}.$ These submanifolds are Legendrian because it
was shown that their tangent spaces, $\tau_{*}\mathcal{D},$ are
$\beta$-planes, i.e., Lagrangian subspaces of the contact distribution which
rule the Segré cone.

The converse part of the theorem can be shown as follows. Let $\omega^{0}$ be
a contact form. Since the Segré cones of type $(2,n-1)$ admit a 1-parameter
ruling by $(n-1)$-planes, one can find 1-forms
$\\{\eta^{a},\zeta^{a}|a=1,\cdots,n-1\\}$ complementing $\omega^{0}$ in
$T^{*}\mathcal{N}$ such that the Segré cone can be written as the vanishing
set of the bilinear forms

$\Upsilon^{ab}=\eta^{a}\circ\zeta^{b}-\eta^{b}\circ\zeta^{a}.$

As a result, the 1-parameter family of ruling $(n-1)$-planes of the Segré cone
can be expressed as
$\textrm{Ker}\,\\{t\eta^{a}+(1-t)\zeta^{a}|a=1,\cdots,n-1\\}$ where
$t\in(0,1).$ Let $\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ denote the
bundle of $\beta$-planes of the Segré cone. A point $p\in\mathcal{E}_{\beta}$
represents a $\beta$-plane of the contact distribution $H$ which is Lagrangian
with respect to the conformal symplectic structure on the contact
distribution. Thus, the tautological bundle $K\subset T\mathcal{E}_{\beta}$
can be defined with fibers

$K_{p}=\\{v\in T_{p}\mathcal{E}_{\beta}\ \vline\ \rho_{*}|_{p}(v)\in p\subset
T_{\rho(p)}\mathcal{N}\\}.$

As a result, the subspaces $\rho_{*}(K_{p})$ rule the Segré cone
$S(2,n-1)\subset H_{\rho(p)}\subset T_{\rho(p)}\mathcal{N}.$ The fibers of
$\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ are one dimensional and
parametrize the ruling subspaces $\rho_{*}(K_{p})$. Let $\mathbf{v}$ be a
vertical vector field of the fibration
$\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ satisfying
$\rho_{*}(\mathbf{v})=\frac{\partial}{\partial t},$ and by abuse of notation,
let a multiple of $\rho^{*}(\omega^{0})$ be denoted by $\omega^{0}$ as well.
Hence, $\omega^{0}\subset T^{*}\mathcal{E}_{\beta}$ is a quasi-contact 1-form.
Defining $\omega^{a}=\rho^{*}(t\eta^{a}+(1-t)\zeta^{a}),$ it follows that
$\mathcal{L}_{\mathbf{v}}\omega^{a}=\rho^{*}(\eta^{a}-\zeta^{a}),$ and,
consequently,

$\omega^{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\omega^{n-1}{\,{\wedge}\;}\mathcal{L}_{\mathbf{v}}\omega^{1}{\,{\wedge}\;}\cdots{\,{\wedge}\;}\mathcal{L}_{\mathbf{v}}\omega^{n-1}\neq
0\quad\textrm{mod}\quad I\\{\omega^{0}\\}.$

The reason that it is non-zero, modulo $I\\{\omega^{0}\\},$ is that the Segré
cone is a subset of the contact distribution.

Hence, locally, the tautological bundle is given by
$K=\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n-1}\\}.$ As a result, the
1-forms $\omega^{0},\omega^{a},$
$\theta_{a}=\mathcal{L}_{\mathbf{v}}\omega^{a}$ which are semi-basic with
respect to the fibration $\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ and
the 1-form $\omega^{n},$ which is vertical, span
$T^{*}_{p}\mathcal{E}_{\beta},$ and one has

$\mathrm{d}\omega^{a}\equiv-\theta^{a}{\,{\wedge}\;}\omega^{n}+T^{a}{}_{bc}\theta^{a}{\,{\wedge}\;}\theta^{b}\quad\textrm{mod}\quad
I\\{\omega^{0},\cdots,\omega^{n-1}\\}.$

Since the Lie contact structure is assumed to be semi-integrable, the lift of
its associated 1-parameter family of Legendrian submanifolds foliate
$\mathcal{E}_{\beta}$ by quasi-Legendrian submanifolds. The tangent
distribution to this foliation is a corank sub-bundle of the tautological
subbundle $K$ given by $\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n}\\}.$ As
a result, $\omega^{i}$’s form a Pfaffian system which implies
$T^{a}{}_{bc}=0.$ Moreover, knowing that
$K=\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n-1}\\}$ and that $\omega^{0}$
is semi-basic, one arrives at

$\mathrm{d}\omega^{0}\equiv-
h_{ab}\theta^{a}{\,{\wedge}\;}\omega^{b},\quad\textrm{mod}\quad
I\\{\omega^{0}\\}.$

In the identity $\mathrm{d}^{2}\omega^{0}=0,$ the vanishing of the 3-form
$\theta^{a}{\,{\wedge}\;}\theta^{b}{\,{\wedge}\;}\omega^{n}$ implies that
$h_{ab}=h_{ba}.$

Let $M$ be the leaf space of the quasi-Legendrian foliation of
$\mathcal{E}_{\beta},$ via the quotient map
$\pi:\mathcal{E}_{\beta}\rightarrow M.$ Because the tangent planes of the
leaves are given by $\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n}\\},$ the
1-forms $\\{\omega^{0},\cdots,\omega^{n}\\}$ are semi-basic with respect to
the fibration $\pi:\mathcal{E}_{\beta}\rightarrow M,$ and the 1-forms
$\theta^{a}$ are vertical. Moreover, the coframe on $\mathcal{E}_{\beta}$ is
adapted to the flag (2.11). Note that the tautological bundle $K$ is
isomorphic to the tautological bundle introduced in Section 2.3.2.

According to Section 2.3, the complete integrability of the Pfaffian system
$\mathcal{I}_{\mathsf{bas}}$ and the relations

$\displaystyle\omega^{0}$
$\displaystyle\equiv-\theta_{a}{\,{\wedge}\;}\omega^{a},\quad\textrm{mod}\quad
I\\{\omega^{0}\\},$ $\displaystyle\omega^{a}$
$\displaystyle\equiv-h^{ab}\theta_{b}{\,{\wedge}\;}\omega^{n},\quad\textrm{mod}\quad
I\\{\omega^{0},\cdots,\omega^{n-1}\\},$

where $\theta_{a}:=h_{ab}\theta^{b}$ suffice to carry out the coframe
adaptations and obtain the $\\{e\\}$-structure that characterizes a causal
structure. Note that the distribution given by
$\textrm{Ker}\,\\{\omega^{0},\cdots,\omega^{n-1}\\}$ characterizes a causal
structure according to Section 2.4.2. The one dimensional fibers of
$\rho:\mathcal{E}_{\beta}\rightarrow\mathcal{N}$ are mapped to the
characteristic curves of the causal structure.

Finally, since the vanishing set of the quadratic forms $\Upsilon^{ab}$
defined in (3.16) is preserved, the Wsf curvature of the obtained causal
structure has to vanish. ∎

##  4\. Four dimensions

In this section a class of causal structures on four dimensional manifolds is
introduced for which a notion of half-flatness can be defined in terms of the
existence of integral surfaces whose tangent planes in each tangent space give
an envelope construction of the null cone at that point. As a result, one
obtains a generalization of half-flat indefinite pseudo-conformal
structures202020In split signature, the terms half-flatness and self-duality
are used to refer to pseudo-conformal structures with a 3-parameter family of
null surfaces. in four dimensions. It is shown that the symmetry group of a
non-flat causal structure of split signature, which does not descend to a
pseudo-conformal structure, is at most 8-dimensional. Finally, the unique
causal structure with 8-dimensional symmetry algebra is introduced.

###  4.1. Causal structures with ruled fibers

Given a causal structure of signature $(2,2),$ suppose the second fundamental
form is normalized to $[\varepsilon_{ab}]=\tiny\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ Since $F_{abc}$ is totally symmetric and trace-free, it
has two independent components $F_{111}$ and $F_{222}$. The Fubini-Pick
invariant, $F:=F_{abc}F^{abc},$ is given by
$F=8(F^{1}{}_{11})^{2}-8(F^{2}{}_{22})^{2}=8\mathsf{F}^{+}\mathsf{F}^{-},$
where

(4.1) $\mathsf{F}^{\pm}={\textstyle{\frac{1}{\sqrt{2}}}}(F^{1}{}_{11}\pm
F^{2}{}_{22}).$

It is known that the vanishing of $F$ implies that the projective surface is
ruled [Sas99, GH79]. In this section, it is assumed that $\mathsf{F}^{+}=0.$
The case $\mathsf{F}^{-}=0$ can be treated analogously.

Define the following quantities

$\mathsf{K}^{\pm}:=K^{11}\pm
K^{12}=-\mathsf{F}^{\pm}_{;{\underline{1}}}+\mathsf{F}^{\pm}_{;{\underline{2}}}\qquad
L^{\pm}:={\textstyle{\frac{1}{\sqrt{2}}}}(L^{1}\pm
L^{2})=\textstyle\frac{1}{2\sqrt{2}}(-\mathsf{K}^{\pm}_{;{\underline{1}}}+\mathsf{K}^{\pm}_{{\underline{2}}}).$

Because $\varepsilon_{ab}$ is indefinite, one should keep in mind that raising
or lowering of the index 2 is accompanied by a negative sign multiplication,
e.g., $\theta_{2}=-\theta^{2},$ or $F^{2}{}_{11}=-F_{211}.$ Define

(4.2)
$\omega^{\pm}:={\textstyle{\frac{1}{\sqrt{2}}}}(\omega^{1}\pm\omega^{2}),\quad\theta^{\pm}:={\textstyle{\frac{1}{\sqrt{2}}}}(\theta_{1}\mp\theta_{2}),\quad\gamma^{\pm}:={\textstyle{\frac{1}{\sqrt{2}}}}(\gamma^{1}\pm\gamma^{2}),\quad\pi_{\pm}:={\textstyle{\frac{1}{\sqrt{2}}}}(\pi_{1}\pm\pi_{2}).$

The structure equations (2.37) can be expressed as

(4.3a) $\displaystyle\mathrm{d}\omega^{0}{}={}$
$\displaystyle-2\phi_{0}{\,{\wedge}\;}\omega^{0}-{}\theta^{-}{\,{\wedge}\;}\omega^{+}-\theta^{+}{\,{\wedge}\;}\omega^{-},$
(4.3b) $\displaystyle\mathrm{d}\omega^{+}{}={}$
$\displaystyle-\gamma^{+}{\,{\wedge}\;}\omega^{0}-\left(\phi^{1}{}_{2}+\phi_{0}+\phi_{3}\right){}{\,{\wedge}\;}\omega^{+}-\theta^{+}{\,{\wedge}\;}\omega^{3},$
$\displaystyle-\mathsf{K}^{+}\,\theta^{-}{\,{\wedge}\;}\omega^{0}-\mathsf{F}^{+}\,\theta^{-}{\,{\wedge}\;}\omega^{-},$
(4.3c) $\displaystyle\mathrm{d}\omega^{-}{}={}$
$\displaystyle-\gamma^{-}{\,{\wedge}\;}\omega^{0}-\left(-\phi^{1}{}_{2}+\phi_{0}+\phi_{3}\right){}{\,{\wedge}\;}\omega^{-}-\theta^{-}{\,{\wedge}\;}\omega^{3},$
$\displaystyle-\mathsf{K}^{-}\,\theta^{+}{\,{\wedge}\;}\omega^{0}-\mathsf{F}^{-}\,\theta^{+}{\,{\wedge}\;}\omega^{+},$
(4.3d) $\displaystyle\mathrm{d}\omega^{3}{}={}$
$\displaystyle-\gamma^{-}{\,{\wedge}\;}\omega^{+}-\gamma^{+}{\,{\wedge}\;}\omega^{-}-2\phi_{3}{\,{\wedge}\;}\omega^{3}-\mathsf{L}^{+}\,\theta^{-}{\,{\wedge}\;}\omega^{0}{}-\mathsf{L}^{-}\,\theta+{\,{\wedge}\;}\omega^{0}$
(4.3e) $\displaystyle\mathrm{d}\theta^{+}{}={}$
$\displaystyle-\pi^{+}{\,{\wedge}\;}\omega^{0}{}-\,\pi_{3}{\,{\wedge}\;}\omega^{+}-\left(\phi^{1}{}_{2}+\phi_{0}-\phi_{3}\right){\,{\wedge}\;}\theta^{+}$
$\displaystyle+\mathsf{W}^{+}\,\omega^{-}{\,{\wedge}\;}\omega^{3}+\mathsf{V}^{+}\,\omega^{+}{\,{\wedge}\;}\omega^{-}-\mathsf{f}^{+}\,\theta^{+}{\,{\wedge}\;}\omega^{+},$
(4.3f) $\displaystyle\mathrm{d}\theta^{-}{}={}$
$\displaystyle-\pi^{-}{\,{\wedge}\;}\omega^{0}{}-\,\pi_{3}{\,{\wedge}\;}\omega^{-}-\left(-\phi^{1}{}_{2}+\phi_{0}-\phi_{3}\right){\,{\wedge}\;}\theta^{-}$
$\displaystyle+\mathsf{W}^{-}\,\omega^{+}{\,{\wedge}\;}\omega^{3}+\mathsf{V}^{-}\,\omega^{+}{\,{\wedge}\;}\omega^{-}-\mathsf{f}^{-}\,\theta^{+}{\,{\wedge}\;}\omega^{+}.$

where

$\begin{gathered}\mathsf{f}^{\pm}=\mathsf{F}^{\pm}_{;n},\quad\mathsf{W}^{\pm}=W_{1313}\mp
W_{1323},\quad\mathsf{V}^{\pm}=W_{1312}\mp W_{2312},.\end{gathered}$

Recall that the symmetric trace-free Wsf curvature, $W_{a3b3},$ has two
independent entries $W_{1313}=W_{2323}$ and $W_{1323}=W_{2313}.$

According to the structure equations (4.3), if in addition to the vanishing of
$\mathsf{F}^{+},$ the quantity $\mathsf{W}^{+}$ vanishes, then the Pfaffian
system $\mathcal{T}$ generated by $\\{\omega^{0},\omega^{+},\theta^{+}\\}$
becomes Frobenius. In fact, if $\mathsf{W}^{+}=0,$ then Bianchi identity
(A.25) for $W_{0abc}$ implies that $\mathsf{V}^{+}=0.$ This observation
motivates the following definition.

######  Definition 4.1.

A four dimensional causal structure is called half-flat, if the torsion
coefficients $\mathsf{F}^{+},\mathsf{W}^{+},$ or
$\mathsf{F}^{-},\mathsf{W}^{-}$ vanish.

######  Remark 4.2.

The above definition only makes sense when the causal structure is of
signature $(2,2),$ since in the Lorentzian case, i.e., signature $(3,1)$, the
Fubini-Pick invariant is given by

$F=8(F^{2}_{111}+F^{2}_{222}),$

which only vanishes if $F_{111},F_{222}=0,$ i.e., the causal structure is
equivalent to a conformal Lorentzian structure.

This term half-flat is justified as follows. In [Pen76], Penrose characterized
self-dual pseudo-conformal structures of split signature, i.e., conformal
structures for which the anti-self-dual part of the Weyl curvature vanishes,
by the existence of a 3 parameter family of 2-surfaces. To explain his
characterization, recall that for such pseudo-conformal structures on $M$, if
the metric is given by $g=\omega^{0}\circ\omega^{3}-\omega^{+}\circ\omega^{-}$
then at $x\in M,$ there is a ruling of the null cone
$\widehat{\mathcal{C}}_{x}\subset T_{x}M$ by the null 2-planes over which one
has

(4.4) $\omega^{0}+\lambda\omega^{-}=0,\qquad\omega^{+}+\lambda\omega^{3}=0,$

for some $\lambda\in\mathbb{R}.$ In fact, $\lambda$ parametrizes such ruling
2-planes (see [AG96]). There is another ruling of $\widehat{\mathcal{C}}_{x},$
given by the null 2-planes over which

(4.5) $\omega^{0}+\mu\omega^{+}=0,\qquad\omega^{-}+\mu\omega^{3}=0,\ $

which are parametrized by $\mu.$ Assuming that $M$ is oriented and that the
null tetrad $(\omega^{0},\omega^{+},\omega^{-},\omega^{3})$ induces a positive
orientation for $M,$ the set of null 2-planes parametrized by $\lambda$ and
$\mu$ is referred to as $\alpha$-planes and $\beta$-planes,
respectively212121In some literature $\alpha$ and $\beta$ planes are defined
the opposite way.. A 2-surface in $M$ is called an $\alpha$-surface or
$\beta$-surface if its tangent space at each point is an $\alpha$-plane or
$\beta$-plane, respectively. A pseudo-conformal structure is called _half-
flat_ if there exists a 3-parameter family of $\alpha$-surfaces in $M,$ or
equivalently, there exists a 3-parameter family of 2-surfaces with the
property that at every point $M,$ any given $\alpha$-plane is tangent to a
unique 2-surface in the family. Such pseudo-conformal structures are also
called right-conformally flat [Pen76], self-dual [LM07], $\alpha$-semi-
integrable [AG96], or half-conformally flat [Bes08]. If $M$ contains a
3-parameter family of $\beta$-surfaces then it is called half-flat as well.
Alternatively, the terms left-conformally flat, anti-self-dual or
$\beta$-semi-integrable can be used as well.

Penrose’s characterization of half-flat pseudo-conformal structures can be
generalized to causal structures in the following way.

######  Proposition 4.3.

Given a half-flat causal structure, there exists a 1-parameter family of
foliations of $M$ by 2-surfaces whose tangent planes at each point $x\in M$
rule the null cone $\mathcal{C}_{x}.$

###### Proof.

As mentioned before, if $\mathsf{W}^{+}=0,$ then the Bianchi identity (A.25)
for $W_{anbc}$ implies that $\mathsf{V}^{+}=0.$ As a result, the Pfaffian
system $\mathcal{T}$ generated by $\\{\omega^{0},\omega^{+},\theta^{+}\\}$
becomes completely integrable. The integral manifolds of $\mathcal{T}$ are
three dimensional submanifolds of $\mathcal{C}$ which project to 2-surfaces in
$M.$ This is due to the fact that the tangent space of the integral manifolds
are spanned by
$\frac{\partial}{\partial\omega^{-}},\frac{\partial}{\partial\omega^{3}},\frac{\partial}{\partial\theta^{-}},$
where the tangent vector $\frac{\partial}{\partial\theta^{-}}$ is vertical
with respect to the fibration $\pi:\mathcal{C}\rightarrow M.$

Now, consider the foliation of $\mathcal{C}$ given by the integral curves of
the Pfaffian system
$\mathcal{U}=\\{\omega^{0},\omega^{+},\omega^{-},\omega^{0},\theta^{+}\\}.$
The tangent directions of these integral curves are asymptotic directions of
the fibers $\mathcal{C}_{x}$ given by $\theta^{+}=0.$ The first step is to
show that at each point $x\in M$ these curves give a ruling of
$\mathcal{C}_{x}$. Fix a point $x\in M$ and an integral curve of
$\theta^{+}=0$ on $\mathcal{C}_{x}$ denoted by $\rho(t)$. Let $\dot{\rho}(t)$
denote the vector $\frac{\partial}{\partial\theta^{-}}$ tangent to $\rho(t).$
The structure equations (4.3) imply

$\mathcal{L}_{\dot{\rho}(t)}\omega^{0}{\,{\wedge}\;}\mathcal{L}_{\dot{\rho}(t)}\omega^{+}=f(x;\rho(t))\omega^{0}{\,{\wedge}\;}\omega^{+},$

for some function $f(x;y)$. As a result, defining
$\mathsf{P}=\textrm{Ker}\,\\{\omega^{0},\omega^{+}\\},$ the 2-plane
$\pi_{*}(\mathsf{P})\subset T_{x}M$ is well-defined and can be represented by
the bi-vector
$\pi_{*}\left(\frac{\partial}{\partial\omega^{-}}{\,{\wedge}\;}\frac{\partial}{\partial\omega^{3}}\right).$
Moreover, because
$\pi_{*}(\mathsf{P})\subset\widehat{T}_{\rho(t)}\mathcal{C}_{x},$ for all
values of $t,$ it follows that the cone over the curve $\rho(t)$, i.e.,
$\widehat{\rho(t)}\subset T_{x}M,$ is a subset of $\pi_{*}(\mathsf{P}).$
Hence, the integral curves of $\theta^{+}=0$ on $\mathcal{C}_{x}$ are 2-plane
in $T_{x}M,$ which give a ruling of $\mathcal{C}_{x}.$

Using the foliation of the fibers $\mathcal{C}_{x}$ by the Pfaffian system
$\mathcal{U}$, one obtains the locally defined quotient map
$\varrho:\mathcal{C}\rightarrow\mathcal{R}.$ Moreover, because the integral
curves of $\mathcal{U}$ are curves in the fibers of the bundle
$\mathcal{C}\rightarrow M,$ the quotient space $\mathcal{R}$ is a fiber bundle
over $M,$ denoted by

(4.6) $\pi_{\mathsf{rul}}:\mathcal{R}\rightarrow M,$

and the fibers $\mathcal{R}_{x}:=\pi_{\mathsf{rul}}^{-1}(x)$ are
1-dimensional. From the previous discussion it follows that $\mathcal{R}_{x}$
can be identified with a generating curve of the ruled fibers
$\mathcal{C}_{x}.$

By the definition of $\mathcal{R},$ the integral manifolds of $\mathcal{T}$
foliate $\mathcal{R}$ by 2-surfaces and are transversal to the fibers
$\mathcal{R}_{x}.$ Under the projection $\pi_{\mathsf{rul}}$ one obtains a
1-parameter family of foliations of $M$ by 2-surfaces. Moreover, the tangent
planes of the 1-parameter family of 2-surfaces passing through each point
$x\in M,$ are given by
$\pi_{*}\left(\frac{\partial}{\partial\omega^{-}}{\,{\wedge}\;}\frac{\partial}{\partial\omega^{3}}\right)\subset
T_{x}M$ which, by our previous discussion, are the ruling 2-planes of
$\mathcal{C}_{x}.$

Similarly, if the pair $\mathsf{F}^{-}$ and $\mathsf{W}^{-}$ vanish, the
2-surfaces are the projection of the integral manifolds of
$\\{\omega^{0},\omega^{-},\theta^{-}\\}.$ ∎

######  Remark 4.4.

Note that at each point $(x;y)\in\mathcal{C}$ the integral manifolds of
$\mathcal{U}$ are null with respect to the quadratic form
$g=\omega^{0}\circ\omega^{3}-\omega^{+}\circ\omega^{-}$ defined in (2.35).
When the Fubini form vanishes, $[g]$ defines a pseudo-conformal structure of
signature $(2,2)$ on $M$ and the obtained 2-surfaces in $M$ become its
$\alpha$-surfaces. In other words, the conditions

$\mathsf{F}^{+}=0,\quad\mathrm{and}\quad\mathsf{W}^{+}=0,$

are the causal analogue of self-duality.

######  Remark 4.5.

It can be shown that the 3-dimensional space of integral 3-folds foliating
$\mathcal{C}$ is equipped with a path structure whose paths are the image of
the generating curve of the ruled surfaces $\mathcal{C}_{x}$ under the
quotient map. Conversely, given a path geometry, its space of paths is
equipped with a half-flat causal structure [Mak16].

Comparing this observation with that of Grossman, [Gro00], it can be shown
that what he refers to as the torsion of a path structure coincides with the
fiber invariants, $\mathsf{F}^{+},\mathsf{K}^{+},\mathsf{L}^{+}$ of a half-
flat causal structure whose vanishing results in a half-flat conformal
structure of split signature.

######  Example 4.6 (Product of two Finsler surfaces).

Similar to Example 2.6.1, given two Finsler surfaces $(N,\Sigma)$ and
$(\tilde{N},\tilde{\Sigma})$ one can consider the induced causal structure of
split signature on their product space, denoted by $M=N\times{}_{f}\tilde{N},$
as defined in (2.60), where $f$ is defined over $N\times\tilde{N}$. According
to structure equations (2.59), Finsler surfaces $(N,\Sigma)$ and
$(\tilde{N},\tilde{\Sigma})$ are equipped with a unique coframing
$(\eta^{0},\eta^{1},\zeta^{1})$ and
$(\tilde{\eta}^{0},\tilde{\eta}^{1},\tilde{\zeta}^{1}),$ and their associated
scalars $(I,J,K)$ and $(\tilde{I},\tilde{J},\tilde{K}).$

The coframe adaptation is rather similar to Example 2.6.1. For simplicity, the
pull-back of coframing $(\tilde{\eta}^{0},\tilde{\eta}^{1},\tilde{\zeta}^{1})$
is denoted by $(\eta^{3},\eta^{2},\zeta^{2}).$ A choice of 2-adapted coframing
is given by

(4.7)
$\begin{gathered}\omega^{0}={\textstyle{\frac{1}{\sqrt{2}}}}(-\eta^{0}-f\eta^{3}),\quad\omega^{1}=\eta^{1},\quad\omega^{2}=f\eta^{2},\quad\omega^{3}={\textstyle{\frac{1}{\sqrt{2}}}}(-\eta^{0}-f\eta^{3}),\\\
\theta_{1}={\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{1}-{\textstyle{\frac{1}{\sqrt{2}}}}f_{,1}\eta^{3},\quad\theta_{2}=-{\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{2}-{\textstyle{\frac{1}{\sqrt{2}}}}\textstyle\frac{f_{,2}}{f}\eta^{3}+{\textstyle{\frac{1}{\sqrt{2}}}}(f_{,0}+\textstyle\frac{f_{,3}}{f})\eta^{2},\\\
\phi_{0}=-\phi_{3}=-\textstyle{\frac{1}{2}}f_{,0}\eta^{3},\quad\gamma_{1}=-{\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{1}-{\textstyle{\frac{1}{\sqrt{2}}}}f_{,1}\eta^{3},\\\
\gamma_{2}={\textstyle{\frac{1}{\sqrt{2}}}}\zeta_{2}+{\textstyle{\frac{1}{\sqrt{2}}}}\textstyle\frac{f_{,2}}{f}\eta^{3}+{\textstyle{\frac{1}{\sqrt{2}}}}(f_{,0}-\textstyle\frac{f_{,3}}{f})\eta^{2},\quad\phi=\textstyle\frac{f_{,1}}{f}\omega^{2},\quad\pi_{3}=0.\end{gathered}$

After carrying out higher coframe adaptations and imposing the transformations
(4.2), it follows that

$\displaystyle\mathsf{F}^{\pm}$
$\displaystyle=\textstyle\frac{1}{2\sqrt{2}}(I\mp\tilde{I}),$
$\displaystyle\mathsf{W}^{\pm}$
$\displaystyle=\textstyle{\frac{1}{4}\left(K-\frac{1}{f^{2}}\tilde{K}+(\ln
f)_{,00}+(\ln f)_{,11}+\frac{1}{f}\left((\ln f)_{,03}+(\ln f)_{,30}-(\ln
f)_{,0}(\ln f)_{,3}\right)\right.}$
$\displaystyle\quad\textstyle{\left.+\frac{1}{f^{2}}\left((\ln f)_{,33}-(\ln
f)_{,22}-\left((\ln f)_{,3}\right)^{2}+\left((\ln
f)_{,2}\right)^{2}\right)\right)}\mp\textstyle\frac{1}{2f}(\ln f)_{,12},$

where $(\ln
f)_{,ij}:=\frac{\partial}{\partial\eta^{j}}\frac{\partial}{\partial\eta^{j}}\ln
f.$ For instance if $f$ is constant, $I=\tilde{I},$ then the condition
$f^{2}K=\tilde{K}$ results in a half-flat causal structure which has vanishing
Wsf curvature.

As another example suppose that $f$ is a function on $N,$ i.e.,
$f_{,2}=f_{,3}=0$, which is also known as the warped product of $N$ and
$\tilde{N}.$ Then, the half-flatness condition implies conformal flatness and
is equivalent to the set of equations

$I=\tilde{I},\quad K=\textstyle\frac{1}{f^{2}}\tilde{K}-(\ln f)_{,00}-(\ln
f)_{,11}.$

######  Example 4.7 (Cayley’s cubic scroll).

The causal structure introduced in Example 2.6.2 has ruled fibers
$\mathsf{F}^{+}=0.$ The expression for $\mathsf{W}^{+}$ is quite long.
However, it can be checked that if

$\begin{gathered}c_{111}=c_{111}(x^{1},x^{3}),\qquad
c_{12}=e_{1}(x^{1})e_{2}(x^{2})e_{3}(x^{3}),\\\ c_{2}=c_{2}(x^{2}),\qquad
c_{1}=c_{1}(x^{1},x^{3}),\qquad c_{0}=c_{0}(x^{1},x^{2}),\end{gathered}$

for some functions $e_{1},e_{2}$ and $e_{3}$ then one has a half-flat causal
structure.

###  4.2. Submaximal causal structures of split signature

A causal structure of split signature on a four dimensional manifold that does
not descend to a pseudo-conformal structure is called submaximal if it is not
flat and has the maximum dimension of symmetry algebra among all non-flat
causal structures with non-zero Fubini cubic form.

The purpose of this section is to show the following proposition, making use
of Definition 2.18 of a locally isotrivially flat causal structure.

######  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.

###### Proof.

First, it is shown that the maximum dimension of the symmetry algebra for a
causal structure of split signature with non-zero Fubini cubic form is 8.
Using the structure equations (4.3) or the infinitesimal group actions
(2.39a), one obtains

(4.8a) $\displaystyle\mathrm{d}\mathsf{F}^{\pm}+\mathsf{F}^{\pm}(\pm
3\phi^{1}{}_{2}+\phi_{0}-\phi_{3})\equiv 0,$ (4.8b)
$\displaystyle\mathrm{d}\mathsf{W}^{\pm}+\mathsf{W}^{\pm}(\pm
2\phi^{1}{}_{2}-4\phi_{3})\equiv 0,$

If $\mathsf{F}^{-}\neq 0,$ it can be normalized to one, resulting in the
reduction

(4.9)
$\phi_{3}=\phi_{0}+3\phi^{1}{}_{2}+r_{i}\omega^{i}+r_{\underline{\mu}}\theta^{\mu}$

where the range of $i$ in this section is $\\{0,+,-,3\\}$ and $\mu$ takes
value in the set $\\{+,-\\}.$ The reduction of $\phi_{3}$ gives rise to the
reduction of $\pi_{i}$’s and $\gamma^{\mu}$’s as follows. First, one obtains
the infinitesimal group actions

$\displaystyle\mathrm{d}r_{3}+\pi_{3}\equiv 0,$
$\displaystyle\mathrm{d}r_{\mu}+\textstyle{\frac{1}{2}}\pi_{\mu}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}$
$\displaystyle\mathrm{d}r_{\underline{\mu}}+\textstyle{\frac{1}{2}}\gamma_{\mu}\equiv
0,$

Consequently, $r_{3},r_{\mu}$ and $r_{\underline{\mu}}$ can be translated to
zero resulting in reductions

$\gamma_{\mu}=q_{\mu
i}\omega^{i}+q_{\mu\underline{\nu}}\theta^{\nu},\quad\pi_{3}=p_{3i}\omega^{i}+p_{3\underline{\mu}}\theta^{\mu},\quad\pi_{\mu}=p_{\mu
i}\omega^{i}+p_{\mu\underline{\nu}}\theta^{\nu}.$

Finally, the infinitesimal action

$\mathrm{d}p_{-\underline{+}}+\pi_{0}\equiv
0\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}},$

allows the reduction
$\pi_{0}=p_{0i}\omega^{i}+p_{0\underline{\mu}}\theta^{\mu}.$

As a result of the reductions above the structure bundle is reduced to 8
dimensions. If $\mathsf{F}^{+}\neq 0,$ due to the reduction (4.9), the
infinitesimal action (4.8a) on $\mathsf{F}^{+}$ is expressed as

$\mathsf{F}^{+}-6\mathsf{F}^{+}\phi^{1}{}_{2}\equiv
0,\quad\textrm{mod}\quad\mathcal{I}_{\mathsf{tot}}.$

Thus, the desired causal structure with 8 dimensional symmetry algebra
satisfies $\mathsf{F}^{+}=0.$ Moreover, if the causal structure has non-zero
Wsf curvature, the relations (4.8b) result in further reductions.

Similar computations can be used to show that reduced structure group acts on
$r_{\mu i},$ $r_{\mu\underline{\nu}},$ $p_{ij},$ $p_{i\underline{\mu}}$ via
scaling. Hence, the only non-zero local invariant of the submaximal causal
structure is $\mathsf{F}^{-}$ whose structure equations, after carrying out
the reductions explained above, are given by

(4.10) $\displaystyle\mathrm{d}\omega^{0}$
$\displaystyle=-2\phi_{0}{\,{\wedge}\;}\omega^{0}-\theta^{+}{\,{\wedge}\;}\omega^{-}-\theta^{-}{\,{\wedge}\;}\omega^{+},$
$\displaystyle\mathrm{d}\omega^{+}$
$\displaystyle=-(2\phi_{0}+4\phi^{1}{}_{2}){\,{\wedge}\;}\omega^{+}-\theta^{+}{\,{\wedge}\;}\omega^{3},$
$\displaystyle\mathrm{d}\omega^{-}$
$\displaystyle=-(2\phi_{0}+2\phi^{1}{}_{2}){\,{\wedge}\;}\omega^{-}-\theta^{-}{\,{\wedge}\;}\omega^{3}-\theta^{+}{\,{\wedge}\;}\omega^{+},$
$\displaystyle\mathrm{d}\omega^{3}$
$\displaystyle=-(2\phi_{0}+6\phi^{1}{}_{2}){\,{\wedge}\;}\omega^{3},$
$\displaystyle\mathrm{d}\theta^{+}$
$\displaystyle=2\phi^{1}{}_{2}{\,{\wedge}\;}\theta^{+},$
$\displaystyle\mathrm{d}\theta^{-}$
$\displaystyle=4\phi^{1}{}_{2}{\,{\wedge}\;}\theta^{-}$

The structure equations above are realized by the locally isotrivially flat
causal structure whose null cones are Cayley’s cubic scroll, i.e. its local
defining function in affine coordinates is

$F(x;y)=y^{1}y^{2}+\textstyle{\frac{1}{3}}(y^{1})^{3}$

The conditions $\mathsf{F}^{+}\neq 0$ and $\mathsf{F}^{-}=0$ are no different
and result in an equivalent causal structure with solvable 8-dimensional
symmetry algebra.

∎

##  Appendix A Full structure equations

The Appendix contains the computations needed to associate an
$\\{e\\}$-structure to a causal geometry. At the end, the symmetries and
Bianchi identities for the torsion elements are expressed.

###  A.1. Prolongation and an {e}-structure

Following the first order structure equations derived in (2.37), the method of
equivalence can be used to associate a preferred choice of coframing to a
causal structure. The first step involves prolonging the first order structure
equations.

####  A.1.1. Prolonged structure group

Using the pseudo-connection in (2.37), one finds that the Cartan
characters222222see [Gar89, Olv95] for an account. are

$c_{1}=2n,\quad c_{2}=n-1,\qquad c_{i}=n-i,\ \mathrm{for}\ 3\leq i\leq n-1.$

With respect to the basis
$\\{\omega^{i},\theta^{a},\phi_{0},\phi^{a}{}_{b},\phi_{n},\gamma_{a},\pi_{n},\pi_{a}\\}$,
the prolonged structure group, $G^{(1)}_{1},$ is

(A.1) $\begin{pmatrix}\mathbb{I}_{2n}&0\\\
\mathbf{R}&\mathbb{I}_{p}\end{pmatrix}\quad\mathrm{such\
that}\quad\mathbf{R}=\begin{pmatrix}\mathbf{d}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&\mathbf{d}&0&0\\\ \mathbf{f}&0&0&0\\\ \mathbf{F}&\mathbf{f}&0&\mathbf{d}\\\
\end{pmatrix},$

where $p=\frac{1}{2}(n^{2}+n+4)$, $\mathbf{E}\in\mathbb{R}^{n-1},$ and
$\mathbf{d},\mathbf{f}\in\mathbb{R}.$ The Maurer-Cartan forms of the prolonged
group can be written as

(A.2) $\begin{pmatrix}0&0\\\
\mathbf{M}&0\end{pmatrix}\quad\mathrm{where}\quad\mathbf{M}=\begin{pmatrix}\pi_{0}&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ 0&\pi_{0}&0&0\\\ \sigma&0&0&0\\\
\tau_{a}&\sigma&0&\pi_{0}\\\ \end{pmatrix},$

Since $\dim G^{(1)}_{1}=n+1,$ the relation

$\dim G^{(1)}_{1}<\sum^{n-1}_{j=1}jc_{j},$

holds for all $n\geq 3.$ Thus, equations (2.37) are not involutive, and
because an $\\{e\\}$-structure is not obtained, according to Cartan’s method
of equivalence, one needs to prolong the structure equations.

####  A.1.2. Part of the second order structure equations

To avoid taking too much space for computations and for later use, the
following 2-forms are introduced

(A.3) $\displaystyle\Omega^{0}$
$\displaystyle=\mathrm{d}\omega^{0}+2\phi_{0}{\,{\wedge}\;}\omega^{0}+\theta_{a}{\,{\wedge}\;}\omega^{a},$
$\displaystyle\Omega^{a}$
$\displaystyle=\mathrm{d}\omega^{a}+\gamma^{a}{\,{\wedge}\;}\omega^{0}+(\phi^{a}{}_{b}+(\phi_{0}+\phi_{n})\delta^{a}{}_{b}){\,{\wedge}\;}\omega^{b}+\theta^{a}{\,{\wedge}\;}\omega^{n},$
$\displaystyle\Omega^{n}$
$\displaystyle=\mathrm{d}\omega^{n}+\gamma_{a}{\,{\wedge}\;}\omega^{a}+2\phi_{n}{\,{\wedge}\;}\omega^{n},$
$\displaystyle\Theta^{a}$
$\displaystyle=\mathrm{d}\theta^{a}+\pi^{a}{\,{\wedge}\;}\omega^{0}+\pi_{n}{\,{\wedge}\;}\omega^{a}+(\phi^{a}{}_{b}-(\phi_{0}-\phi_{n})\delta^{a}{}_{b}){\,{\wedge}\;}\theta^{b}$
$\displaystyle\Phi_{0}$
$\displaystyle=\mathrm{d}\phi_{0}+\pi_{0}{\,{\wedge}\;}\omega^{0}+\textstyle{\frac{1}{2}}\pi_{a}{\,{\wedge}\;}\omega^{a}-\textstyle{\frac{1}{2}}\gamma_{a}{\,{\wedge}\;}\theta^{a}$
$\displaystyle\Phi^{a}{}_{b}$
$\displaystyle=\mathrm{d}\phi^{a}{}_{b}+\phi^{a}{}_{c}{\,{\wedge}\;}\phi^{c}_{b}+\theta^{a}{\,{\wedge}\;}\gamma_{b}-\theta_{b}{\,{\wedge}\;}\gamma^{a}-\omega^{a}{\,{\wedge}\;}\pi_{b}+\omega_{b}{\,{\wedge}\;}\pi^{a},$
$\displaystyle\Phi_{n}$
$\displaystyle=\mathrm{d}\phi_{n}+\textstyle{\frac{1}{2}}\pi_{a}{\,{\wedge}\;}\omega^{a}+\pi_{n}{\,{\wedge}\;}\omega^{n}+\textstyle{\frac{1}{2}}\gamma_{a}{\,{\wedge}\;}\theta^{a},$
$\displaystyle\Gamma_{a}$
$\displaystyle=\mathrm{d}\gamma_{a}-\omega_{a}{\,{\wedge}\;}\pi_{0}-\omega^{n}{\,{\wedge}\;}\pi_{a}-(\phi^{b}{}_{a}+(\phi_{0}-\phi_{n})\delta^{b}{}_{a}){\,{\wedge}\;}\gamma_{b},$
$\displaystyle\Pi_{n}$
$\displaystyle=\mathrm{d}\pi_{n}-\theta^{a}{\,{\wedge}\;}\pi_{a}-2\phi_{n}{\,{\wedge}\;}\pi_{n},$
$\displaystyle\Pi_{a}$
$\displaystyle=\mathrm{d}\pi_{a}-\theta_{a}{\,{\wedge}\;}\pi_{0}-(\phi^{b}{}_{a}+(\phi_{0}+\phi_{n})\delta^{b}{}_{a}){\,{\wedge}\;}\pi_{b}-\gamma_{a}{\,{\wedge}\;}\pi_{n}.$

As a result, structure equations (2.37) can be written as

(A.4) $\displaystyle\Omega^{0}$ $\displaystyle=0,$ $\displaystyle\Omega^{a}$
$\displaystyle=-F^{a}{}_{bc}\,\theta^{b}{\,{\wedge}\;}\omega^{c}-K^{a}{}_{b}\,\theta^{b}{\,{\wedge}\;}\omega^{0},$
$\displaystyle\Omega^{n}$
$\displaystyle=-L_{a}\,\theta^{a}{\,{\wedge}\;}\omega^{0},$
$\displaystyle\Theta_{a}$
$\displaystyle=-f_{abc}\,\theta^{b}{\,{\wedge}\;}\omega^{c}+\textstyle{\frac{1}{2}}W_{anbc}\,\omega^{b}{\,{\wedge}\;}\omega^{c}+W_{bncn}\,\omega^{c}{\,{\wedge}\;}\omega^{n}.$

From the Lie algebra of the prolonged group in (A.2), it follows that

$\Pi_{n}=-\sigma{\,{\wedge}\;}\omega^{0}+P_{n},\qquad\Pi_{a}=-\tau_{a}{\,{\wedge}\;}\omega^{0}-\sigma{\,{\wedge}\;}\omega_{a}+P_{a},$

where the 2-forms $P_{n}$ and $P_{a}$ are generated by
$\\{\omega^{i},\theta^{a},\phi_{0},\phi^{a}{}_{b},\phi_{n},\gamma_{a},\pi_{n},\pi_{a}\\}.$
Note that the terms involving the 1-form $\pi_{0}$ in (A.2), are included
within the definition of $\Gamma_{a},\Pi_{n}$ and $\Pi_{a}.$

Differentiating (2.37a), gives

(A.5)
$2\Phi_{0}{\,{\wedge}\;}\omega^{0}+\Theta_{a}{\,{\wedge}\;}\omega^{a}=0.$

As a result of (2.37d) and the symmetries in (2.38), it follows that
$\Theta_{a}{\,{\wedge}\;}\omega^{a}=0,$ and therefore,
$\Phi_{0}=\zeta{\,{\wedge}\;}\omega^{0}$ for some 1-form $\zeta.$ By replacing

$\pi_{0}\longmapsto\pi_{0}-\zeta,$

one arrives at

(A.6) $\Phi_{0}=0.$

Differentiating (2.37c), and using the infinitesimal group action (2.39c), it
is obtained

(A.7a) $\displaystyle\Gamma_{a}={}$
$\displaystyle\textstyle{\frac{1}{2}}W_{0aij}\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{0ab0}\theta^{b}{\,{\wedge}\;}\omega^{0}+\textstyle{\frac{1}{2}}C_{abc}\theta^{b}{\,{\wedge}\;}\theta^{c}$
$\displaystyle-
F_{abc}\gamma^{b}{\,{\wedge}\;}\theta^{c}-L_{a}\pi_{n}{\,{\wedge}\;}\omega^{0}+\tau_{ab}{\,{\wedge}\;}\omega^{b}+\xi_{a}{\,{\wedge}\;}\omega^{n},$
(A.7b) $\displaystyle\Phi_{n}={}$ $\displaystyle
W_{nnij}\omega^{i}{\,{\wedge}\;}\omega^{j}-\textstyle{\frac{1}{2}}\ell_{a}\theta^{a}{\,{\wedge}\;}\omega^{0}+\nu{\,{\wedge}\;}\omega^{n}+\textstyle{\frac{1}{2}}\xi_{a}{\,{\wedge}\;}\omega^{a},$

for some coefficients $E_{nab0},C_{abc},$ where $\ell_{a}=L_{a,n},$ and the
yet undetermined 1-forms $\nu,\xi_{a},$ and $\tau_{ab}=\tau_{ba}$ which are
congruent to zero modulo $\mathcal{I}_{\mathsf{tot}}.$ The reason for using
$W_{nnij}$ instead of $\textstyle{\frac{1}{2}}W_{nnij}$ in (A.7b) becomes
clear when the replacements (A.13) are imposed.

By differentiating (2.37b), it follows that

$\xi_{a}=k_{ab}\,\theta^{b},$

where $k_{ab}:=K_{ab;n}.$

Now $\Phi^{a}{}_{b}$ can be determined as a result of differentiating (2.37b)
and replacing $\Phi_{0}$ and $\Phi_{n}$ by the expressions (A.6) and (A.7b).
Using the infinitesimal group action on $K_{ab},F_{abc}$ given by (2.39a) and
(2.39b), and their symmetries in (2.38), and (2.37d), the derivative of
(2.37b) yields

(A.8) $\displaystyle 0$
$\displaystyle\equiv\left(\Phi^{a}{}_{b}+\Phi_{n}\delta^{a}{}_{b}\right){\,{\wedge}\;}\omega^{b}+K_{ab}\,\theta^{b}{\,{\wedge}\;}\theta_{c}{\,{\wedge}\;}\omega^{c}-F_{abc}\,f^{b}{}_{de}\,\theta^{d}{\,{\wedge}\;}\omega^{e}{\,{\wedge}\;}\omega^{c}$
$\displaystyle\quad+F_{abc;d}\,\omega^{d}{\,{\wedge}\;}\theta^{b}{\,{\wedge}\;}\omega^{c}+F_{abc;{\underline{d}}}\,\theta^{d}{\,{\wedge}\;}\theta^{b}{\,{\wedge}\;}\omega^{c}+F_{abc}\,F^{c}{}_{de}\,\theta^{b}{\,{\wedge}\;}\theta^{d}{\,{\wedge}\;}\omega^{e}$
$\displaystyle\quad+\textstyle{\frac{1}{2}}F_{abc}\,W^{b}{}_{nde}\,\omega^{d}{\,{\wedge}\;}\omega^{e}{\,{\wedge}\;}\omega^{c},\quad\textrm{mod}\quad
I\\{\omega^{0},\omega^{n}\\}.$

Multiplied by the $(n-1)$-form
$\eta=\theta^{1}{\,{\wedge}\;}\dots\,{\wedge}\;\theta^{n-1}$, the above
equation results in

$\left(\Phi^{a}{}_{b}+\Phi_{n}\delta^{a}{}_{b}\,\right){\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\eta\equiv\textstyle{\frac{1}{2}}-F_{aeb}W^{b}{}_{ncb}\,\omega^{c}{\,{\wedge}\;}\omega^{d}{\,{\wedge}\;}\omega^{b}{\,{\wedge}\;}\eta,\quad\textrm{mod}\quad
I\\{\omega^{0},\omega^{n}\\}.$

By Cartan’s lemma, it follows that

$\left(\Phi^{a}{}_{b}+\Phi_{n}\delta^{a}{}_{b}\right){\,{\wedge}\;}\eta\equiv-\left(\zeta^{a}{}_{bc}{\,{\wedge}\;}\omega^{c}+\textstyle{\frac{1}{2}}F_{abe}W^{e}{}_{ncd}\omega^{c}{\,{\wedge}\;}\omega^{d}\right){\,{\wedge}\;}\eta,\quad\textrm{mod}\quad
I\\{\omega^{0},\omega^{n}\\},$

for some 1-form $\zeta^{a}{}_{bc},$ where
$\zeta^{a}{}_{bc}\equiv\zeta^{a}{}_{cb}$ modulo $\mathcal{I}_{\mathsf{tot}}.$

Fixing $c=C,$ after taking the wedge product of
$\Phi^{a}{}_{b}+\Phi_{n}\delta^{a}{}_{b}$ with all the 1-forms $\omega^{i}$,
such that $i\neq C$, and using (A.7b), the result is

$\Phi^{a}{}_{b}{\,{\wedge}\;}\omega^{0}{\,{\wedge}\;}\dots\,{\wedge}\;\widehat{\omega^{C}}{\,{\wedge}\;}\dots\,{\wedge}\;\omega^{n}{\,{\wedge}\;}\eta=\zeta^{a}{}_{bC}{\,{\wedge}\;}\omega^{C}{\,{\wedge}\;}\omega^{0}{\,{\wedge}\;}\dots\,{\wedge}\;\widehat{\omega^{C}}{\,{\wedge}\;}\dots\,{\wedge}\;\omega^{n}{\,{\wedge}\;}\eta.$

Carrying out the same procedure for $\Phi^{b}{}_{a}$, using
$\Phi_{ab}+\Phi_{ba}=0,$ it follows that

$(\zeta_{abC}+\zeta_{baC}){\,{\wedge}\;}\omega^{C}{\,{\wedge}\;}\omega^{0}{\,{\wedge}\;}\dots\,{\wedge}\;\widehat{\omega^{C}}{\,{\wedge}\;}\dots\,{\wedge}\;\omega^{n}{\,{\wedge}\;}\eta=0,$

and therefore, $\zeta_{abc}+\zeta_{bac}\equiv 0$ modulo
$\mathcal{I}_{\mathsf{tot}}.$ However, since
$\zeta^{a}{}_{bc}\equiv\zeta^{a}{}_{cb}$, one obtains $\zeta^{a}{}_{bc}\equiv
0$ modulo $\mathcal{I}_{\mathsf{tot}}.$

Consequently, equation

(A.9)
$\Phi_{ab}=\textstyle{\frac{1}{2}}W_{abij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{abci}\,\theta^{c}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C_{abcd}\theta^{c}{\,{\wedge}\;}\theta^{d}+\lambda_{ab}{\,{\wedge}\;}\omega^{0}+\kappa_{ab}{\,{\wedge}\;}\omega^{n},$

is obtained where $\lambda_{ab}=-\lambda_{ba},\kappa_{ab}=-\kappa_{ba}$ and
$\lambda_{ab},\kappa_{ab}\equiv 0$ modulo $\mathcal{I}_{\mathsf{tot}}.$

Differentiating (2.37b), and using (A.7), and (A.9), it is straightforward to
find

(A.10)
$\begin{gathered}\kappa_{ab}=0,\quad\nu=0,\quad,\tau_{ab}=E_{0acb}\,\theta^{c}+K_{ab}\,\pi_{n}-F_{ab}{}^{c}\,\pi_{c}\quad\lambda_{ab}=0,\\\
\end{gathered}$

for some $E_{0abc}$.

Equations (A.10), (A.7), (A.9) and (A.6) can be combined to give

(A.11) $\displaystyle\Phi_{0}$ $\displaystyle=0,$ $\displaystyle\Phi_{ab}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{abij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{abci}\,\theta^{c}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C_{abcd}\,\theta^{c}{\,{\wedge}\;}\theta^{d}$
$\displaystyle\Phi_{n}$
$\displaystyle=W_{nnij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}-\textstyle{\frac{1}{2}}\ell_{a}\,\theta^{a}{\,{\wedge}\;}\omega^{0}+\textstyle{\frac{1}{2}}k_{ab}\,\theta^{b}{\,{\wedge}\;}\omega^{a},$
$\displaystyle\Gamma_{a}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{0aij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{0abi}\,\theta^{b}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C_{abc}\,\theta^{b}{\,{\wedge}\;}\theta^{c}$
$\displaystyle\quad-
F_{abc}\,\gamma^{b}{\,{\wedge}\;}\theta^{c}-L_{a}\,\pi_{n}{\,{\wedge}\;}\omega^{0}+K_{ab}\,\pi_{n}{\,{\wedge}\;}\omega^{b}-F_{ab}{}^{c}\,\pi_{c}{\,{\wedge}\;}\omega^{b}.$

The quantities $W_{ijkl},$ $E_{ijkl},C_{abcd}$ and $C_{abc}$ are not expressed
until the replacement (A.13) is made which makes the computation slightly
easier.

####  A.1.3. Reduction of the prolonged group

The infinitesimal action of $G^{(1)}_{1}$ on $W_{nn0n}$ and $W_{nn0a}$ is
given by

$\mathrm{d}W_{nn0n}+\textstyle{\frac{1}{2}}\sigma\equiv
0\qquad\mathrm{d}W_{nn0a}+\textstyle{\frac{1}{4}}\tau_{a}\equiv 0$

modulo the ideal generated by the 1-forms
$\\{\omega^{i},\theta_{a},\phi_{0},\phi_{n},\phi^{a}{}_{b},\gamma_{a},\pi_{n},\pi_{a}\\}_{i=0,a=1}^{n,\
n-1}.$

Consequently, the normalizations $W_{nn0n}=0,W_{nn0a}=0,$ reduce $G^{(1)}_{1}$
to the group $G^{(1)}_{2}$ containing the one parameter $\mathbf{d}$ in (A.1),
which corresponds to the 1-form $\pi_{0}.$ As a result, the expression for
$\Phi_{n}$ changes to

(A.12)
$\Phi_{n}=W_{nnab}\,\omega^{a}{\,{\wedge}\;}\omega^{b}+2W_{nnan}\,\omega^{a}{\,{\wedge}\;}\omega^{n}-\textstyle{\frac{1}{2}}\ell_{a}\,\theta^{a}{\,{\wedge}\;}\omega^{0}+\textstyle{\frac{1}{2}}k_{ab}\,\theta^{b}{\,{\wedge}\;}\omega^{a}$

One sees that the action of the parameter $\mathbf{d}$ on the torsion
coefficients in the 2-forms $\Theta_{0},\cdots,\Phi_{0},\cdots,\Gamma_{n-1}$ a
priori cannot be used to reduce the prolonged structure group to the trivial
group, unless certain non-vanishing conditions are assumed.

The only part left, is to see whether any reduction is possible via the
torsion coefficients in $\Pi_{n},\cdots,\Pi_{0}.$

####  A.1.4. An {$e$}-structure

In order to make computing $\Pi_{0},\cdots,\Pi_{n}$ easier the replacements

(A.13)
$\pi_{0}\mapsto\pi_{0}+\textstyle{\frac{1}{2}}\ell_{a}\,\theta^{a},\quad\pi_{a}\mapsto\pi_{a}-\textstyle{\frac{1}{2}}k_{ab}\,\theta^{b}+W_{nnab}\,\omega^{b},\quad\pi_{n}\mapsto\pi_{n}-2W_{nnan}\,\omega^{a}.$

are imposed which result in

$\Phi_{0}+\Phi_{n}=0.$

Similarly to Section A.1.2, by differentiating the 2-forms
$\Gamma_{a},\Phi^{a}{}_{b}$ in (A.11), $\Phi_{n}$ in (A.12) and
$\Phi_{0}=-\Phi_{n},$ and carrying out tedious but straightforward
calculations, it follows that

(A.14) $\displaystyle\Pi_{n}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{nij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{nai}\,\theta^{a}{\,{\wedge}\;}\omega^{i}+\sigma_{ni}{\,{\wedge}\;}\omega^{i},$
$\displaystyle\Pi_{a}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{aij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{abi}\,\theta^{b}{\,{\wedge}\;}\omega^{i}-F_{ab}{}^{c}\,\pi_{c}{\,{\wedge}\;}\theta^{b}+\sigma_{ai}{\,{\wedge}\;}\omega^{i},$

where $\sigma_{ab}=\sigma_{ba},$ $\sigma_{na}=\sigma_{an}$ are congruent to
zero modulo $\mathcal{I}_{\mathsf{tot}}.$

The expressions for the torsion coefficients are listed in (A.20). It turns
out that without assuming certain non-vanishing conditions on the torsion
coefficients, the infinitesimal action of the parameter $\mathbf{d}$ on the
torsion terms does not reduce the prolonged group.

Since, the prolongation of the reduced group $G^{(1)}_{2}$ is trivial, it
follows that the set of 1-forms

$\\{\omega^{i},\theta^{a},\phi_{0},\phi^{a}{}_{b},\phi_{n},\gamma_{a},\pi_{i}\\}_{i=0,a=1}^{n,\
n-1}$

constitutes an {e}-structure for $\mathcal{C}.$

####  A.1.5. The third order structure equations

Similarly to (A.3), defining the 2-form

(A.15)
$\Pi_{0}=\mathrm{d}\pi_{0}-2\phi_{0}{\,{\wedge}\;}\pi_{0}-\gamma^{a}{\,{\wedge}\;}\pi_{a},$

helps making the expressions more compact.

It is a matter of computation, similar to Section A.1.4, to find

(A.16) $\displaystyle\Pi_{0}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{0ij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{0ai}\,\theta^{a}{\,{\wedge}\;}\omega^{i}-K_{ab}\,\pi^{a}{\,{\wedge}\;}\theta^{b}-L_{a}\pi_{n}{\,{\wedge}\;}\theta^{a}+\sigma_{0i}{\,{\wedge}\;}\omega^{i},$

where 1-forms $\sigma_{ij}$ defined in (A.14) satisfy
$\sigma_{ij}=\sigma_{ji}$. Note that the symmetry of $\sigma_{ij}$ is a result
of imposing $\Phi_{0}+\Phi_{n}=0.$ Expressions for $\sigma_{ij}$ are listed in
(A.22).

####  A.1.6. Full structure equations

The full structure equations after replacement (A.13) using the 2-forms (A.3)
are expressed as follows.

(A.17) $\displaystyle\Omega^{0}$ $\displaystyle=0,$ $\displaystyle\Omega^{a}$
$\displaystyle=-F^{a}{}_{bc}\,\theta^{b}{\,{\wedge}\;}\omega^{c}-K^{a}{}_{b}\,\theta^{b}{\,{\wedge}\;}\omega^{0},$
$\displaystyle\Omega^{n}$
$\displaystyle=-L_{a}\,\theta^{a}{\,{\wedge}\;}\omega^{0},$
$\displaystyle\Theta_{a}$
$\displaystyle=-f_{abc}\,\theta^{b}{\,{\wedge}\;}\omega^{c}-\textstyle{\frac{1}{2}}k_{ab}\theta^{b}{\,{\wedge}\;}\omega^{0}-W_{nnab}\omega^{0}{\,{\wedge}\;}\omega^{b}$
$\displaystyle\
+\textstyle{\frac{1}{2}}W_{anbc}\,\omega^{b}{\,{\wedge}\;}\omega^{c}+W_{bncn}\,\omega^{c}{\,{\wedge}\;}\omega^{n},$
$\displaystyle\Phi_{0}$
$\displaystyle=-\textstyle{\frac{1}{2}}W_{nnab}\omega^{a}{\,{\wedge}\;}\omega^{b}+\textstyle{\frac{1}{2}}\ell_{a}\theta^{a}{\,{\wedge}\;}\omega^{0}-\textstyle{\frac{1}{4}}k_{ab}\theta^{b}{\,{\wedge}\;}\omega^{a},$
$\displaystyle\Phi_{ab}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{abij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{abci}\,\theta^{c}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C_{abcd}\,\theta^{c}{\,{\wedge}\;}\theta^{d}$
$\displaystyle\Phi_{n}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{nnab}\,\omega^{a}{\,{\wedge}\;}\omega^{b}-\textstyle{\frac{1}{2}}\ell_{a}\,\theta^{a}{\,{\wedge}\;}\omega^{0}+\textstyle{\frac{1}{4}}k_{ab}\,\theta^{b}{\,{\wedge}\;}\omega^{a},$
$\displaystyle\Gamma_{a}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{0aij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{0abi}\,\theta^{b}{\,{\wedge}\;}\omega^{i}+\textstyle{\frac{1}{2}}C_{abc}\,\theta^{b}{\,{\wedge}\;}\theta^{c}$
$\displaystyle\quad-
F_{abc}\,\gamma^{b}{\,{\wedge}\;}\theta^{c}-L_{a}\,\pi_{n}{\,{\wedge}\;}\omega^{0}+K_{ab}\,\pi_{n}{\,{\wedge}\;}\omega^{b}-F_{ab}{}^{c}\,\pi_{c}{\,{\wedge}\;}\omega^{b},$
$\displaystyle\Pi_{n}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{nij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{nai}\,\theta^{a}{\,{\wedge}\;}\omega^{i}+\sigma_{ni}{\,{\wedge}\;}\omega^{i},$
$\displaystyle\Pi_{a}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{aij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{abi}\,\theta^{b}{\,{\wedge}\;}\omega^{i}-F_{ab}{}^{c}\,\pi_{c}{\,{\wedge}\;}\theta^{b}+\sigma_{ai}{\,{\wedge}\;}\omega^{i},$
$\displaystyle\Pi_{0}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{0ij}\,\omega^{i}{\,{\wedge}\;}\omega^{j}+E_{0ai}\,\theta^{a}{\,{\wedge}\;}\omega^{i}-K_{ab}\,\pi^{a}{\,{\wedge}\;}\theta^{b}-L_{a}\pi_{n}{\,{\wedge}\;}\theta^{a}+\sigma_{0i}{\,{\wedge}\;}\omega^{i},$

The expressions of the quantities on the right hand side are given in (A.20),
(A.22) and (A.23). When differentiation (2.37b), the vanishing of 3-forms
$\theta^{b}{\,{\wedge}\;}\theta^{c}{\,{\wedge}\;}\omega^{d},\theta^{b}{\,{\wedge}\;}\theta^{c}{\,{\wedge}\;}\omega^{0}$
combined with the expression of $C_{abcd},$ and $C_{abc}$ in (A.20), yield the
relations

(A.18a) $\displaystyle F_{abc,{\underline{d}}}-F_{abd,{\underline{c}}}$
$\displaystyle=\textstyle{\frac{1}{2}}(K_{ac}\varepsilon_{bd}+K_{bc}\varepsilon_{ad}-K_{ad}\varepsilon_{bc}-K_{bd}\varepsilon_{ac}),$
(A.18b) $\displaystyle K_{ab,{\underline{c}}}-K_{ac,{\underline{b}}}$
$\displaystyle=2(\varepsilon_{ab}L_{c}-\varepsilon_{ac}L_{b})+F_{abd}K^{d}{}_{c}-F_{acd}K^{d}{}_{b},$

Moreover, the vanishing of the 3-forms
$\theta_{a}{\,{\wedge}\;}\theta_{b}{\,{\wedge}\;}\omega^{0},$ in the exterior
derivative of (2.37c) implies $L_{a;{\underline{b}}}=L_{b;{\underline{a}}}.$
This equation together with the contraction of Equations (A.18) give the
following Bianchi identities among the fiber invariants of $\mathcal{C}$

(A.19a) $\displaystyle K_{ab}$
$\displaystyle=\textstyle{\frac{2}{n-1}}F^{c}{}_{ab;{\underline{c}}},$ (A.19b)
$\displaystyle L_{a}$
$\displaystyle=\textstyle{\frac{1}{2(n-2)}}\left(F_{abc}K^{bc}-K_{a}{}^{b}{}_{;{\underline{b}}}\right),$
(A.19c) $\displaystyle L_{a;{\underline{b}}}$
$\displaystyle=L_{b;{\underline{a}}}.$

###  A.2. Quantities in (A.17)

The expression for the torsion coefficients are as follows:

(A.20) $\displaystyle E_{abc0}$
$\displaystyle=\textstyle{\frac{1}{2}}\left(K_{bc;a}-K_{ac;b}-K_{ad}f^{d}{}_{bc}+K_{bd}f^{d}{}_{ac}\right),\qquad
E_{abcn}=0$ $\displaystyle E_{abcd}$
$\displaystyle=F_{bcd;a}-F_{acd;b}+F_{bde}f^{e}{}_{ca}-F_{ade}f^{e}{}_{cb}$
$\displaystyle C_{abcd}$
$\displaystyle=F_{bce}F^{e}{}_{ad}-F_{ace}F^{e}{}_{bd}+\textstyle{\frac{1}{2}}\left(K_{bc}\varepsilon_{ad}+K_{ad}\varepsilon_{bc}-K_{ac}\varepsilon_{bd}-K_{bd}\varepsilon_{ac}\right)$
$\displaystyle E_{0ab0}$ $\displaystyle=-L_{b;a}-L_{d}f^{d}{}_{ab},\qquad
E_{0abn}=\textstyle{\frac{1}{2}}k_{ab},\qquad
C_{abc}=L_{c}\varepsilon_{ab}-L_{b}\varepsilon_{ac}$ $\displaystyle E_{0abc}$
$\displaystyle=\textstyle{\frac{1}{2}}\left(K_{ab;c}+K_{bc;a}+K_{ad}f^{d}{}_{bc}+K_{cd}f^{d}{}_{ab}\right)-F_{abc;0}-\textstyle{\frac{1}{2}}k_{bd}F^{d}{}_{ac}$

The quantities $W_{ijkl}=-W_{ijlk}$ satisfy the following identities:

(A.21) $\begin{gathered}W_{an0b}=-W_{nnab},\quad W_{an0n}=0,\quad
W_{anbn}=W_{bnan},\quad W{}^{a}{}_{nan}=0,\quad W_{[a|n|bc]}=0,\quad\\\
W_{abcn}=W_{cnab}+F_{acd}W^{d}{}_{nbn}-F_{bcd}W^{d}{}_{nan},\quad
W_{a[bcd]}+F_{ae[b|}W_{en|cd]}=0,\\\
W_{abcd}-W_{cdab}=F_{aec}W^{e}{}_{nbd}-F_{aed}W^{e}{}_{bnc}-F_{bec}W^{e}{}_{nad}+F_{bed}W^{e}{}_{nac}\\\
\begin{aligned}
W_{ab0c}&=-W_{0cab}+F_{acd}W^{d}{}_{n0b}-F_{bcd}W^{d}{}_{n0a}+\textstyle{\frac{1}{2}}K_{cd}W^{d}{}_{nba}\\\
&\quad+\textstyle{\frac{1}{2}}\left(K_{bd}W^{d}{}_{nca}-K_{ad}W^{d}{}_{ncb}\right)\end{aligned}\\\
W_{0abn}=-W_{nnab}-K_{(a|d}W^{d}{}_{n|b)n},\quad
W_{ab0n}=-2W_{nnab}+K_{[a|d}W^{d}{}_{n|b]n}\\\ W_{0[abc]}=0,\quad
W_{0a0n}=L_{c}W^{c}{}_{nan},\quad
W_{0[a|0|b]}=\textstyle{\frac{1}{2}}L_{c}W^{c}{}_{nab},\quad
W_{0}{}^{a}{}_{0a}=0.\end{gathered}$

According to the symmetries above, the independent components of $W_{ijkl}$
are $W_{abcd},$ $W_{anbc},$ $W_{0abc},$ $W_{nabn},$ $W_{0a0b}$ and $W_{nnab}.$
It is shown in the A.3 that $W_{ijkl}$ is generated by $W_{anbn}.$

The 1-forms $\sigma_{ij}=\sigma_{ji}$ are

(A.22) $\displaystyle\sigma_{nn}$ $\displaystyle=0,$
$\displaystyle\sigma_{an}$ $\displaystyle=W_{anbn}\gamma^{b},$
$\displaystyle\sigma_{a0}$
$\displaystyle=3W_{nnab}\gamma^{b}-\ell_{a}\pi_{n}-\textstyle{\frac{1}{2}}k_{ab}\pi^{b},$
$\displaystyle\sigma_{0n}$ $\displaystyle=0,$ $\displaystyle\sigma_{ab}$
$\displaystyle=-W_{(a|n|b)c}\gamma^{c}+\textstyle{\frac{1}{2}}k_{ab}\pi_{n}-f_{abc}\pi^{c},$
$\displaystyle\sigma_{00}$
$\displaystyle=\textstyle\frac{1}{n-2}\left(2W_{0bba}+K^{cd}W_{dnca}-L^{c}W_{cnan}\right)\gamma^{a}$
$\displaystyle\
-\textstyle\frac{2}{n-1}L^{a}{}_{;a}\pi_{n}-\textstyle\frac{1}{n-1}\left(K^{ac}{}_{;a}+K_{ab}f^{abc}\right)\pi_{c}.$

The quantities $E_{iaj}=E_{jai}$ are given by

(A.23) $\displaystyle E_{nan}$
$\displaystyle=\textstyle\frac{2}{n-1}\varepsilon^{bc}W_{0bca},$
$\displaystyle E_{nab}$
$\displaystyle=3W_{nnab}+\textstyle{\frac{1}{2}}k_{ab;n}+K_{(a|d}W^{d}{}_{n|b)n},$
$\displaystyle E_{0ab}$
$\displaystyle=-W_{0a0b}+2W_{nnbd}K^{d}{}_{a}-\ell_{a;b}-\ell_{d}f^{d}{}_{ab}-\textstyle{\frac{1}{2}}k_{ab;0}-\textstyle{\frac{1}{4}}k_{a}{}^{c}k_{bc},$
$\displaystyle E_{abc}$
$\displaystyle=W_{(a|b0|a)}-W_{(a|n|c)d}K_{db}-W_{ancn}L_{b}+\textstyle{\frac{1}{2}}K_{(a|b;n|c)}-F_{abc;n0}$
$\displaystyle\qquad-\textstyle{\frac{1}{2}}F_{acd;n}K_{db;n}-\textstyle{\frac{1}{2}}K_{(a|d;n}F_{db|c);n}+(\textstyle{\frac{1}{2}}L_{c}W_{cnbn}+\textstyle{\frac{1}{2}}\ell_{b;n})\varepsilon_{ac}$
$\displaystyle E_{0an}$
$\displaystyle=-\textstyle{\frac{1}{2}}L_{c}W_{cnan}-\textstyle{\frac{1}{2}}\ell_{a;n},$
$\displaystyle E_{0b0}$
$\displaystyle=\textstyle\frac{1}{n-1}\left(-2F_{abc}W_{0d0c}+W_{0aad}K_{db}-K^{ad}W_{nand}L_{b}+L_{a}E_{nba}-L_{a;d}f^{ad}{}_{b}\right.$
$\displaystyle\qquad-\left(f^{a}{}_{bc}L^{c}-\varepsilon^{ca}L_{b;c}\right)_{;a}L^{c}f_{acd}f^{ad}{}_{b}-\textstyle{\frac{1}{2}}\left(K^{a}{}_{d;a}+K^{ae}f_{ade}\right)k^{d}{}_{b}$
$\displaystyle\qquad-\left.\left(K^{a}{}_{b;a}+K^{ad}f_{adb}\right)_{;0}+L^{d}W_{nndb}\right)$

The quantities $W_{ijk}$ satisfy

$W_{ijk}+W_{kij}+W_{jki}=0,\quad W_{ijk}=-W_{ikj}$

and are obtained from $W_{ijkl},F_{abc}$ as follows

(A.24) $\displaystyle W_{nan}$
$\displaystyle=\textstyle-\frac{1}{n-2}(W_{anba;n}-W_{anbn;a}-F_{adb;n}W_{dnan}),$
$\displaystyle W_{nab}$ $\displaystyle=\textstyle
2W_{nnab}+k_{[a|c}W_{cn|b]n}$ $\displaystyle W_{n0n}$
$\displaystyle=\textstyle\frac{1}{(n-2)(n-1)}(W_{cdcd;n}-E_{cdec}W_{dnen}-E_{cded}W_{encn}-W_{adan;d}+W_{addn;a})$
$\displaystyle\quad\textstyle+\frac{1}{2(n-1)}k_{cd}W_{ancn}.$ $\displaystyle
W_{abn}$
$\displaystyle=\varepsilon_{ab}W_{n0n}+W_{anbn;0}-W_{nnab;n}-\textstyle{\frac{1}{2}}k_{ac}W_{cnbn}$
$\displaystyle\textstyle W_{a0n}$
$\displaystyle=\textstyle\frac{1}{n-2}\left(-W_{nnba;b}-W_{bnba;0}+\textstyle{\frac{1}{2}}k_{ed}W_{dnea}+W_{0dda;n}-W_{0ddn;a}\right.$
$\displaystyle\quad\left.+W_{0dan;d}-E_{eda}W_{dnen}+E_{ede}W_{dnan}+E_{0edn}W_{dnea}\right.$
$\displaystyle\quad\left.-K_{ad}W_{ndn}+F_{ade}W_{den}\right)$ $\displaystyle
W_{n0a}$
$\displaystyle=\textstyle{\frac{1}{2}}W_{a0n}-\textstyle{\frac{1}{2}}\ell_{b}W_{bnan},$
$\displaystyle W_{abc}$
$\displaystyle=\textstyle\varepsilon_{ab}W_{n0c}-\varepsilon_{ac}W_{n0b}-W_{nnab;c}+W_{nnac;b}+W_{anbc;0}$
$\displaystyle\quad-\textstyle{\frac{1}{2}}K_{ad}W_{dnbc}+f_{acd}W_{nndb}-f_{abd}W_{nndc},$
$\displaystyle W_{0ab}$
$\displaystyle=\textstyle-2W_{nnab;0}+\ell_{c}W_{cnab}-k_{[a|d}W_{nnc|b},$
$\displaystyle W_{00n}$
$\displaystyle=\textstyle\frac{1}{(n-1)(n-2)}\left(W_{ab0[b;a]}-\textstyle{\frac{1}{2}}W_{abab;0}+\textstyle{\frac{1}{2}}E_{abe0}W_{enba}+\textstyle{\frac{1}{2}}E_{abe0}W_{nnea}\right)$
$\displaystyle\quad\textstyle+\frac{1}{n-1}\left(E_{0adn}W_{nnda}-E_{0ad0}W_{dnad}+L_{a}W_{nan}\right),$
$\displaystyle W_{a0b}$
$\displaystyle=\varepsilon_{ab}W_{00n}+W_{0a0b;n}+W_{0abn;0}-E_{0adn}W_{nndb},$
$\displaystyle\quad+E_{0ad0}W_{dnbn}-L_{a}W_{nbn}-K_{ab}W_{n0n}+F_{abc}W_{c0n},$
$\displaystyle W_{00b}$
$\displaystyle=-W_{0aab;0}+W_{0a0b;a}+E_{0ab0}W_{dba}+E_{0ada}W_{d0b}$
$\displaystyle\quad-E_{0adb}W_{d0a}-L_{a}W_{nba}-K_{ab}W_{n0a}+F_{abd}W_{d0a}$

In the above expressions the indices are lowered using $\varepsilon^{ab}$ and
the summation is taken over repeated indices.

###  A.3. Bianchi identities for $W_{ijkl}$

The following relations hold among the torsion elements $W_{ijkl}$ in (A.17).

(A.25) $\displaystyle W_{anbc}$
$\displaystyle=\textstyle{\frac{2}{3}}\left(W_{an[b|n;|{\underline{c}}]}+F_{a[c}{}^{d}W_{b]ndn}\right)$
$\displaystyle\quad+\textstyle\frac{4}{3(n+2)}\left(\varepsilon_{a[c|}W^{d}{}_{n|b]n;{\underline{d}}}+\varepsilon_{a[b}F^{de}{}_{c]}W_{dnen}\right)$
$\displaystyle W_{nnab}$
$\displaystyle=\textstyle\frac{1}{2(n+3)}\left(-W^{d}{}_{nab;{\underline{d}}}+F^{cd}{}_{a}W_{cndb}-F^{cd}{}_{b}W_{cnda}\right),$
$\displaystyle W_{abcd}$
$\displaystyle=W_{[a|ncd;|{\underline{b}}]}-f^{e}{}_{ac}f_{bde}+f^{e}{}_{ad}f_{bce}+W_{ane[c}F_{b]d}{}^{e}-W_{dne[c}F_{b]a}{}^{e}$
$\displaystyle\qquad-\varepsilon_{a[c|}\left(W_{nn|d]b}-\textstyle{\frac{1}{2}}k_{|d]b;n}+W_{d]nen}K_{b}{}^{e}\right)$
$\displaystyle\qquad+\varepsilon_{b[c}\left(W_{nnd]a}-\textstyle{\frac{1}{2}}k_{|d]a;n}+W_{d]nen}K_{a}{}^{e}\right)$
$\displaystyle W_{0abc}$
$\displaystyle=2W_{nnbc;{\underline{a}}}-4W_{nnd[c}F_{b]a}{}^{d}+k_{a[b;c]}+f^{d}{}_{a[c}k_{b]d}$
$\displaystyle W_{0a0b}$
$\displaystyle=\textstyle\frac{-1}{n-2}W_{0abc;{\underline{d}}}\varepsilon^{cd}-\frac{1}{n-2}W_{0ecb}F^{ec}{}_{a}-\varepsilon^{cd}E_{0cd}\varepsilon_{ab}+E_{0ab}$
$\displaystyle\
+\varepsilon^{cd}(E_{0acd;b}-E_{0acb;d})+E_{0aec}f^{ec}{}_{b}-F_{abe}E^{ec}{}_{c}+F_{aec}E^{ec}{}_{b}$
$\displaystyle\ +L^{c}W_{nbca}+K_{ab}E_{nc}{}^{c}-K_{a}{}^{d}E_{ndb}$

It follows from the relations above that the infinitesimal action of the
structure group on $W_{ijkl}$ is obtained from the appropriate chains of
coframe derivatives of $W_{anbn}.$

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