Differential Geometric Aspects of Causal Structures

This article is concerned with causal structures, which are defined as a field of tangentially non-degenerate projective hypersurfaces in the projectivized tangent bundle of a manifold. The local equivalence problem of causal structures on manifolds of dimension at least four is solved using Cartan's method of equivalence, leading to an $\{e\}$-structure over some principal bundle. It is shown that these structures correspond to parabolic geometries of type $(D_n,P_{1,2})$ and $(B_{n-1},P_{1,2})$, when $n\geq 4$, and $(D_3,P_{1,2,3})$. The essential local invariants are determined and interpreted geometrically. Several special classes of causal structures are considered including those that are a lift of pseudo-conformal structures and those referred to as causal structures with vanishing Wsf curvature. A twistorial construction for causal structures with vanishing Wsf curvature is given.

In the General Theory of Relativity, space-time is represented by a four dimensional manifold M equipped with a Lorentzian metric g. An important feature of a Lorentzian structure is its associated field of null cones in T M given by the set of vectors v ∈ T M satisfying g(v, v) = 0. As a result, it is possible to define a relation of causality between the points of M, i.e., a partial ordering x ≺ y defined by the property that x can be joined to y by a curve that is either time-like or null. If x and y represent two events in space-time then x ≺ y means that the occurrence of x has an effect on y. In [KP67] Kronheimer and Penrose examined causality in a manifold M of dimension n on an axiomatic basis. The starting point of their study is a continuous assignment of null cones in the tangent space of each point of M so that it is possible to define the set of points that are inside, outside and on the null cone in each tangent space. 1 The importance of the causal relationship between points of space-time became more significant in the context of the singularity theorems of Penrose and Hawking [Pen65,Pen72,HE73]. The study of causal relations in spaces equipped with a field of convex null cones in T M received a major contribution from the Alexandrov school in Russia under the name of chronogeometry [Ale67,Gut95], i.e., a geometry determined by the chronological relations between the points of the manifold. The objective was to use the principles of causality and symmetry to give an axiomatic treatment of special relativity. In [Seg76], Segal attempted to give such a treatment for general relativity (see also [DS01,Lev93,Wor84,Tau77,EPS72].) Subsequently, structures defined by assigning a convex cone in each tangent space, termed as causal structures, came to play a role in the Lie theory of semi-groups [HÓ97], causal boundaries [GPS05], and hyperbolic systems of partial differential equations [Sch13]. Rather than to investigate causality relations between the points of a manifold with a given causal structure, the main idea of this article is to gain a deeper understanding of the geometry of causal structures through the solution of the Cartan equivalence problem. This approach enables one to construct a complete set of local invariants and also to obtain some results on symmetries of causal structures. A global rigidity result can also be obtained through this approach.
Before going any further, a more precise definition of causal structures is in order. Given an (n + 1)-dimensional manifold M, with n ≥ 3, a causal structure (M, C) of signature (p + 1, q + 1) with p, q ≥ 0, and p + q = n − 1 is given by a sub-bundle of the projectivized tangent bundle π ∶ C ⊂ PT M → M, whose fibers C x ⊂ PT x M are projective hypersurfaces 2 with projective second fundamental form of signature (p, q) (see Definition 2.4). Note that in this definition the convexity of the null cones C x is not assumed.
In order to describe a causal structure in an open set U ⊂ T M, one can use a defining function L ∶ T M → R, (x i ; y j ) ↦ L(x i ; y j ), where (x 0 , . . . , x n ) are coordinates for M and (y 0 , . . . , y n ) are fiber coordinates. The function L is assumed to be homogeneous of some degree r 1 in the fiber variables and is referred to as a Lagrangian for the causal structure. Then, the vanishing set of L over U coincides with U ∩Ĉ whereĈ ⊂ T M is the cone over C ⊂ PT M. The functions L(x i ; y i ) and S(x i ; y i )L(x i ; y i ), where S(x i ; y i ) is nowhere vanishing and homogeneous of degree r 2 in y i 's, define the same causal structure.
As an example, if the fibers C x are hyperquadrics defined by a pseudo-Riemannian metric g of signature (p + 1, q + 1), then the causal structure (M, C) given by the family of null cones of g corresponds to the pseudo-conformal structure induced by the metric g. 3 In other words, there exists a local defining function of this causal structure of the form F (x i ; y j ) = g ij (x)y i y j .
The above description makes it clear that the relation of causal structures to pseudoconformal structures is an analogue of what Finsler structures are to Riemannian structures. Recall that a Finsler structure (M n+1 , Σ 2n+1 ) is given by a codimension one subbundle Σ ⊂ T M, whose fibers Σ x ⊂ T x M are strictly convex affine hypersurfaces, i.e., the second fundamental form of each fibers is positive definite. The sub-bundle Σ is called the indicatrix bundle (see [Bry02]). 4

→ M
where the map π is the quotient map of J 2 (R, R) by the foliation defined by the 2-jets of the solutions of the ODE. This map endows M with a causal structure. The null geodesics of M are given by π ○ τ −1 (p) where p ∈ J 1 (R, R), while the contact geodesics are given by τ ○ π −1 (q) where q ∈ M.
Equivalently, given a causal structure (M, C) in dimension three, Holland and Sparling showed that the double fibration above is equivalent to where N is the space of characteristic curves (or null geodesics) 7 of the causal structure. The precise definition of these curves is given in Section 2.3.5. It turns out that starting with a causal structure (M, C), one can show that locally C ≅ J 2 (R, R) and N ≅ J 1 (R, R). Therefore, all the results mentioned above can be translated in terms of local invariants of the causal structure in three dimensions.
The correspondence above motivates the study of the local invariants of causal geometries in higher dimensions and differential equations that can be associated to them. As will be explained, this study turns out to be related to a number of current themes of research.
In [HS11], Holland and Sparling studied causal structures in higher dimensions and showed that an analogue of the Weyl sectional curvature can be defined. Also, they 5 In this article J k (R l , R m ) denotes the space of k-jets of functions from R l to R m . 6 A contact path geometry over a contact manifold N is given by a family of contact paths with the property that at each point x ∈ N , a unique curve of the family passes through each contact direction in TxM. A contact projective geometry is a contact path geometry whose contact paths are geodesics of some affine connection. 7 In this article the term characteristic curves is used to refer to the lift of null geodesics to the null cone bundle.
showed for causal structures the Raychaudhuri -Sachs equations from General Relativity remains valid for higher dimensional causal structures (also see [AJ16,Min15]). Before describing the nature of the complete set of invariants for causal structures it is helpful to make parallels with what happens in Finsler geometry. On an (n+1)-dimensional manifold M the indicatrix bundle Σ of a Finsler structure is (2n+1)-dimensional equipped with a contact 1-form, i.e., a 1-form ω ∈ T * Σ satisfying ω ∧ (dω) n ≠ 0. The 1-form ω is called the Hilbert form of the Finsler structure. As indicated by Cartan [Car34,Car30] and Chern [Che48], in Finsler geometry there are two essential invariants comprising the total obstruction to local flatness, namely the flag curvature and the centro-affine cubic form of the fibers C x (also referred to as the Cartan torsion). The vanishing of the Cartan torsion implies that the Finsler structure is a Riemannian structure, in which case the flag curvature coincides with the sectional curvature.
Regarding causal structures, the main step is to realize that the 2n-dimensional space C carries a quasi-contact structure (also known as even contact structure), i.e., a 1-form ω 0 satisfying ω 0 ∧ (dω 0 ) n−1 ≠ 0 and (dω 0 ) n = 0. The 1-form ω 0 is referred to as the projective Hilbert form. 8 It is shown that, for n ≥ 3, there are two essential local invariants which form the total obstruction to local flatness. As one would expect, the Fubini cubic form of each fiber C x as a projective hypersurfaces in PT x M constitutes one of these obstructions whose vanishing implies that the causal structure is a pseudo-conformal structure. The second essential invariant is a generalization of the sectional Weyl curvature of a pseudoconformal structure restricted to the shadow space 9 as found by Holland and Sparling in [HS11] (see Section 2.5). By analogy with Finsler geometry, the second essential invariant is called the Weyl shadow flag curvature (referred to as Wsf curvature.) Furthermore, from the first order structure equations for causal geometries it follows that a causal structure (M, C) can be equivalently formulated as having a manifold C 2n equipped with a distribution D of rank n which contains an involutive corank one subdistribution ∆ ⊂ D and satisfies a certain non-degeneracy condition. The (small) growth vector of D is (n, 2n − 1, 2n), i.e., rank[D, D] = 2n − 1, and rank[D, [D, D]] = 2n. 10 (see Section 2.4.2.) In this case, M n+1 is the space of integral leaves of ∆. 11 This description identifies causal geometries as parabolic geometries modeled on B n P 1,2 and D n P 1,2 for n ≥ 4 and D 3 P 1,2,3 , with their natural 3 -grading. As will be shown, they have some similar features to path geometries [Gro00] and contact path geometries [Fox05a], which correspond respectively to parabolic geometries modeled on A n P 1,2 and C n P 1,2 .
Another essential part of any causal structure is given by its set of characteristic curves. Note that as a quasi-contact manifold, C is equipped with the characteristic line field of the projective Hilbert form, i.e., the unique degenerate direction of dω 0 lying in Ker ω 0 , defined by vector fields v satisfying The integral curves of a characteristic vector field v is called a characteristic curve. In the case of pseudo-conformal structures the projection of these curves to M coincide with null geodesics.
The description above shows that, unlike Finsler geometry, the definition of characteristic curves does not depend on a choice of metric. Recall that in Finsler geometry geodesics can be defined as the extremals of the arc-length functional where γ ∶ (a, b) =∶ I → C is a curve in C and γ * φ = L(x i ; y i )dt. The function L(x i ; y i ) is called the Finsler metric which in the calculus of variations is referred to as the Lagrangian.
The Finsler metric is homogeneous in y i 's and has non-degenerate vertical Hessian 12 . When addressing the geodesics of a Finsler structure, or more generally extremal manifolds in the calculus of variations, it is generally assumed that the Lagrangian does not vanish along them [Gri83]. The study of extremal curves along which the Lagrangian vanishes seems to have received much less attention in the literature (see [Run83,Run84]). The exception is the case of Lagrangians that are quadratic in y i 's. This is due to the fact that they arise in the context of pseudo-conformal structures. For instance, this problem is briefly discussed by Guillemin [Gui89], where he studies the deformations of the product metric π * 1 dt 2 ⊕ −π * 2 g on S 1 × S 2 where π 1 and π 2 are projections onto S 1 , and S 2 and g is the round metric on S 2 , which is Zollfrei, i.e., all of its null geodesics are closed with the same period. In Section 2.5.3 it is discussed how causal structures provide a geometric setup for the variational problems involving certain classes of Lagrangians L(x i ; y i ) which are homogeneous in y i 's and have vertical Hessian of maximal rank over their vanishing set.
The foliation of C defined by the characteristic curves allows one to define the quotient map τ ∶ C → N where N is (2n − 1)-dimensional and is called the space of characteristic curves of C. A priori, N is only locally defined around generic points and can be used to give a geometric description of the theory of correspondence spaces for causal structure in the sense ofČap [Čap05] (see Section 3.3). Note that the space N , when globally defined, is of interest in Penrose's twistor theory [LeB90] and a twistorial desciption of causal relations among the points of the space-time [Low06].
Finally, an interesting instance of studying causal structure can be found in [Hwa13] where Hwang considers complex causal structures arising on certain uniruled projective manifolds, M n+1 , satisfying the property that through a generic point x ∈ M there passes an (n − 1)-parameter family of rational curves of minimal degree. The hypersurface C x ⊂ PT x M obtained from the tangent directions to such curves at the point x is called the variety of minimal rational tangent (or VMRT). Under the assumption that C x is smooth of degree ≥ 3, he shows that the causal structure defined by them is locally isotrivially flat (see Definition 2.18) 13 . Furthermore, it turns out that the characteristic curves for such causal structures coincide with rational curves of minimal degree 14 . Finally, assuming that M has Picard number 1 and n ≥ 2, he shows that M is biregular to a hyperquadric equipped with its natural causal structure (see Section 2.2). This theorem is a generalization of what has been considered previously in [Bel01,Ye94] for complex conformal structures.
More broadly, Hwang's study of causal structures is a part of the differential geometric characterization of uniruled varieties initiated by Hwang and Mok [Hwa12] via geometric structures referred to as cone structures (see Definition 2.1). The starting point for Hwang and Mok is a given family of rational curves of minimal degree whose tangent directions at a generic point x ∈ M give rise to the VMRT C x ⊂ PT x M . For instance, if C x = PT x M one obtains the complex analogue of path geometry [Gro00] and if C x is a hyperplane which induces a contact distribution on M, one obtains a complex contact path geometry [Fox05a]. The program of Hwang and Mok seems interesting and calls for further exploration from the perspective of the method of equivalence, especially through the lens of flag structures as developed in [DZ12].

Main results.
In all the statements of this article smoothness of underlying manifolds and maps is always assumed and the dimension of M is always taken to be at least four unless otherwise specified.
In Section 2, after reviewing the flat model for causal structures, the first order structure equations for a causal geometry (M n+1 , C) of signature (p+1, q +1) are derived. It is shown that the bundle of null cones C is foliated by characteristic curves and is endowed with the conformal class of a quadratic form which is degenerate along the vertical directions 12 It is usually assumed that the Finsler metric when restricted to TxM only vanishes at the origin and is positive elsewhere [BCS00]. However, as was mentioned previously, the condition can be relaxed [She01]. 13 Hwang refers to such structures as flat cone structures of codimension one. 14 Motivated by the notion of conformal torsion introduced by LeBrun [LeB83] and contact torsion introduced by Fox [Fox05a], one can define causal torsion whose vanishing implies that the rational curves of minimal degree coincide with the characteristic curves of the causal structure they define. The precise definition of causal torsion is not given in this article.
of the fibration C → M. Subsequently, the first order structure equations are used in the Appendix to obtain structure equations leading to an {e}-structure.
The essential theorem of this article that is stated in Section 2.3.13 is the following.
Theorem 2.12. To a causal structure (M n+1 , C 2n ), one can associate an {e}-structure on a principal bundle P 1,2 → P → C of dimension (n+3)(n+2) 2 where P 1,2 ⊂ O(p + 2, q + 2) is the parabolic subgroup defined in (2.5). The essential invariants of a causal structure are the Fubini cubic form of the fibers and the Wsf curvature. The vanishing of the essential invariants implies that the {e}-structure coincides with the one defined by the Maurer-Cartan forms of o(p + 2, q + 2).
The proof of the above theorem takes up most of the Appendix. Using the {e}-structure thus obtained, it is straightforward to deduce that the essential invariants are the Fubini cubic form and the Wsf curvature. Section 2.4 deals with the parabolic nature of causal structures. In Section 2.5 some geometrical interpretations of the essential invariants are given. The first Section ends with examples of causal structures.
In Section 3.1, the structure equations for causal geometries with vanishing Wsf curvature are examined. It is shown that if the Lie derivative of the Fubini cubic form along the vector fields tangent to the characteristic curves is proportional to itself, then the Wsf curvature has to vanish, unless the causal geometry is a pseudo-conformal structure. Such spaces are the causal analogues of Landsberg spaces in Finsler geometry 15 . Moreover, having defined the scalar shadow flag curvature (referred to as the ssf curvature), and the notion of completeness of characteristic curves for a causal structure, the following theorem is proved in Section 3.2.
Theorem 3.5. Let (M, C) be a causal structure with vanishing Wsf curvature over a compact manifold with non-zero Fubini cubic form. If the characteristic curves of C are complete, then the scalar flag curvature of (M, C) is non-negative.
The above theorem implies the following corollary.
Corollary 3.6. Let (M, C) be a compact causal structure with vanishing Wsf curvature and complete characteristic curves. If it has negative ssf curvature, then it is a flat pseudo-conformal structure.
The corollary above can be taken as a rigidity theorem which is a causal analogue of the seminal work of Akbar-Zadeh in Finsler geometry [AZ88]. Section 3.3 involves the induced structure on the space of characteristic curves, N , of a causal structure with vanishing Wsf curvature. The (2n − 1)-dimensional manifold N is shown to have a contact structure and is endowed with an (n + 1)-parameter family of Legendrian submanifolds with the property that through each point γ ∈ N there passes a 1-parameter family of them. Moreover, the sub-bundle of T N defined by the contact distribution contains a Segré cone of type (2, n − 1). The theorem below is proved in which the property of being semi-integrable means that at every point γ ∈ N , the Segré cone is ruled by the tangent spaces of the corresponding 1-parameter family of Legendrian submanifolds passing through γ.
Theorem 3.10. Given a causal structure with vanishing Wsf curvature (C, M ), its space of characteristic curves admits a semi-integrable Lie contact structure. Conversely, any semi-integrable Lie contact structure on a manifold N induces a causal structure with vanishing Wsf curvature on the space of its corresponding Legendrian submanifolds.
It should be noted that the theorem above is reminiscent of Grossman's work [Gro00] on torsion-free path geometries and the term semi-integrability is borrowed from there. He showed that given a torsion-free path structure on N n+1 , the space of paths, S 2n , has a Segré structure, i.e., each tangent space is equipped with a Segré cone of type (2, n). Furthermore, S contains an (n + 1)-parameter family of n-dimensional submanifolds such that through each point γ ∈ S there passes a 1-parameter family of them. Grossman showed that at each point γ ∈ S, the Segré cone is ruled by the tangent spaces of the 1-parameter family of submanifolds passing through γ. He called such Segré structures semi-integrable. Furthermore, twistorial constructions such as that of Theorem 3.10 fit exactly intoČap's 15 See [BCS00] for an account.
theorem of correspondence spaces [Čap05]. However,Čap's theory does not contain the geometric interpretation given here in terms of existence of submanifolds that rule the null cone or the Segré cone. Such interpretations are clear from the constructions presented here.
In Section 4.1 a notion of half-flat causal structures of signature (2, 2) is introduced using the existence of a 3-parameter family of surfaces whose tangent planes at each point of M give a ruling of the projective surfaces C x . This notion of half-flatness coincides with the notion of half-flatness for conformal structures of signature (2, 2).
Finally, in Section 4.2 a study of the symmetry algebra of certain non-flat causal structures of split signature that do not induce a conformal structure on M is given from which the proposition below follows.
Proposition 4.8. The submaximal causal structure of split signature that does not descend to a pseudo-conformal structure has 8 dimensional symmetry algebra and is locally equivalent to the locally isotrivially flat causal structure whose null cones are projectively equivalent to Cayley's cubic scroll.
Locally isotrivially flat causal structures, defined in 2.18, are the causal analogue of locally Minkowskian Finsler spaces.
The Appendix contains the details of the equivalence problem calculations. Since causal structures are examples of parabolic geometries of type (B n , P 1,2 ) and (D n , P 1,2 ) if n ≥ 4 and (D 3 , P 1,2,3 ), they admit a regular and normal Cartan connection ( [ANN15,ČS09]). Using the machinery of Tanaka, the correct change of coframe can be given recursively, in order to introduce a Cartan connection on the structure bundle of causal geometries which is normal in the sense of Tanaka. However, for computational reasons, writing down the precise form of the Cartan connection leads to very long expressions which are not particularly enlightening (see [Mak16]). Furthermore, the first order structure equations suffice for the purposes of this article.
Acknowledgments. This work was carried out under the guidance of Professor N. Kamran, whom I would like to thank greatly for his clear explanation of Cartan's method of equivalence and introducing me to the work of Holland and Sparling [HS10]. His encouragement, help and support during my PhD made this work possible. I am also very much indebted to Dennis The for the many helpful conversations that we had and for patiently explaining some aspects of parabolic geometry. The funding for this work was provided by McGill University.

Local description of causal structures
In this section, after introducing the necessary definitions and notations, the first order structure equations of causal structures are derived. It is assumed that the reader is familiar with Cartan's method of equivalence for G-structures (see [Gar89,Olv95] for an account). The essential torsion terms appearing in the first structure equations are interpreted geometrically. The derivation of the full structure equations is carried out in the Appendix A. In Section 2.5 a discussion on the fundamental invariants of a causal structure is given. Finally, the section ends with a few examples of causal structures.
2.1. Definitions and notational conventions. Since the main purpose of this article is the local study of causal geometries, the smoothness of manifolds and maps that are considered is always assumed. In other words, the manifolds considered are assumed to be the maximal open sets over which the necessary smoothness conditions are satisfied.
Let M be a smooth (n + 1)-dimensional (real or complex) manifold, n ≥ 3 and PT M denote its projectivized tangent bundle with projection π ∶ PT M → M , and fibers π −1 (x) = PT x M. Given a tangent vector y ∈ T x M , its projective equivalence class is denoted by Throughout this article, the indices range as and are subjected to the summation convention. When a set of 1-forms is introduced as I = I{ω i } n i=0 , it is understood that I is the ideal algebraically generated by {ω 0 , . . . , ω n } in the exterior algebra of M .
The symmetrization and anti-symmetrization operations for tensors are denoted using Penrose's abstract index notation. For example, consider a tensor with coefficients A abcd . The tensor A (ab c d) is symmetric with respect to all indices except the third, and A [a bc d] is anti-symmetric in the first and last indices. When dealing with symmetric forms such as g = 2ω 0 ○ ω n − ε ab ω a ○ ω b , or F abc θ a ○ θ b ○ θ c with F abc = F (abc) , the symbol ○ denotes the symmetric tensor product. Bold-face letters such as i, j, a, b are used to denote a specific index, e.g. the set {ω a } only contains the 1-form ω a . The summation convention is not applied to bold-face letters.
At this stage, before stating the definition of a causal structure, a more general geometric structure, namely that of a cone structure, is defined [Hwa12]. The reason for this is that in the process of coframe adaption for cone structures of codimension one, causal structures are distinguished as a sub-class of such geometries via the characterizing property of having tangentially non-degenerate fibers (see Section 2.3.3).
Definition 2.1. A cone structure of codimension k ≤ n on M n+1 is given by an immersion ι ∶ C → PT M where C is a connected, smooth manifold of dimension 2n + 1 − k with the property that the fibration π ○ ι ∶ C → M is a submersion whose fibers C ) for all y ∈ M In this article, only cone structures of codimension one are considered i.e., the fibers ι(C x ) ⊂ PT x M are projective hypersurfaces. Since the map ι ∶ C → PT M is assumed to be an immersion and this article concerns the local nature of geometric structures, it is safe to drop the symbol ι in ι(C) and ι(C x ) and denote them as C and C x , respectively. As a result, by restricting to small enough neighborhoods, one can safely assume C ⊂ PT M is a sub-bundle defined in an open set U ⊂ PT M, where the fibers C x are smooth submanifolds. Consequently, a cone structure of codimension one over a manifold M can be represented by (C, M ).
In local coordinates, a point of T M is represented by (x; y) where x = (x 0 , . . . , x n ) are the base coordinates and y = (y 0 , . . . , y n ) are the fiber coordinates. In a sufficiently small neighborhood of a point p = (x; z) ∈ C, , the homogeneous coordinates [y 0 ∶ . . . ∶ y n ] can be chosen so that z = [0 ∶ . . . ∶ 0 ∶ 1] and the coneĈ x is tangent to the hyperplane {y 0 = 0} alongẑ. The affine coordinates (y 0 , . . . , y n−1 ) obtained from setting y n = 1, are used to derive the explicit expressions of the invariants. Using these preferred coordinates in a sufficiently small neighborhood of p, it is assumed that C is given in terms of homogeneous coordinates as a locus (2.1) L(x 0 , . . . , x n ; y 0 , . . . , y n ) = 0 where L is homogeneous of degree r in y i and, in terms of affine coordinates, as a graph (2.2) y 0 = F (x 0 , . . . , x n ; y 1 , . . . , y n−1 ).
Example 2.2. A cone structure of codimension one on M n+1 whose fibers are hyperplanes given by Ker ω for some ω ∈ T * M , and are maximally non-integrable, i.e., ω ∧ (dω) k−1 ≠ 0, is called a contact structure if n = 2k − 1, and a quasi-contact structure (or even contact structure) if n = 2k. These can be considered as "degenerate" examples of codimension one cone structures and have no local invariants. In these cases, the function F in (2.2) is of the form F (x; y) = c a (x)y a + c n (x), with ω = dx 0 + c a (x)dx a + c n (x)dx n .
Example 2.3. A pseudo-conformal structure, i.e., a conformal class of a pseudo-Riemannian metric g ij of signature (p + 1, q + 1), p, q ≥ 0, can be introduced by its field of null cones, i.e., the set of quadrics C x ∶= [y] ∈ PT x g ij y i y j = 0 , As a result, a pseudo-conformal structure is an example of a cone structure of codimension one, which, in contrast to the previous example, is "non-degenerate" (see section 2.3.3).
In terms of the local form (2.2), a representative g ij of the conformal class, at a point s = (x; y) ∈ C, can be put in the form 2y 0 y n − ε ab y a y b , where and therefore F (s) = ε ab y a y b .
2.2. The flat model. When setting up the equivalence problem for a given geometric structure, finding the flat model can be very helpful for identifying the structure bundle. Naturally, one expects the flat model for causal structures to coincide with the flat model for conformal geometry. In causal geometry, the flat model is obtained from the natural pseudo-conformal structure on a quadric. Consider the quadric Q n+1 p+1,q+1 ⊂ P n+2 , where p + q = n − 1, given by has signature (p+2, q+2). It is equipped with a pseudo-conformal structure of signature (p+ 1, q + 1), expressed as 2y 0 y n − ε ab y a y b where y i 's denote the fiber coordinates of T Q n+1 p+1,q+1 . The null cone bundle of this pseudo-conformal class is denoted by C ⊂ PT Q n+1 p+1,q+1 . Consider the flag manifold denotes the space of k-dimensional subspaces of R m which are null with respect to the inner product S, e.g., Gr 0 1 (n + 3) ≅ Q n+1 p+1,q+1 . Taking a basis {e 1 , ⋯, e m } for R m , the transitivity of the action of G p,q ∶= O(p+2, q+2) on Gr 0 k (m) results in isomorphisms Gr 0 k (m) ≅ G p,q P k where P k is the parabolic subgroup isomorphic to the stabilizer of the subspace spanned by {e 1 , ⋯, e k }.
Let m = n + 3, n ≥ 4 and define the parabolic subgroup P k1,⋯,ki ∶= P k1 ∩ ⋯ ∩ P ki of G. Consequently, if follows that (2.5) NF 4 0,1 ≅ G p,q P 1,2,3 , p + q = 2, NF n+1 0,1 ≅ G p,q P 1,2 , p + q = n − 1 ≥ 3 The relevant fibrations are shown in the diagram below. (2.6) 0,1 is isomorphic to C. The Maurer-Cartan form on G p,q determines a canonical Cartan connection over C which, as will be shown, is associated to the flat causal structure. Moreover, the fibration of G p,q over Gr 0 2 (n + 3) gives the flat model for Lie contact structures studied in [SY89,Miy91]. As will be seen, these structures arise on the space of characteristic curves of certain classes of causal geometries.
The Maurer-Cartan form of G p,q can be written as with dϕ+ϕ ∧ ϕ = 0. The symbol ε ab is used to lower and raise indices. The set of semi-basic 1-forms with respect to the projections υ, υ ○ π and υ ○ τ are respectively I 1 = {ω 0 , ⋯, ω n , θ 1 , ⋯, θ n−1 }, . The tangent spaces T p C are equipped with a filtration defined as follows. At p ∈ C, the line over p, denote byp ⊂ T π(p) Q n+1 p+1,q+1 , is null. Let ω 0 denote a representative of the dual projective class ofp with respect to the conformally defined inner product S. For instance, if p = [e n ], then take ω 0 = e ♭ n . Define K 2 ∶= Ann(π * ω 0 ) ⊂ T C. Using the tautological bundle can be associated to the filtration of T p C. Note that at each p ∈ C, m(p) ≅ m. The Lie bracket on T C induces a bracket on m which is tensorial and makes m a graded nilpotent Lie algebra (GNLA), i.e., [K i , K j ] = K i+j with K i = 0 for all i ≥ 4 and K −1 the generating component. Let G 0 denote the subgroup of automorphisms of T C, i.e., O(p + 2, q + 2), which preserves the grading of m via the adjoint action. Let K 0 denote the Lie algebra of G 0 . Then, the Lie algebra of automorphisms of C, i.e., g = o(p+2, q +2), admits a Z-grading defined by Note that K 1 = E⊕Ver where Ver is the vertical distribution with respect to the fibration π and E = π * (g −1 ). Moreover, T C is equipped with a conformally defined quadratic form obtained by lifting the canonical one defined on Q n+1 p+1,q+1 , g = 2ω 0 ○ ω n − ε ab ω a ○ ω b , which is degenerate on Ver. The group of automorphisms of T C, which preserves the grading of m, also preserves the conformal class of the lifted quadratic form g on T C. The Lie algebra g 0 ⊂ g which corresponding to the group of automorphisms of m that preserves the grading, is isomorphic to co(p, q) ⊕ R.
Using the Maurer-Cartan form ϕ, the subspaces g −3 , g −2 , and g −1 are dual to the Pfaffian systems I −3 ∶= {ω 0 }, I −2 ∶= {ω a } n−1 a=1 , and I −1 ∶= {ω n , θ a } n−1 a=1 . 2.3. The first order structure equations. In this section the first order structure equations for causal geometries are derived via the method of equivalence. The procedure involves successive coframe adaptations, as a result of which various important aspects of causal geometries are unraveled, e.g., the quasi-contact structure of C, the projective invariants of the fibers C x , and the Wsf curvature. Also, it is noted that a causal structure can be equivalently formulated as a codimension one sub-bundle of PT * M, which is called a dual formulation of causal geometries. Such formulation is traditionally preferred in the context of geometric control theory [AZ02].
The derivation of the first order structure equations is similar to that of Finsler structures as presented in [Bry02].
2.3.1. The projective Hilbert form. Given a cone structure of codimension one (C, M ), the subset C x ⊂ PT x M is a projective hypersurface, for all x ∈ M . At a point y ∈ C x , the affine tangent spaceT y C x is defined as the hyperplane T zĈx ⊂ T x M tangent to the conê C x ⊂ T x M, containing z ∈ŷ. Note that this subspace is independent of the choice of z. It is important here to distinguish between the affine tangent spaceT y C x ⊂ T x M and the vertical tangent space VT (x;y) C corresponding to the fibration π ○ ι ∶ C → M.
The 1-form ω 0 ∈ T * C, at (x; y) ∈ C is defined as a multiple of (π ○ ι) * (α) where α ∶= Ann(T y C x ) ∈ T * x M. By definition, ω 0 is defined up to a multiplication by a nowhere vanishing smooth function on C and is therefore semi-basic with respect to the fibration π ∶ C → M. In this article ω 0 is referred to as the projective Hilbert form.
2.3.2. The adapted flag. At each point (x; y) ∈ C, there is a natural basis ( ∂ ∂ω 0 , . . . , ∂ ∂ω n ) for T x M adapted to the flagŷ ⊂T y C x ⊂ T x M, where { ∂ ∂ω n } and { ∂ ∂ω 1 , . . . , ∂ ∂ω n }, respectively, spanŷ andT y C x . In terms of the graph (2.2), the local expressions are The dual coframe (ω 0 , . . . , ω n ) can be written as (2.10) The pull-back of these 1-forms to C spans the semi-basic 1-forms with respect to the fibration π ∶ C → M. At each point p ∈ C, given by p = (x i ; y i ), these 1-forms are adapted to the flag where VT p M = Ker {ω 0 , ⋯, ω n } is the vertical tangent bundle, K p = Ker {ω 0 , ⋯, ω n−1 } is the fiber of the tautological vector bundle given by where p = (x; y), the lineŷ corresponds to the projective class of y and H p = Ker {ω 0 } is the quasi-contact distribution (see Section 2.3.5) To clarify the notation, if p = (x; y) then the fiber VT p C is also denoted by T y C x and the quasi-contact distribution can be expressed as the splitting T y C x ⊕T y C x which is not unique.
2.3.3. Causal structures and the first coframe adaptation. Recall that a projective hypersurface V ⊂ P n is called tangentially non-degenerate if its projective second fundamental form has maximal rank (see [Sas99,AG93]). Now the stage is set to define causal structures.
Definition 2.4. A causal structure of signature (p + 1, q + 1) on M n+1 is a cone structure of codimension one, (M, C), where the fibers C x are tangentially non-degenerate projective hypersurfaces and their projective second fundamental form has signature (p, q) everywhere.
In section 2.3.6 it is shown that the tangential non-degeneracy assumption of the fibers C x implies that the signature of h ab at any point of C determines its the signature of h ab everywhere.
Let d V denote the vertical exterior derivative, i.e., the exterior differentiation with respect to fiber variables y i . Using the expressions (2.10), it follows that As a result of (2.12) one obtains which implies that ω 0 has rank 2n − 1, i.e., ω 0 ∧ (dω 0 ) n−1 ≠ 0. The 1-forms θ a define vertical 1-forms with respect to the fibration π ∶ C → M. The 1-forms θ a are linearly independent everywhere when the fibers C x are tangentially nondegenerate at every point, i.e., det(∂ a ∂ b F ) ≠ 0.
Since the vertical 1-forms θ 1 , . . . , θ n−1 are linearly independent, at each point (x; y) ∈ C, one obtains a coframe I tot ∶= {ω 0 , . . . , ω n , θ 1 , . . . , θ n−1 } for T (x;z) C adapted to the flag defined previously in (2.11). Such a coframe is called 1-adapted. For a quadric, this filtration coincides with the one introduced in (2.8). In analogy with the flat model, the following algebraic ideals are defined: (2.15) I tot = I{ω 0 , ⋯, ω n , θ 1 , ⋯, θ n−1 }, Using the expressions (2.10), equation (2.13) gives From the expression above it follows that the linear independence of θ a 's modulo ω i 's is equivalent to the non-degeneracy of the vertical Hessian ∂ a ∂ b F . The fact that such a coframing is adapted to the flag (2.14) and satisfies (2.13) implies that different choices of 1-adapted coframes are related by In the language of G-structures, the structure group of the bundle of 1-adapted coframes, Remark 2.5. The reason for using a 2 and b 2 in G 1 is as follows. Let L(x i ; y i ) be a Lagrangian function for a causal structure, i.e., the vanishing set of L defines C. As is conventional in the case of pseudo-conformal structures, it is desirable to restrict to the set of causal transformations preserving the sign of L in a neighborhood of the null cones, i.e., restricting to the set of Lagrangians of the form S 2 L for some nowhere vanishing homogeneous function S ∶ T M → R. Recall that the projective Hilbert form in terms of L is given by ω 0 = ∂ i Ldx i . Replacing L by S 2 L and restricting to the zero locus of L, the projective Hilbert form transforms to S 2 (L i dx i ) = S 2 ω 0 . This observation justifies the use of a 2 in G 1 . Moreover, recall that a vector bundle of rank 1 is called orientable if it has a non-vanishing global section. In the case of contact manifolds, a contact distribution is called co-orientable or transversally orientable if it has a globally defined contact form [LM87]. Similarly, a quasi-contact structure is called co-orientable if it has a globally defined quasi-contact form ω 0 . If the quasi-contact distribution on C is given a co-orientation, it is preserved under the action of G 1 . Similarly, if the characteristic field of a quasi-contact structure (see Section 2.3.5) is given an orientation, then having b 2 in (2.17) implies that the structure group G 1 preserves the orientation of the characteristic field. Moreover, the conformal scaling of the quadratic form g defined in 2.3.11 by the action of G 1 is given by a 2 b 2 . As a result, it is possible to define space-like and time-like horizontal vector fields v in T C via the signature of g(v, v).
To implement Cartan's method of equivalence, one lifts the 1-adapted coframe to the G 1 -bundle in the following way (see [IL03,Gar89,Olv95] for the background). Recall that the set of all coframes on C, defines a GL(2n)-principal bundleς ∶P → C, with the right action where ϑ p ∈ ς −1 (p) is a coframe at p ∈ C, g ∈ GL(2n), and the action on the right hand side is the ordinary matrix multiplication. Among the set of all coframes on C, let ϑ represent 1-adapted coframes t (ω 0 , ⋯, θ n−1 ), which are well-defined up to the action of G 1 . As a result, the 1-adapted coframes are sections of a G 1 -bundle ς 1 ∶ P 1 → C. On P 1 ≅ C × G 1 one can define a canonical set of 1-forms by setting The exterior derivatives of the 1-forms in ϑ are given by where α(p, g) = g −1 ⋅ dg is g 1 -valued 1-form where g 1 denotes the Lie algebra of G 1 . The 2forms T (p, g) can be expressed in terms of The terms constituting T are called the torsion terms. They are semi-basic with respect to the projection ς 1 . Via α, one can identify the vertical tangent spaces of the fibers of ς 1 ∶ P 1 → C with g 1 . The g 1 -valued 1-form α is called a pseudo-connection on P 1 which is not yet uniquely defined. Recall that the difference between a connection and a pseudo-connection on G 1 -bundle lies in the fact that a connection formα satisfies the equivariance condition R * gα = Ad(g −1 )α, where g ∈ G 1 , while a pseudo-connection in general does not.
A choice of pseudo-connection in (2.18) gives rise to structure equations for the 1-forms ϑ. For 1-adapted coframes, the pseudo-connection forms are of the form, where π ab = π ba . It follows that where the 1-forms σ a , σ b a , are the torsion terms and vanish modulo I tot . Their components are determined in Section 2.3.9.
The equation of dω 0 follows directly from (2.13). Moreover, the Pfaffian system I bas , is completely integrable with the fibers C x being its integral manifolds. It follows that dω a = τ a i ∧ ω i , and dω n = τ n i ∧ ω i , where τ a i , τ n i ≡ 0 modulo I tot . Using the fact that the action of G 1 preserves I bas , and that two pseudo-connection differ by a semi-basic g 1 -valued 1-form, all the 1-forms τ a i , τ n i can be absorbed into the pseudo-connection forms. The only torsion term left is h ab θ b ∧ ω n , for some function h ab , which is discussed in Section 2.3.6. From now on, since structure equations are always written for the lifted coframe ϑ, the underline in ω i and θ a 's will be dropped.
Remark 2.6. The oriented projectivized tangent bundle P + T M is defined so that at x ∈ M the fiber P + T x M is isomorphic to the space of rays in T x M , or, equivalently, the set of oriented real lines in T x M passing through the origin. As a result, The above definition of cone structures, and more specifically causal structures, can be changed so that C is a sub-bundle of P + T M. In the case of causal structures immersed in P + T M , the assumption that the fibers C x = ι −1 (x) be connected is not desirable. This is due to the fact that C x should be thought of as the light cones which are comprised of the future and past light cones whose projectivizations are disjoint. Thus, one expects This way, in analogy with Finsler geometry, a cone structure is called reversible ifĈ = −Ĉ. Otherwise, it is called non-reversible. It should be noted that among oriented causal structures, non-reversible ones can be of interest in certain problems due to possible differences in their future and past light cones. This is similar to the case of non-reversible Finsler structures containing the important class of Randers spaces [BRS04]. In [KP67], Kronheimer and Penrose use the term self-dual for a reversible causal structure. This is due to the fact that they define the dual of a causal structure to be the causal structure obtained from replacing all the causal relations between the points of the space by their inverses, i.e., interchanging the words "future" and "past".

2.3.4.
A dual formulation of causal structures. In the context of geometric control theory, geometric structures arising from distributions are typically formulated in the cotangent bundle [AZ02]. Firstly, the definition of a Legendre transformation is recalled.
Definition 2.7. Given a sufficiently smooth real-valued function L, usually referred to as a Lagrangian, on T M , the associated Legendre transformation Leg ∶ T M → T * M, is defined by ∂y i , and (p 0 , . . . , p n ) are local coordinates for the fibers of T * M .
The Legendre transformation is a local diffeomorphism provided that Leg * = ∂ 2 L ∂y i ∂y j is non-degenerate everywhere. The inverse Legendre transformation Leg −1 can be obtained by the same procedure using the Hamiltonian function Q ∶= A○Leg −1 where A ∶= dL(A)−L with A being the Liouville vector field, i.e., A = y i ∂L ∂y i − L in local coordinates. When the degree of homogeneity of L is r ≠ 1, Euler's theorem gives Q = (r − 1)L ○ Leg −1 .
Changing the defining function to L ′ (x, y) = S(x, y)L(x, y), the function Leg remains invariant over C, hence the vanishing set of Q in T * M remains invariant. Consequently, one can equivalently study the dual causal structure C * ∶= Leg(C) ⊂ PT * M whose fibers Consequently, the coframe adaptation can be carried out using λ for the dual cone structure. Hence, this dual formulation can be taken as the starting point for studying causal structures.
2.3.5. Characteristic curves and a quasi-Legendrian foliation. A causal structure comes with a distinguished set of curves which can be defined using the quasi-contact nature of C. Recall from Example 2.2 that a quasi-contact structure (or even contact structure) on a 2n-dimensional manifolds was defined locally in terms of a 1-form of rank 2n − 1. As a result, there is a unique direction along which ω 0 and dω 0 are degenerate i.e., [v] ⊂ T PT M such that v ⌟ ω 0 = 0 and v ⌟ dω 0 = 0, which is called the characteristic field of the quasicontact structure and its integral curves are called characteristic curves (c.f. [Hsu92]). A section of the characteristic field is called a characteristic vector field. In local coordinates using the defining function F in (2.2), one obtains that the vector field represents a characteristic vector field on C.
The bundle C is 2n-dimensional and ω 0 ∈ T * C has maximal rank 2n − 1, therefore, C has a quasi-contact structure. The fibers C x form a quasi-Legendrian foliation of C since the vertical tangent spaces T y C x are annihilated by I bas = {ω 0 , . . . , ω n }.
Remark 2.9. Recall that the geodesics of a Finsler structure are defined as the projection of integral curves of the Reeb vector field associated to the contact structure on the indicatrix bundle induced by the Hilbert form [Bry02]. Here, the projection of the characteristic curves gives an analogue of null geodesics in pseudo-conformal structures.
2.3.6. Projective second fundamental form and the second coframe adaptation. The next coframe adaptation involves the projective fundamental form of the fibers C x .
Let ω a ≡ −h ab θ b ∧ ω n modulo I{ω 0 , ⋯, ω n−1 }. Using the identity d 2 ω 0 = 0, it follows that i.e., h ab = h ba . Using the local expressions of ω a in (2.10), one obtains that Using the structure equations (2.20), the infinitesimal action of the structure group G 1 on h ab is obtained from the identity d 2 ω a = 0, and is given by The relation above implies that the functions h ab (p, g 0 ) andh ab (p, g 1 ) where g 1 = gg 0 , with g ∈ G 1 represented by the matrix (2.17), are related by It follows that the bilinear form φ 2 = h ab θ a ○ θ b when pulled-back to the fibers C x coincides with its projective second fundamental form.
Remark 2.10. The action of the structure group on h ab can also be obtained directly from the structure equations (2.20). However, in implementing Cartan's method of equivalence finding the infinitesimal group action by differentiating the structure equations is more straightforward and computationally easier. This is why from now on the focus will be primarily on the infinitesimal action of the structure group on the torsion terms.
Note that by definition 2.4, tangential non-degeneracy of the fibers C x implies that h ab is non-degenerate everywhere in C. As a result, the signature of h ab , as a well-defined tensor over C, remains the same everywhere.
Assuming that h ab has signature (p, q), the relation (2.22) implies that A a b can be chosen so that h ab is normalized to ε ab . By restricting the G 1 -bundle to the coframes for which h ab = ε ab , and using the relation (2.21), it follows that As a result, for such coframes one can replace ψ a b by φ a b + (φ 0 + φ n ) δ a b where the matrixvalued 1-forms φ a b take values in o(p, q) modulo I tot Coframings with normalized h ab are 2-adapted, and they are local sections of a G 2bundle over C where Remark 2.11. By a result of Griffiths and Harris [GH79], if the projective second fundamental form of a projective hypersurface C x ⊂ PT x M has degeneracy of rank k, then its projective dual C * x ⊂ PT * x M has codimension k + 1. In other words, let (M, C) be a cone structure (M, C) of codimension one, whose fibers have projective second fundamental form with degeneracy of rank k. Then, its dual cone structure (M, C * ) has fibers of dimension n − k − 1. Therefore, it is more natural to study cone structure of codimension one whose fibers have degenerate second fundamental form via their dual formulation since dim C = 2n, and dim C * = 2n − k.
Hence, with respect to 2-adapted coframes, the structure equations are bc and E a 0c vanish, and E a nb is trace-free. Differentiating dω a , and collecting the terms of the form τ a b ∧ ω b ∧ ω n for some 1-form satisfying τ ab = τ ba and τ a a ≡ 0 modulo I tot , one obtains Contracting the above equations with ε ab , and using the properties As a result, the infinitesimal action of the structure group on E a nb is given by Similarly to Section 2.3.6, it is possible to restrict to coframes with vanishing E a nb , and consequently, by relation (2.27), the 1-forms π ab are pure trace modulo I tot , i.e., π ab = π n ε ab , modulo I tot , for some 1-form π n . Such coframes are called 3-adapted.
The 3-adapted coframes are local sections of a G 3 -bundle, where G 3 consists of matrices 2.3.8. Symmetries of F abc . In order to show F abc is completely symmetric, one first differentiates equation (2.26b) to obtain As a result, F abc is totally symmetric.
2.3.9. The fourth coframe adaptation, Fubini cubic form and the Wsf curvature. The next coframe adaptation reveals the essential invariants of a causal structure, which are the Fubini cubic forms of the fibers C x and the Weyl shadow flag curvature (also referred to as the Wsf curvature).
Differentiating (2.26b) and collecting terms of the form τ ab Contacting by ε ab , and defining Putting this back in (2.29), and contracting with ε bc gives As before, the infinitesimal action above makes it possible to translate F a to zero. Therefore, the class of 4-adapted coframes can be defined by the apolarity relation The resulting reduction in the structure group is realized via the relation γ a 0 ≡ ε ab γ n b modulo I tot .
It can be shown that the cubic form when pulled-back to the fiber C x , coincides with Fubini cubic form of C x . This becomes evident in the derivation of the local expression of F abc in (2.65). It is noted that because the second fundamental form is normalized, for any choice of 4-adapted coframe, the Fubini cubic form is well-defined tensor on C.
The structure group G 4 is reduced to the group of matrices of the form , the structure equations after the fourth coframe adaptation are of the form a n bc] = 0 By making the following changes , it can be assumed K a i0 vanishes By replacing π n → π n + 1 n−1 ε ac E anbc θ b − 1 n−1 ε cd W cndn ω n , the traces of W anbn and E a nbc are absorbed in π n , and therefore, W anbn and E ancb can be chosen to be trace-free in (a, b).
The symmetric trace-free tensor W anbn is called the Weyl shadow flag curvature (the Wsf curvature), which, as will be shown, is the second essential invariant of causal structures.
By differentiating (2.32b), and collecting 3-forms A ab θ a ∧ ω b ∧ ω n such that A ab = A ba , and A a a = 0, it follows that where f abc = F abc;n , i.e., the derivative of F abc along the characteristic curves.
2.3.10. The fifth coframe adaptation. There is one more coframe adaptation that can be imposed on any causal structure, which reduces the structure group by one dimension. First, note that the derivative of (2.32b) gives modulo I{ω 1 , ⋯, ω n }. To find the infinitesimal group action on K ab via the relation above one needs to find the 2-form components ν ab ∧ θ b of dγ a where ν ab ≢ 0 modulo I tot . Differentiating (2.32c), it follows that Inserting the above expression of dγ a into (2.33), one sees that the infinitesimal action of the structure group G 5 on K ab is given by is not a tensor as its infinitesimal transformation rule in terms of the structure group involves F ab c . Using the apolarity relation (2.30), and the fact that K ab and φ ab are symmetric and skew-symmetric respectively, the infinitesimal action of the structure group G 4 on K ∶= ε ab K ab is found to be Now, the 5-adapted coframes can be defined by the property that K = 0. Hence, one obtains The 5-adapted coframes are local sections of a G 5 -bundle where G 5 is the matrix group defined by 2.3.11. Invariant conformal quadratic form. According to (2.34), the structure group G 5 preserves the conformal class of the quadratic form which is semi-basic with respect to the fibration π ∶ C → M. As will be shown in Section 2.3.14, the vanishing of the Fubini form implies that the null cones of the causal structure coincide with those of g. The o(p, q)-valued pseudo-connection associated to the structure group G 5 acting on g, is given by where t γ and t θ are column vectors of 1-forms γ a ∶= ε ab γ b and θ a ∶= ε ab θ b , respectively. From now on, ε ab will be used to raise and lower the indices a, b, c.
2.3.12. The first order structure equations. Having carried out five coframe adaptations, the first structure equations for causal structures can now be expressed as follows.
The pseudo-connection form for the first order structure group is Consequently, the structure equations can be written as The infinitesimal group action on the torsion coefficients is equivalent to saying that the following 1-forms are semi-basic with respect to the fibration π ∶ C → M.
By the infinitesimal actions (2.39), it follows that no more reduction of the structure group is possible, unless certain non-vanishing conditions on the torsion coefficients are assumed. Such special cases are considered in subsequent sections.
2.3.13. An {e}-structure. Using the first order structure equations, an {e}-structure can be associated to causal structures. The necessary computations are carried out in the Appendix.
Theorem 2.12. To a causal structure (M n+1 , C 2n ), n ≥ 4, one can associate an {e}structure on the principal bundle P 1,2 → P → C of dimension (n+3)(n+2) 2 where P 1,2 ⊂ O(p + 2, q + 2) is the parabolic subgroup defined in (2.5). If n = 3, then P 1,2 is replaced by P 1,2,3 . The essential invariants of a causal structure are the Fubini cubic form of the fibers and the Wsf curvature. The vanishing of the essential invariants implies that the {e}structure coincides with the Maurer-Cartan forms of (p + 2, q + 2). The structure equations can be written as where φ is defined in (2.7) and the right hand side of (2.40) is expressed in (A.17).
The proof of the above theorem will be continued in the Appendix and involves lots of tedious calculations. Of course, it is desirable to find the right change of basis so that the {e}-structure defines a Cartan connection. This issue is discussed in Section 2.4.3. It is noted the number of torsion elements on the right hand side of the first order structure equations is minimal due to the special absorption process that was undertaken here. When obtaining the {e}-structure in the Appendix, to simplify computations, one term will be added to Equation (2.37d) (see Equation A.1.4).
An immediate corollary of Theorem 2.12 is the following.
Corollary 2.13. The maximal dimension of the symmetry algebra of causal structures on a (n + 1)-dimensional manifolds is (n+3)(n+2) Proof. As explained in Section 2.2 the flat model has the maximal symmetries. Now, suppose there is non-flat model that is maximally symmetric. Then, either the Fubini form or the Wsf have a non-zero entry. However, by the infinitesimal group actions (2.39), the structure group acts by scaling on both of the essential invariants. As a result, the symmetry group of the causal structure has to be a proper sub-group of O(p + 1, q + 1) obtained after normalizing the value of that non-zero entry of the essential invariants. Thus, it cannot be maximally symmetric.
The corollary above is clear from the parabolic point of view, since any parabolic geometry (G, P ) has maximal symmetry dimension equal to dim G which can only be reached in the flat case [ČS09].
2.3.14. The case of conformal structures. Using the first order structure equations, an observation is made regarding (pseudo-)conformal structures viewed as causal structures.
Proposition 2.14. A causal structure (M, C) of signature (p + 1, q + 1) with vanishing Fubini cubic form induces a pseudo-conformal structure of the same signature on M. The resulting pseudo-conformal structure is flat if the initial causal structure has vanishing Wsf curvature, and hence flat. Conversely, any pseudo-conformal structure defines a causal structure of the same signature on its bundle of null cones. The resulting causal structure is flat if the initial pseudo-conformal structure is flat.
Proof. To show the first part, note that if the Fubini form vanishes then, by the Bianchi identities (A.19), the quantities K ab and L a vanish as well. The first order structure equations (2.36) can be used to show that the Lie derivatives of the quadratic form g, defined in Section 2.3.11, along vertical vector fields ∂ ∂θ1 , ⋯, ∂ ∂θn−1 are given by (2.41) for some non-vanishing function λ on C. The vector fields ∂ ∂ω i , and ∂ ∂θa are defined as dual to the coframe ω 0 , ⋯, θ n−1 . The vertical vector fields are integrable with the fibers C x as their integral manifolds, and M as the quotient space. Consequently, the conformal class of the quadratic form g descends to M and because it is non-degenerate and of maximal rank on M, a pseudo-conformal structure on M is obtained with the same signature as that of g. If the causal structure is flat, i.e., the {e}-structure is given by the Maurer-Cartan forms of O(p + 2, q + 2), then the structure bundle fibers over M as well, implying that the resulting pseudo-conformal structure has to be flat too.
The converse part is shown similarly. Given a pseudo-conformal structure of signature (p+1, q +1), choose a coframe (ω 0 , ⋯, ω n ) so that a representative of the pseudo-conformal structure takes the form g = 2ω 0 ○ ω n − ε ab ω a ○ ω b . On its null cone bundle π ∶ C → M, the metric g pulls back to define a quadratic form on C satisfying (2.41). The 1-forms ω i define a set of semi-basic 1-forms on C. Using conformal transformations, ω i 's can be chosen to become adapted to the flag (2.11). This can be seen as follows. The affine tangent spacê T y C x ⊂ T x M is a null hyperplane and by a conformal transformation ω 0 can be taken to be its annihilator. Moreover, the dual of ω 0 with respect to the metric g lies onŷ. More explicitly, let y ∈ŷ ⊂Ĉ. Because, g ij y i y j = 0, the 1-form ω 0 as the annihilator ofT y C x can be expressed as ω 0 = g ij y j dx i . Now, it is apparent that the dual of ω 0 lies inŷ. Since the metric is expressed as g = 2ω 0 ○ ω n − ε ab ω a ○ ω b , it follows that ω a (ŷ) = 0, and ω n (ŷ) ≠ 0. Hence, ω i 's are adapted to the flag (2.11).
Since g satisfies (2.41), the resulting causal structure have vanishing Fubini form. If the conformal structure is flat, i.e., isomorphic to the flat model, then as discussed in Section 2.2, the corresponding causal structure is flat as well. Because the construction is reversible, the result follows from the first part of the proposition.
The following corollary follows immediately.
Corollary 2.15. If a pseudo-conformal structure lifts to a causal structure with vanishing Wsf curvature, then the resulting causal structure is flat.
Proof. By Proposition 2.14, a pseudo-conformal structure defines a causal structure with vanishing Fubini cubic form. If the resulting causal structure has vanishing Wsf curvature, then all of its essential invariants are zero and is therefore flat. By Proposition 2.14, one obtains that the initial pseudo-conformal structure has be flat as well.
Remark 2.16. As was mentioned previously, the discussion above goes through in the case of conformal structures of positive definite signature. The extra step would be to complexify the inner product and the tangent bundle and work with complex causal structures endowed with a reality condition. See [LeB83] for a treatment of complex conformal structures.
2.4. Causal structures as parabolic geometries. This section provides the necessary definitions and theorems needed to characterize causal structures as parabolic geometries. A main reference on parabolic geometries is [ČS09], but, for the purposes of this section the articles [KT14, ANN15, Yam93, Zel09] suffice.
The main step is the characterization of causal structures by a bracket generating distribution with certain properties, described in Section 2.4.2. As a result, causal structures on an (n + 1)-dimensional manifold can be viewed as regular normal infinitesimal flag structures of type (B n , P 1,2 ) or (D n , P 1,2 ) if n ≥ 4, or (D 3 , P 1,2,3 ) if n = 3. By a result of Tanaka-Morimoto-Čap-Schichl, such structures can be canonically identified as regular normal parabolic geometries of the same type.
2.4.1. Parabolic geometries and Tanaka Theory. A parabolic geometry is a Cartan geometry (P, ω) of type (G, P ) where G is a real or complex semi-simple Lie group and P ⊂ G is a parabolic subgroup. Recall that a Cartan geometry (P, ω) of type (G, H) is a principle bundle H → P → M, endowed with a Cartan connection ω. The 1-form ω takes value in g, the Lie algebra of G, and satisfies the following conditions.
(1) ω u ∶ T u P → g is an isomorphism of vector space for all u ∈ P. ( A canonical way of identifying a geometric structure as a parabolic geometry is by characterizing it in terms of certain bracket generating distributions possibly equipped with a tensor field, e.g., conformal structures or semi-Riemannian structures. Recall the following definitions. To any distribution D ⊂ T N , one can associate a Pfaffian system One obtains that On a manifold M, a bracket generating distribution, D, of depth µ induces a filtration of T x M for all x ∈ M, by Let m(x) denote the grading of T x M that corresponds to the filtration above, i.e., At each x ∈ M, the Lie bracket induces a bracket on m(x) called the Levi bracket which is tensorial and turns m(x) into a graded nilpotent Lie algebra (GNLA). The Lie algebra m(x) is called the symbol algebra of D(x). A distribution D of constant type has the property that m(x) ≅ m for some GNLA m for all x ∈ M. An interesting class of GNLA arises from pairs (G, P ) where G is a real or complex semi-simple group and P ⊂ G is a parabolic subgroup in the following way. A µ -grading for the Lie algebra of G, denoted by g, is a vector space decomposition for all i ≤ −1, i.e., the negative part of the grading, is generated by g −1 ; and (iii) g ±µ ≠ 0.
The subalgebra g − is GNLA and is always a parabolic subalgebra of g. Its corresponding subgroup, P ⊂ G, is a subgroup that lies between the normalizer N G (p) and its identity component. The subgroup G 0 ⊂ P defined as is called the Levi subgroup corresponding to P.
For such a choice of (G, P ), there is a canonical construction of a bracket generating distribution whose symbol algebra is g − . Let N g be the simply connection Lie group with Lie algebra g − . The left-invariant vector fields obtained from g −1 induces a distribution of constant type on N g whose symbol algebra is isomorphic to g − .
For a semi-simple Lie algebra g every derivation is inner. Therefore given a µ -grading of g it admits a unique grading element i.e., e ∈ g 0 such that [e, g] = jg for all g ∈ g j and −µ ≤ j ≤ µ.
Consequently, the space of q-forms on g − with values in g, denoted by Λ q (g − , g), admits a decomposition ⊕ p Λ p,q (g − , g) where Λ p,q (g − , g) is the set of q-forms ω, with weight p, i.e., L e ω = pω. 16 The decomposition can be written as According to a theorem of Tanaka [Yam93], if H 1,p (g − , g) = 0, for all p ≥ 0, then via a process called the Tanaka prolongation the Lie algebra of infinitesimal symmetries of a distribution with symbol algebra g − is isomorphic to g, i.e., inf(D) = g.
The above theorem can be applied to regular infinitesimal flag structures of type (G, P ), which is defined as follows. Consider a manifold M endowed with a bracket generating distribution of constant type whose symbol algebra is m. The graded frame bundle is a right principal bundle Aut gr (m) → F gr (M ) → M, where Aut gr (m) is the group of automorphisms of m that preserve the grading. 17 If the symbol algebra of the distribution is isomorphic to g − and Aut gr (m) is isomorphic 18 to Ad(G 0 ) ⊂ Aut gr (g − ), where G 0 is the Levi subgroup of (G, P ) and Ad is the adjoint action, then one says that the distribution induces a regular infinitesimal flag structure of type (G, P ).
By a result of Tanaka [Tan79], Morimoto [Mor93] andČap-Schichl [ČS00], there is an equivalence of categories between regular, normal parabolic geometries and regular infinitesimal flag structures via Tanaka prolongation. Regularity and normality of a parabolic geometry (G, P ) is defined in terms of its curvature function of its Cartan connection (P, ω) as follows.
Therefore, at each point s ∈ P, the curvature Ω = dω + [ω, ω] of the Cartan connection can be used to define the curvature function K ∶ P → ∧ 2 g * − ⊗ g as where the space g * − can be identified with g + ∶= ⊕ i>0 g i via the Killing form. The Spencer cohomology groups H p,q ⊂ g ⊗ ∧ q (g * − ), are defined using the Kostant operators ∂ ∶ Λ p,q → 16 In [Yam93], the space Λ p,q is referred to as C p−q+1,q . 17 In case M is endowed with additional structure, e.g., conformal structures or semi-Riemannian structures, this principal bundle can be reduced further. 18 If there exists additional structure on the distribution then Autgr(m) reduces to G 0 .
Λ p−1,q+1 , and ∂ * ∶ Λ p,q → Λ p+1,q−1 , given by where c ∈ Λ p,q (g) and [e * i , X j ] − denotes the g − -component of [e * i , X j ], with respect to the decomposition g = g − ⊕ p. The operators above result in the orthogonal decomposition where the operator ◻ ∶= ∂ * ∂ + ∂∂ * is referred to as the Kostant Laplacian and ◻H p,q = 0. The Λ p,2 -component of the curvature function K is denoted by K (p) . Consequently, K can be decomposed as where K (p) ∶ g i × g j → g i+j+p is the component of K with homogeneity p. Additionally, one can decompose the curvature function as The components of the curvature lying in H p,2 are called harmonic curvature which constitute the set of essential invariants.
A Cartan connection is called torsion-free if K − = 0, normal if ∂ * K = 0, and regular if it is normal and K (i) = 0, for all i ≤ 0.

2.4.2.
Characterizing causal structures by a bracket generating distribution. The fact that [ε ab ] is non-degenerate allows one to give a characterization of causal structures in terms of a bracket generating distribution with a certain small growth vector obtained as follows.
Define the distributions ∆ and D such that (∆) ⊥ = I bas and D ⊥ = {ω 0 , ⋯, ω n−1 }. The rank of D is n and ∆ is a completely integrable sub-distribution of D of corank 1 whose integral manifolds are given by the fibers C x . Let v be a characteristic vector field. It follows that v ∈ D and v ∉ ∆.
Let {f 1 , ⋯, f n−1 } be a dual basis for ∆ and e n be a basis for D ∆. Since the growth vector is (n, 2n − 1, 2n), it follows that rank(D 2 D) = n − 1. Because ∆ is Frobenius, one obtains that the set of vectors e a = [e n , f a ] are linearly independent and span D 2 D. This relation among e a , e n and f a is equivalent to the structure equation dω a ≡ −ε ab θ b ∧ ω n modulo I{ω 0 , ⋯, ω n−1 }.
Furthermore, the equation dω 0 ≡ θ a ∧ ω a modulo I{ω 0 }, implies that there is a nondegenerate pairing between e a 's and f a 's, i.e., [e a , f a ] = e 0 (no summation involved ) for all 1 ≤ a ≤ n − 1, where e 0 spans D 3 D 2 .
One can investigate the action induced by the structure group G 5 in (2.34) on the symbol algebra of D as follows. Each tangent space T x C admits a filtration D(x) ⊂ D 2 (x) ⊂ D 3 (x) which results in the grading m( Using the Lie bracket, m(x) becomes a GNLA which would be the symbol algebra of D. Since g −1 is given by the kernel of I 1 ∶= I{ω 0 , ⋯, ω n−1 }, the action of the structure group G 5 on I tot = I{ω 0 , ⋯, ω n , ⋯, θ n−1 }, modulo I 1 , is given by scaling on ω n and by conformal transformations on θ a , i.e. , This action is isomorphic to the action of the Levi subgroup of O(p+2, q +2) corresponding to the parabolic subgroup P 1,2 if n ≥ 3 or P 1,2,3 if n = 3, on the coframe bundle. In other words, in causal structures the group of automorphisms of the coframe bundle which preserves the grading of T x C is reduced to G 0 .

Normal Cartan connection.
Using the flat model 2.2, it follows that the symbol algebra arising from the bracket generating distribution D, defined in Section 2.4.2 as {ω 0 , ⋯, ω n−1 } ⊥ , is isomorphic to the negative part of the 3 -grading for g = o(p + 2, q + 2), i.e., The parabolic subalgebra that corresponds to this 3 -grading is denoted as p 1,2 when n ≥ 4 and p 1,2,3 when n = 3. The corresponding parabolic subgroup is given in Section 2.2, i.e., the parabolic subgroup of O(p + 2, q + 2) that is isomorphic to the stabilizer of the subspace spanned by {e 1 , e 2 } where {e 1 , ⋯, e n+3 } is some basis for R n+3 and p + q = n − 1. Alternatively, p 1,2 can be described by crossing the first two nodes in the Dynkin diagrams B n and D n when n ≥ 4 and all three nodes when n = 3. Furthermore, the distribution D induces a regular infinitesimal flag structure. This is due to the fact that the condition H p,1 (g − , g) = 0 is satisfied for all p ≥ 0. More directly, by the discussion in Section 2.4.2 and the first order structure equations, one obtains that Aut gr (g − ) is reduced to the Levi subgroup. Thus, by Tanaka prolongation one can view causal structures as regular, normal parabolic geometries of type (B n , P 1,2 ) or (D n , P 1,2 ) if n ≥ 4, or (D 3 , P 1,2,3 ). In Section 2.5.1 it is discussed that when n ≥ 4, causal structures are canonically isomorphic to regular, normal parabolic geometries of type (B n , P 1,2 ) or (D n , P 1,2 ). When n = 3, causal structures are "subgeometries" of regular, normal parabolic geometries of type (D 3 , P 1,2,3 ), i.e., the integrability of the fibers C x , implies the vanishing of one of the harmonic curvature components of such parabolic geometries.
Knowing the existence of a normal Cartan connection for causal structures, the normality conditions of Tanaka are straightforward to obtain and can be used to recursively define the correct change of basis which transforms the {e}-structure to a Cartan connection (see [Mak16]). Since finding the explicit expressions for the change of basis is very tedious and the result is not particularly illuminating, finding the normal Cartan connection is not carried out here.

Essential invariants and the Raychaudhuri equation.
In this section the essential invariants whose vanishing imply the flatness of the causal structure are introduced. Their geometric interpretation is given. A variational problem is naturally associated to a causal geometry which allows one to define the notion of null Jacobi fields in the causal setting. Finally, the section ends with a discussion on associated Weyl connections of a causal structure and a derivation of the Raychaudhuri equation for causal structures. In this section there are references to the Appendix, particularly to the {e}-structure obtained in Section A. For an understanding of W anbn , consider the 2-plane given by λ ∧ µ where λ = λ i e i , µ = µ i e i ∈ T M for some basis {e 0 , ⋯, e n }. The sectional conformal curvature of a pseudoconformal structure evaluated at λ ∧ µ is defined by As a result, at the point (x, [y]) ∈ C, the invariant W nabn can be interpreted as the part of the conformal sectional curvature acting on 2-planes ξ a = ∂ ∂ω n ∧ ∂ ∂ω a . According to the construction of the coframes in (2.11), ξ is a subset ofT y C x and containsŷ where [ŷ] = y ∈ C x . It follows that ξ a = µ a ∧ y i ∂ i , where µ a is an element of the shadow space As a result, W anbn can be thought of as the restriction of the bilinear form W ijkl y j y l to the shadow space. This restriction is well-defined up to scale. Infinitesimally, this scaling action is realized by the 1-form φ n as is apparent from (2.39d). Finally, it can be argued that the Fubini cubic form and the Wsf curvature coincide with the harmonic curvatures of the parabolic geometries that correspond to causal structures. By the discussion in Section 2.4.3, causal structures on an (n + 1)-dimensional manifold correspond to parabolic geometries of type (B n , P 1,2 ) and (D n , P 1,2 ) if n ≥ 4, and (D 3 , P 1,2,3 ). As discussed in Section 2.4.1, a grading is induced on the cohomology classes of parabolic geometries. Moreover, the set of essential invariants for a parabolic geometry is given by the harmonic curvature. It turns out that for parabolic geometries of type (B n , P 1,2 ) and (D n , P 1,2 ) if n ≥ 4, the harmonic curvature has two irreducible components.
Using the grading (2.45) of the curvature, one component has homogeneity one whose g 0irreducible representation is given as a totally symmetric and trace-free cubic tensor. The other irreducible component has homogeneity two whose g 0 -irreducible representation is given as a symmetric and trace-free bilinear form (see [KT14,ANN15,Yam93]). The structure equations of causal geometries imply that the essential invariant F abc corresponds to a cohomology class of homogeneity one and the Wsf curvature corresponds to a cohomology class of homogeneity two. Also, from the infinitesimal group actions (2.39a) and (2.39d) one obtains that the Fubini cubic form and the Wsf curvature are closed under the action of the Levi subgroup. Moreover, one can impose the Tanaka normality conditions on the pseudo-connection form of the {e}-structure to derive a normal Cartan connection (see [Mak16]). It turns out that the necessary change of coframe does not affect the Fubini cubic form and the Wsf curvature, i.e., the essential invariants of the {e}-structure coincide with the harmonic curvature.
The situation is different in four dimensions. Four dimensional causal structures are a special class of parabolic geometries of type (D 3 , P 1,2,3 ). The harmonic curvature of these parabolic geometries is five dimensional, three of which have homogeneity one and two have homogeneity two. Consider a four dimensional causal structure of split signature. The Fubini cubic form of projective surfaces and the Wsf curvature are comprised of two entries (see Section 4). By Equation (4.8) all these four entries are irreducible under the action of the Levi subgroup. As a result, four dimensional causal structures are special cases of parabolic geometries of type (D 3 , P 1,2,3 ) in which one of the harmonic curvatures of homogeneity one is zero.
2.5.2. The ssf curvature. The action of the structure group G 5 defined in (2.34) on the bilinear form (2.35) is a scaling action g → a 2 b 2 g. Fix a representative g and consider the sub-bundle P 1 = {p ∈ P a 2 b 2 = 1}, where P denotes the structure bundle associated to the derived {e}-structure (see Section A.1.4). It follows that over P 1 one has for some functions s i and s a . Differentiating this equation again, it follows that (2.49) ds i ≡ π i + δ 0 i (s a γ a + s a π a ) + δ a i s b φ b a + s n γ a + s a π n mod I tot .
Consequently, s i can be translated to zero by the action of the structure group. It follows that (2.50) π i = p ij ω j + p ia θ a .
Substituting (2.50) and φ n = −φ 0 + s a θ a into the structure equations, equation (2.37d) becomes In conformal geometry, the symmetric part of p ij is referred to as the Schouten tensor. The quantities R a nbn and p nn are called the shadow flag curvature and the scalar shadow flag curvature (or the ssf curvature) of the causal structure respectively. 2.5.3. Null Jacobi fields and the tidal force. As was mentioned in section 2.3.5, the characteristic curves of causal structures is intrinsically defined by the quasi-contact structure of C. Nevertheless, one can associate a variational problem to causal geometries via the Griffiths formalism [Gri83] (see also [Hsu92]). As a result, the notion of null Jacobi fields naturally arises. The corresponding Jacobi equations result in another interpretation of the Wsf curvature. The fact that Jacobi equations of null Jacobi fields involve the Wsf curvature has to do with the second Lie derivative of g in (2.35), which is referred to as the tidal force [O'N83].
Recall that for the Pfaffian system I char = {ω 0 , . . . , ω n−1 , θ 1 , . . . , θ n−1 }, after fixing an interval I = (a, b), one can define the space of smooth immersions of its integral curves as The space U (I) can be given a Whitney C ∞ -topology. After making a choice of orientation for the characteristic curves, the space of the characteristic curves of C, U (I), is defined as U (I) Diff(I), i.e., the space obtained from U (I) by identifying the characteristic curves that are reparametrizations of each other.
Consider the functional Φ ∶ U → R, which at a point [γ] ∈ U (I) is defined by The triple (C, I, ω 0 ) is referred to as a variational problem on C, and is intrinsic to any causal structure. Since ω 0 (γ) = 0 everywhere, the above definition of Φ is independent of the choice of representative γ ∈ [γ]. This is in contrast with Finsler geometry where the geodesics are defined as the extremal curves of the arc-length functional which is always positive and are equipped with a specific parametrization. Intuitively, the infinitesimal variations of a characteristic curve γ can be identified with the tangent space T γ U (I) which is the space of smooth variational vector fields along γ. To give a more precise definition consider a parametrization for I = (a, b) and a map Γ ∶ (−ǫ, ǫ) → U (I). As a result of such parametrization, the curves given by Γ(s) can be parametrized. Using this parametrization, one can defineΓ ∶ (−ǫ, ǫ) × I → C, asΓ(s, t) = (Γ(s)) (t). The map Γ is called a compact variation of the curve γ if the corresponding mapΓ is smooth,Γ(0, t) = γ(t),Γ * (s 0 , t)(I char ) = 0, for any fixed s 0 , andΓ is compactly supported in I, i.e., for all values of s, the curvesΓ(s, t) andΓ(0, t) coincide outside of a compact subset of I. Then, T γ U (I) is expressed as The variational equations of a characteristic curve can be thought of as the first order approximation of T γ U (I). A practical derivation of the variational equations can be given in terms of the Pfaffian system I char .
Let η ∈ I char , and Γ a variation of the characteristic curve γ. By the definition ofΓ(s, t), one hasΓ * (s 0 , t)η = 0, and therefore L ∂ ∂sΓ * (s, t)η ≡ 0 modulo I{ds}. Setting s = 0, the latter expression implies that is the infinitesimal variations associated to Γ(s). In the expression of the variational equations one can drop the pull-back γ * and write them asĴ ⌟ dη + d(Ĵ ⌟ η) ≡ 0 mod I char . The tangential component of the vector fieldĴ(t) along γ(t) has no effect on the variational equations. Therefore,Ĵ along the curve γ(t) can be expressed aŝ The vector field J = π * Ĵ is called a null Jacobi field along the null geodesics π(γ(t)).
The variational equations forĴ(t) can be written aŝ Equation (2.53a) gives no new information. The last two equations give 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 where v is the characteristic vector field satisfying ω n (v) = 1.
Using equations (2.54), it follows that where sf W is viewed as an automorphism of the shadow space. Recall from Section 2.5.1 that W a nbn is given by the null Jacobi equation can be expressed as 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 . 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 ∶ C → P, define the null Jacobi fieldŝ J 1 (t), ⋯,Ĵ n−1 (t) along the characteristic curve γ(t), t ∈ [a, b] by the initial conditionŝ where t 0 ∈ [a, b] and v(t) is tangent vector to γ(t). Define the (1, 1) tensor field called the null Jacobi tensor as which is a (n−1)×(n−1) matrix whose bth column is the null Jacobi field J b (t) = π * (Ĵ 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 ′ A −1 , where A is invertible. The expansion, θ, vorticity tensor, ω and shear tensor, σ, are defined as 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 Using the fact that tr(ω) = tr(σ) = tr(ωσ) = 0 and tr( sf R) = (n − 1)p nn one arrives at which is the causal analogue of the Raychaudhuri equation along null geodesics.  (N, Σ), where N is n-dimensional and Σ ⊂ T N is of codimension one whose fibers Σ p ⊂ T p N are strictly convex hypersurfaces. The fiber bundle Σ is often referred to as the indicatrix bundle over which the Finsler metric has unit value. It is known [Car34,Bry02] that Σ is equipped with unique coframing η = (η 0 , η 1 , ⋯, η n−1 , ζ 1 , ⋯, ζ n−1 ) satisfying (2.59) c ζ b ∧ η c for some invariants I abc , J abc , R 0a0b and R 0abc . The 1-form η 0 is called the Hilbert form of the Finsler structure which, at the point v ∈ Σ p , is uniquely characterized by the conditions where H v Σ p ⊂ T p N is the tangent hyperplane to Σ p at the point v. As a result of the structure equation, η 0 induces a contact structure on Σ whose associated Reeb vector field defines the geodesic flow of the Finsler structure on Σ. Moreover, the quadratic form ω 0 ○ ω 0 + δ ab ω b ○ ω c is well-defined for the Finsler structure. It turns out that the quantity I abc = δ ad δ be I de c can be used to define the cubic form I abc ζ a ○ ζ b ○ ζ c , which, when restricted to a fiber Σ p , coincides with the centro-affine cubic form of the hypersurface Σ p ⊂ T p N. The quantities J abc measure the rate of change of I abc along the geodesics, i.e., J abc = ∂ ∂η 0 I abc . The quantity R 0a0b is symmetric using which the flag curvature of the Finsler structure can be represented by Let (N, Σ) and (Ñ ,Σ) be two Finsler manifolds. Given a strictly positive function f on their product space, at a point p ∈ M = N × fÑ , one can define a projective hypersurface whose associated cone is given by As a result, M is equipped with a causal structure of signature (n,ñ) whereñ = dimÑ . Note that it is assumed that the Finsler structures are reversible.
If G andG represent the Finsler metrics on N andÑ , respectively, then the causal structure on M is locally given as the zero locus of G − fG. If f only depends on N, then the above causal structure corresponds to the warped product of the Finsler structures on N andÑ .
A choice of 3-adapted coframing for the causal structure on M is given by (2.61) 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 (ω i ,θ a ) is given bỹ A 1 = − 1 2 (I 111 + I 122 ), A 2 = 1 2 (I 112 + I 222 ), ) are the components of the Fubini cubic form of the causal structure and A a ,b denotes the coframe derivative ∂ ∂ζ b A a . A choice of connection forms for the first order structure equations is given bỹ )ω 3 . Consequently, the Wsf curvature is given by the trace-free part of (2.62) ∂ ∂η a ln f. Moreover, the quantities K ab and L a are given by To see the symmetric relation K 12 = K 21 one needs to realize A 1 ,2 = A 2 ,1 . It is rather straightforward to generalize this example to obtain a causal structure on M = N × f 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 ) = 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⋯k are functions of x i 's.
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 = 1 3 (y 1 ) 3 +y 1 y 2 in P 3 . In Section 4.2, this example is used to characterize the submaximal causal structure of split signature. 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 ω i and θ a can be used to express the Fubini cubic form and Wsf curvature. Firstly, note that θ a = ∂ a ∂ b F dy b . To express F abc , consider the following transformation (2.64)ω 0 = a 2 ω 0 ,ω a = E a ω 0 + ω a ,ω n = eω 0 + ω n ,θ a = a 2 θ a .
Differentiatingω 0 , it follows that Consequently, in Equation (2.20b) for 1-adapted coframes one obtains By composing Equations (2.21) and (2.25), it follows that the Fubini cubic form F abc is defined via the equation dh ab + h ac ψ b c + h bc ψ a c − 2h ab (φ 0 + φ n ) ≡ F abc θ c mod I bas . Transformations (2.64) imply that φ n = 0. Using the above expressions for φ 0 and ψ a b , without normalizing h ab to ε ab , one can impose the apolarity condition h ab F abc = 0, as in (2.30), by setting Finally, using ∂ a ∂ b F to lower indices, it follows that 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 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 which by isotrivially flatness is zero. Hence, the reduction of π ab via (2.27), gives π ab − 1 n−1 ε cd π cd ε ab = 0. Combining this with the fact that θ a = 0 modulo I{θ 1 , ⋯, θ n−1 }, it follows that W a nbn = 0, i.e., locally isotrivially flat causal structures have vanishing Wsf curvature.

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 DF abc in (2.39a) (2.39a) is semi-basic, After taking the Lie derivative of (3.1) along characteristic curves and using the structure equations, one arrives at Assume f abc = λF abc , for some function λ. Then, if F abc ≠ 0, the infinitesimal group action on λ is given by dλ − 2λφ n + π n ≡ 0 mod I tot . By translating λ to zero, the 1-form π n reduces to (3.2) π n = p ni ω i + p na θ a .
for some coefficients p ni and p na . Replacing π n with the above expression in equation (2.37d), one finds the infinitesimal action of the group G 5 on p na to be given by Translating p na to zero results in the reduction Consequently, equation (2.37d) changes to W anbn = p nn , W anbc = −p nb ε ac + p nc ε ab , W an0n = p a0 , W an0b = p ab , E anb0 = −p ab .
Similarly, one finds that the normalization ε ab p ab = 0 results in π 0 ≡ 0 modulo I tot . It follows that Assume furthermore that p nn ≠ 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) dp nn − 4p nn φ n ≡ 0, mod I tot .
Hence, the normalization p nn = ±1 can be made, resulting in φ n reduced to (3.7) φ n = r i ω i + r a θ a .
Furthermore, the infinitesimal group action of the reduced structure group on r a is given by As a result, by translating r a to zero, the 1-forms γ a are reduced, i.e., After these reductions, the proposition below can be proved.
Proposition 3.1. A causal structure with vanishing Wsf curvature satisfying f abc = λF abc has vanishing ssf curvature.
Proof. As was discussed, the condition f abc = λF abc results in the reduction of the 1-forms π i according to equations (3.2), (3.3) and (3.5). In the case p nn ≠ 0, equation (3.6) it can be shown that φ n can be reduced to (3.7) by normalizing p nn to a non-zero value c.
Consequently, the 1-forms γ a can be reduced to (3.8). Now that all the 1-forms π i , φ n and γ a are reduced equation (3.4) can be written as where c denotes the normalized value of p nn . It is noted that the translations r a , p na , ε ab p ab to zero are used in the above equation. Similarly, express dω a and dω n by replacing γ a and φ n with their reduced forms in (2.37). The reduced structure equations can be used to express the identity d 2 θ a = 0. Collecting the coefficients of the 3-form θ a ∧ ω b ∧ ω n , it follows that (3.9) cF abc + (r c;n − r n;c + 1 2 p cn − p nc − r n q cn − r c r n )ε ab + p nb ε ac + p an ε bc + W abcn = 0, where W abcn = −W bacn is a torsion term for Φ 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 π a and γ a , it is found that If c ≠ 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 = λF 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.,İ 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 = λF 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 = λF 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 φ n is replaced by its reduced form −φ 0 + s a θ a and π 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 d 2 θ 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) dp nn + 4p nn φ 0 ≡ 0 mod I tot .
Hence, if it is non-zero, depending on its signature, it can be normalized to −1 or 1, resulting in the reduction Using the reduced structure equations, the identity dφ 0 , can be used to give the infinitesimal action of the reduced structure group on t a as dt a − t b φ b a − 1 2 γ a ≡ 0 mod I tot . Therefore, by translating t a to zero, the 1-forms γ a are reduced to Remark 3.3. Note that as a result of the conditions F ≠ 0 and p nn ≠ 0 and the subsequent reductions, the first structure equations take the form T ǫ = 1 2 T ǫ ij ω i ∧ ω j + T ǫ ai θ a ∧ ω i + 1 2 T ǫ ab θ a ∧ θ b , for some coefficients T ǫ ij , T ǫ ai , T ǫ ab , with ǫ representing i and 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 Therefore, the bilinear forms (3.14) 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 d 2 θ a = 0, using the structure equations after the reductions, the vanishing of the coefficient of θ a ∧ ω b ∧ ω n implies the equation 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 ω 0 , ω a , θ a . Now, let s ∶ C → P, be a section of the structure bundle, inducing the coframe (s * ω i , s * θ a ). The corresponding frame is denoted by . The vector field ∂ ∂(s * ω n ) is a tangent vector to the characteristic curves.
Definition 3.4. A characteristic curve γ, of a causal structure is called complete if for each section s ∶ C → P, the parametrization of γ with respect to ∂ ∂(s * ω n ) can be extended to R. Now the stage is set to state the following theorem.
Theorem 3.5. Let (M, C) be a causal structure with vanishing Wsf curvature over a compact manifold with non-zero Fubini cubic form. If the characteristic curves of C are complete, then the scalar flag curvature of (M, C) is non-negative.
Proof. Restrict a section of the structure bundle, s ∶ C → 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 ′′ − 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, 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 N . The construction is analogous to Grossman's notion of semi-integrable Segré structures arising from torsion-free path geometries [Gro00].
Let p ∈ C be a generic point, that is a point admitting an open neighborhood U ⊂ C foliated by characteristic curves. As a result, the quotient map τ ∶ U → N 2n−1 defines the space of characteristic curves in U .
Recall from Section 2.3.5 that a characteristic vector field v spans the unique degenerate direction of ω 0 and dω 0 , at every point i.e., It follows that L v ω 0 = 0. Therefore, there exists a 1-formω 0 ⊂ Γ(T * N ) of rank of 2n − 1 such that τ * ω0 = ω 0 . As a result, the quasi-contact structure on C defined by ω 0 induces a contact distribution on N . By abuse of notation, ω 0 is used to denoteω 0 as the distinction is clear from the context. Because v ⌟ ω a = v ⌟ θ a = 0, the 1-forms ω 0 , ⋯, θ n−1 introduce a coframe on N , and the {e}-structure F fibers over N .
To understand the resulting structure on 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 ⊂ 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 ≥ 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 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, N) is given. Let V be a 2N-dimensional vector space with a decomposition V ≅ R 2 ⊗R N .
An element p ∈ V is said to be of rank one if and only if p = a ⊗ b. A Segré cone of type (2, N), denoted by S (2, N), is defined as the set of rank one elements of this decomposition.
In terms of local coordinates (z 1 1 , ⋯, z 1 N , z 2 1 , ⋯, z 2 N ) adapted to this decomposition, the Segré cone is given by the points satisfying the homogeneous quadratic equations As a result, a Segré cone of type (2, N) has a double ruling. A (N − 1)parameter ruling by 2-planes E v ≅ R 2 ⊗ v where v ∈ R N . Such 2-planes are called α-planes [AG96]. Another ruling of S(2, N) is given by the 1-parameter family of N-planes u ⊗ R N where u ∈ R 2 . Such N-planes are called β-planes. 19 The proposition below is used to show how a Lie contact structure is induced on N .
Proposition 3.7. Given a causal structure with adapted coframe (ω i , θ a ), the span of the bilinear forms restricted to H = Ker ω 0 is invariant under the action of the structure group G 5 . Moreover, C has vanishing Wsf curvature, if and only if the Span{Υ 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 Υ ab modulo I{ω 0 } is invariant under the action of G 5 .
Let v denote a characteristic vector field. To verify the second claim it suffices to note that (2.37) implies Thus, if the condition W a nbn = 0 is satisfies, i.e., causal structure has vanishing Wsf curvature, then the span{Υ ab } is invariant along the characteristic curves.
As a result of the proposition above, the set of tangent vectors u ∈ H satisfying Υ ab (u, u) = 0 is well-defined and gives a Segré cone of type (2, n − 1) in the contact distribution H ⊂ T N .
Definition 3.8. A manifold N 2N+1 with a contact distribution H is said to have a Lie contact structure of signature (p, q), p + q = N if there is a smoothly varying family of Segré cones S(2, N) ⊂ H x such that at every point x ∈ N , its β-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 ∈ N and a β-plane in S(2, N) ⊂ H x , there passes a unique member of the family through x, tangent to that β-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 bundleρ ∶ E β → N where E β denotes the bundle of β-planes of the Segré cones. Because β-planes are Lagrangian subspaces of the contact distribution, E β is a sub-bundle of LG(N ) where ρ ∶ LG(N ) → N is the bundle of Lagrangian-Grassmannians of the contact distribution H. Since, the bundle mapρ is the restriction of ρ to the sub-bundle E β , by abuse of notation, it is denoted by ρ 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 (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 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 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 Υ ab , it follows that the β-planes, i.e., the 1-parameter ruling of the Segré cone, are given by (3.17) Ker {Aθ a + Bω a a = 1, . . . , n − 1}, 19 The relation between these α-and β-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).
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 ̟ ∶ C → E β between the fiber bundles τ ∶ C → N and ρ ∶ E β → N ,.
Proof. Let γ t = (x(t); y(t)) be a characteristic curve of C which corresponds to the point [γ] ∈ N . Let D be the completely integrable distribution of T C given by the vertical tangent spaces T y C x ∶= Ker I bas with D t ∶= T y(t) C x(t) . By the discussion above, it follows that, at t = t 0 , τ * D t0 ⊂ H [γ] is isotropic, (n − 1)-dimensional and lies in the zero locus of Υ ab , i.e. τ * D t0 is a β-plane. By defining one obtains the desired diffeomorphism.
Continuing the proof of Theorem 3.10, note that the fibers C x , as the integral manifolds of D, give a foliation of E β via the diffeomorphism ̟. Because of the transversality of the fibers C x and characteristic curves γ passing through x, it follows that the subsets τ (C x ) ⊂ N are immersed as (n − 1)-dimensional submanifolds of N . These submanifolds are Legendrian because it was shown that their tangent spaces, τ * D, are β-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 ω 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 {η a , ζ a a = 1, ⋯, n − 1} complementing ω 0 in T * N such that the Segré cone can be written as the vanishing set of the bilinear forms As a result, the 1-parameter family of ruling (n − 1)-planes of the Segré cone can be expressed as Ker {tη a + (1 − t)ζ a a = 1, ⋯, n − 1} where t ∈ (0, 1). Let ρ ∶ E β → N denote the bundle of β-planes of the Segré cone. A point p ∈ E β represents a β-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 ⊂ T E β can be defined with fibers As a result, the subspaces ρ * (K p ) rule the Segré cone S(2, n − 1) ⊂ H ρ(p) ⊂ T ρ(p) N . The fibers of ρ ∶ E β → N are one dimensional and parametrize the ruling subspaces ρ * (K p ).
Let v be a vertical vector field of the fibration ρ ∶ E β → N satisfying ρ * (v) = ∂ ∂t , and by abuse of notation, let a multiple of ρ * (ω 0 ) be denoted by ω 0 as well. Hence, ω 0 ⊂ T * E β is a quasi-contact 1-form. Defining ω a = ρ * (tη a + (1 − t)ζ a ), it follows that L v ω a = ρ * (η a − ζ a ), and, consequently, The reason that it is non-zero, modulo I{ω 0 }, is that the Segré cone is a subset of the contact distribution.
Hence, locally, the tautological bundle is given by K = Ker {ω 0 , ⋯, ω n−1 }. As a result, the 1-forms ω 0 , ω a , θ a = L v ω a which are semi-basic with respect to the fibration ρ ∶ E β → N and the 1-form ω n , which is vertical, span T * p E β , and one has dω a ≡ −θ a ∧ ω n + T a bc θ a ∧ θ b mod I{ω 0 , ⋯, ω 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 E β by quasi-Legendrian submanifolds. The tangent distribution to this foliation is a corank sub-bundle of the tautological subbundle K given by Ker {ω 0 , ⋯, ω n }. As a result, ω i 's form a Pfaffian system which implies T a bc = 0. Moreover, knowing that K = Ker {ω 0 , ⋯, ω n−1 } and that ω 0 is semi-basic, one arrives at In the identity d 2 ω 0 = 0, the vanishing of the 3-form θ a ∧ θ b ∧ ω n implies that h ab = h ba .
Let M be the leaf space of the quasi-Legendrian foliation of E β , via the quotient map π ∶ E β → M. Because the tangent planes of the leaves are given by Ker {ω 0 , ⋯, ω n }, the 1forms {ω 0 , ⋯, ω n } are semi-basic with respect to the fibration π ∶ E β → M, and the 1-forms θ a are vertical. Moreover, the coframe on E β 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 I bas and the relations where θ a ∶= h ab θ b suffice to carry out the coframe adaptations and obtain the {e}-structure that characterizes a causal structure. Note that the distribution given by Ker {ω 0 , ⋯, ω n−1 } characterizes a causal structure according to Section 2.4.2. The one dimensional fibers of ρ ∶ E β → N are mapped to the characteristic curves of the causal structure.
Finally, since the vanishing set of the quadratic forms Υ ab defined in (3.16) is preserved, the Wsf curvature of the obtained causal structure has to vanish.

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 structures 20 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 pseudoconformal 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 [ε ab ] = 1 0 0 −1 . 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 It is known that the vanishing of F implies that the projective surface is ruled [Sas99,GH79]. In this section, it is assumed that F + = 0. The case F − = 0 can be treated analogously. Define the following quantities Because ε 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., θ 2 = −θ 2 , or F 2 11 = −F 211 . Define (4.2) ω ± ∶= 1 √ 2 (ω 1 ± ω 2 ), θ ± ∶= 1 √ 2 (θ 1 ∓ θ 2 ), γ ± ∶= 1 √ 2 (γ 1 ± γ 2 ), π ± ∶= 1 √ 2 (π 1 ± π 2 ). 20 In 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.
The structure equations (2.37) can be expressed as 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 F + , the quantity W + vanishes, then the Pfaffian system T generated by {ω 0 , ω + , θ + } becomes Frobenius. In fact, if W + = 0, then Bianchi identity (A.25) for W 0abc implies that V + = 0. This observation motivates the following definition.
Definition 4.1. A four dimensional causal structure is called half-flat, if the torsion coeffi- 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 = ω 0 ○ ω 3 − ω + ○ ω − then at x ∈ M, there is a ruling of the null coneĈ x ⊂ T x M by the null 2-planes over which one has (4.4) ω 0 + λω − = 0, ω + + λω 3 = 0, for some λ ∈ R. In fact, λ parametrizes such ruling 2-planes (see [AG96]). There is another ruling ofĈ x , given by the null 2-planes over which which are parametrized by µ. Assuming that M is oriented and that the null tetrad (ω 0 , ω + , ω − , ω 3 ) induces a positive orientation for M, the set of null 2-planes parametrized by λ and µ is referred to as α-planes and β-planes, respectively 21 . A 2-surface in M is called an α-surface or β-surface if its tangent space at each point is an α-plane or β-plane, respectively. A pseudo-conformal structure is called half-flat if there exists a 3-parameter family of α-surfaces in M, or equivalently, there exists a 3-parameter family of 2-surfaces with the property that at every point M, any given α-plane is tangent to a unique 2-surface in the family. Such pseudo-conformal structures are also called rightconformally flat [Pen76], self-dual [LM07], α-semi-integrable [AG96], or half-conformally flat [Bes08]. If M contains a 3-parameter family of β-surfaces then it is called half-flat as well. Alternatively, the terms left-conformally flat, anti-self-dual or β-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 ∈ M rule the null cone C x .
Proof. As mentioned before, if W + = 0, then the Bianchi identity (A.25) for W anbc implies that V + = 0. As a result, the Pfaffian system T generated by {ω 0 , ω + , θ + } becomes completely integrable. The integral manifolds of T are three dimensional submanifolds of 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 ∂ ∂ω − , ∂ ∂ω 3 , ∂ ∂θ − , where the tangent vector ∂ ∂θ − is vertical with respect to the fibration π ∶ C → M. Now, consider the foliation of C given by the integral curves of the Pfaffian system U = {ω 0 , ω + , ω − , ω 0 , θ + }. The tangent directions of these integral curves are asymptotic directions of the fibers C x given by θ + = 0. The first step is to show that at each point x ∈ M these curves give a ruling of C x . Fix a point x ∈ M and an integral curve of θ + = 0 on C x denoted by ρ(t). Letρ(t) denote the vector ∂ ∂θ − tangent to ρ(t). The structure equations (4.3) imply for some function f (x; y). As a result, defining P = Ker {ω 0 , ω + }, the 2-plane π * (P) ⊂ T x M is well-defined and can be represented by the bi-vector π * ∂ ∂ω − ∧ ∂ ∂ω 3 . Moreover, because π * (P) ⊂T ρ(t) C x , for all values of t, it follows that the cone over the curve ρ(t), i.e., ρ(t) ⊂ T x M, is a subset of π * (P). Hence, the integral curves of θ + = 0 on C x are 2-plane in T x M, which give a ruling of C x .
Using the foliation of the fibers C x by the Pfaffian system U, one obtains the locally defined quotient map ̺ ∶ C → R. Moreover, because the integral curves of U are curves in the fibers of the bundle C → M, the quotient space R is a fiber bundle over M, denoted by (4.6) π rul ∶ R → M, and the fibers R x ∶= π −1 rul (x) are 1-dimensional. From the previous discussion it follows that R x can be identified with a generating curve of the ruled fibers C x .
By the definition of R, the integral manifolds of T foliate R by 2-surfaces and are transversal to the fibers R x . Under the projection π 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 ∈ M, are given by π * ∂ ∂ω − ∧ ∂ ∂ω 3 ⊂ T x M which, by our previous discussion, are the ruling 2-planes of C x .
Similarly, if the pair F − and W − vanish, the 2-surfaces are the projection of the integral manifolds of {ω 0 , ω − , θ − }.
Remark 4.4. Note that at each point (x; y) ∈ C the integral manifolds of U are null with respect to the quadratic form g = ω 0 ○ ω 3 − ω + ○ ω − 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 α-surfaces. In other words, the conditions F + = 0, and 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 C is equipped with a path structure whose paths are the image of the generating curve of the ruled surfaces 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, F + , K + , L + of a half-flat causal structure whose vanishing results in a half-flat conformal structure of split signature.
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 andÑ . Then, the half-flatness condition implies conformal flatness and is equivalent to the set of equations . Example 4.7 (Cayley's cubic scroll). The causal structure introduced in Example 2.6.2 has ruled fibers F + = 0. The expression for W + is quite long. However, it can be checked that if c 111 = c 111 (x 1 , x 3 ), c 12 = e 1 (x 1 )e 2 (x 2 )e 3 (x 3 ), 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.
As a result of the reductions above the structure bundle is reduced to 8 dimensions. If F + ≠ 0, due to the reduction (4.9), the infinitesimal action (4.8a) on F + is expressed as F + − 6F + φ 1 2 ≡ 0, mod I tot . Thus, the desired causal structure with 8 dimensional symmetry algebra satisfies 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 µi , r µν , p ij , p iµ via scaling. Hence, the only non-zero local invariant of the submaximal causal structure is F − whose structure equations, after carrying out the reductions explained above, are given by (4.10) 2 ) ∧ ω 3 , dθ + = 2φ 1 2 ∧ θ + , dθ − = 4φ 1 2 ∧ θ − 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 + 1 3 (y 1 ) 3 The conditions F + ≠ 0 and F − = 0 are no different and result in an equivalent causal structure with solvable 8-dimensional symmetry algebra.
A.3. Bianchi identities for W ijkl . The following relations hold among the torsion elements W ijkl in (A.17 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 .