Mathematische Zeitschrift (2024) 306:15 https://doi.org/10.1007/s00209-023-03412-5 Mathematische Zeitschrift On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability Adrià Marín-Salvador1,2 · Roberto Rubio2 Received: 27 June 2023 / Accepted: 1 November 2023 / Published online: 8 December 2023 © The Author(s) 2023 Abstract We  compute the 2 contact  manifold of null geodesics of the family of spacetimes S × S , g◦ − c2 dt 2 1 d 2 + , with g◦ the round metric on S2 and t the S1 - d,c∈N coprime coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on ST S2 ∼ = L(2, 1) under the natural projection L(2, 1) → L(2c, 1). We extend this computation to Z × S1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the man- ifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime. 1 Introduction A spacetime is a Lorentzian manifold together with a choice of a global timelike vector field, that is, a vector field of negative length at all points. For a spacetime, its space of null geodesics N consists of the family of unparametrized geodesics with null tangent vectors at all points [16, 20]. When N is a manifold, it can be equipped with a canonical contact structure H (see [17, 18]). The first author has been supported by the Spanish MEFP under the Beca de Colaboración BDNS 512590 and by “la Caixa” Foundation (ID 100010434) under the fellowship LCF/BQ/EU21/11890122. The second author has been supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No. 750885 GENERALIZED and by the Spanish State Research Agency under the grants PID2019-109339GA-C32 and RYC2020-030114-I. B Roberto Rubio roberto.rubio@uab.es Adrià Marín-Salvador adria.marin@maths.ox.ac.uk 1 University of Oxford, Oxford OX2 6GG, UK 2 Universitat Autònoma de Barcelona, 08193 Barcelona, Spain 123 15 Page 2 of 22 A. Marín-Salvador, R. Rubio The contact structure H has proved to be essential in the theory, yielding important results on causality, providing, for instance, obstructions to two events (points) being on the same non-spacelike curve [6, 7, 14]. The spaces of null geodesics and their contact structures were computed explicitly in some noncompact cases [2, 10] and the question of whether two spacetimes with diffeomorphic spaces of null geodesics must be diffeomorphic was addressed, under the name of reconstruction, in [3]. Until very recently, the only explicit cases where N was known to be a manifold were globally hyperbolic spacetimes, which are diffeomorphic to C × R for a Cauchy hypersurface C [4] and for which N ∼ = ST C [17], and Zoll (or Zollfrei) manifolds [11, 23], whose null geodesics are all periodic. In the last years this subject has attracted more attention with the negative answer to Guillemin’s conjecture that every Zoll 3-dimensional spacetime is covered by S2 × R [22], and new classes of examples for which N is a manifold [12, 13]. However, to the best of our knowledge, there are hardly any explicit calculations of spaces of null geodesics and their contact structures for compact spacetimes or results on the possibility of retrieving the spacetime from its space of null geodesics. Firstly, we consider the spacetimes (S2 × S1 , gc/d ) for the family of metrics gc/d = 2 g◦ − dc2 dt 2 with c, d ∈ N+ , where g◦ is the round metric on S2 and t is the angle coordinate on S1 . Its space of null geodesics Nc/d and its contact structure are described in terms of the lens spaces L(2c, 1) for gcd(c, d) = 1, by using a quaternionic approach to the Hopf fibration and ST S2 that we develop in Sect. 3.2. We prove, Theorem 3.5 For any c, d ≥ 1 with gcd(c, d) = 1, Nc/d ∼ = L(2c, 1). Theorem 3.14 The canonical contact structure H on ST S2 ∼ = N1/d is the canonical contact structure χ on ST S2 . In general, the canonical contact structure H on L(2c, 1) ∼ = Nc/d for c > 1 and gcd(c, d) = 1, is r∗ χ, where r : ST S2 → L(2c, 1) is the projection. Note that, for c > 2, the manifold L(2c, 1) is not presented as the unit tangent bundle of a manifold. Moreover, the study of S2 × S1 allows us to prove an analogue of Theorem 3.14 2 for the class of compact spacetimes {(Z × S1 , g Z − dc2 dt 2 )}c,d∈N+ , with Z a Zoll manifold (Proposition 3.17). Secondly, we bring methods of Engel geometry to deal with the spaces of null geodesics of three-dimensional spacetimes. An Engel structure on a four-dimensional manifold Q is a rank-two distribution D that generates a rank-three distribution E := [D, D] satisfying [E, E] = T Q, where we are referring to the bracket of sections. The distribution D defines a unique line distribution W, known as kernel, by the property [W, E] ⊆ E, which completes a flag W ⊂ D ⊂ E ⊂ T Q. There exists a canonical (Cartan) prolongation from a contact three-manifold (N , ξ ) to obtain an Engel distribution on S(ξ ). Similarly, given a three- dimensional Lorentzian manifold (M, g), one can canonically define an Engel structure on the projectivization of the bundle of null vectors PC of M. In Proposition 4.7, we compute the kernel of the Lorentz prolongation of an arbitrary Lorentzian three-manifold. We have later learnt that this computation is also made, with different techniques, in the preprint [24, Thm. 1.3], but with no mention to the contact structure (which we discuss in Theorem 4.8 below). Next, by considering the Cartan deprolongation of the Lorentz prolongation of a spacetime, we show, 123 On the space of null geodesics... Page 3 of 22 15 Theorem 4.8 Let M be a three-dimensional spacetime. Then, N∼ = PC/W. In addition, if N is a manifold and p : PC → PC/W ∼ = N is a submersion, the canonical contact structure on N is H∼ = p∗ E. Finally, we make use this theorem to make a first step towards the characterization of the three-dimensional contact manifolds that are the space of null geodesics of a spacetime by the existence of a certain Engel manifold together with a foliation, and retrieve the spacetime. Theorem 4.9 A three-dimensional contact manifold (N , ξ ) is contactomorphic to the space of null geodesics of a spacetime if and only if there exists an Engel manifold (Q, D) with Engel flag W ⊂ D ⊂ E ⊂ T Q such that N∼ = Q/W and ξ∼ = p∗ E, (10) for p : Q → Q/W the projection, and Q admits an oriented foliation by circles F such that (i) for all S ∈ F and x ∈ S, we have Tx S ⊕ Wx = Dx , (ii) the space of leaves M := Q/F is a manifold and the projection q : Q → M is a submersion, (iii) for every S ∈ F, the image q∗ D| S is a cone in the vector space Tq(S) M and the map x ∈ S → q∗ Dx is injective. In addition, if the above conditions are satisfied, (N , ξ ) is contactomorphic to the space of null geodesics of (Q/F, g), where g is a metric on Q/F with bundle of cones q∗ D. The main objects at play and our results are visually presented as follows. ∼ Cor.4.14 (PC, D) (S(ξ ), D) Lorentz prolongation Cartan prolongation Thm.4.10 Thm.4.8 (M, g) (N , ξ ) Space of null geodesics Corollary 4.14 deals with the relation between the Cartan and Lorentz prolongations, and opens the interesting question of the relation between the several Engel manifolds having the same Cartan deprolongation. 2 Definitions and basic properties 2.1 The space of null geodesics of a spacetime We work throughout the paper in the category of smooth manifolds. For a Lorentzian manifold (M, g), a nonzero vector v ∈ T M is said to be timelike, spacelike or null if g(v, v) is, respectively, negative, positive or zero. A smooth curve γ : I → M is timelike, spacelike or null if its velocity vector γ̇ is so everywhere. Likewise, we talk about timelike, spacelike and null submanifolds or vector fields. The set of null vectors on a Lorentzian manifold M has the structure of a smooth bundle π : C → M, whose fibres consist of two hemicones. We denote such a choice by C + . A differentiable choice of one of such hemicones, when possible, makes M time-oriented. Time-orientability is equivalent to the existence of a global timelike vector field X ∈ X(M). 123 15 Page 4 of 22 A. Marín-Salvador, R. Rubio Definition 2.1 A spacetime is a time-oriented connected Lorentzian manifold of dimension ≥ 3. Definition 2.2 The space of null geodesics N of a spacetime (M, g) is N := {γ (I ) | γ : I → M is a maximal geodesic with γ̇ ⊂ C + }. The space N can be constructed as the leaf space for a distribution on C + . Recall that the geodesic spray X g ∈ X(T M) is the vector field on T M whose integral lines are γ̇ (t) ∈ Tγ (t) M for γ : I → M a geodesic, whereas the Euler vector field  ∈ X(T M) is defined as (v) = T0 c(∂s ), with v ∈ Tx M and c : R → Tx M given by c(s) = es v, and whose differential at 0 we denote by T0 c. Note that c is an integral line of . The geodesic spray and the Euler field are tangent to the bundle C + and define an integrable distribution X g ,  (see [2]). Note that, by quotienting C + by the Euler field, we obtain the projectivization of the bundle C + , which is relevant as we only care about unparametrized null geodesics. Then, by quotienting by the geodesic spray, we identify directions in different projectivized cones for which there exists a geodesic in M going through both of them. Thus, we have N∼ = C +/ X g ,  . (11) From now on, we will consider the case in which N is a manifold. Remark 2.3 A sufficient condition for N to be a manifold is found in [2] (namely, when the spacetime (M, g) is strongly causal and null-pseudo-convex). 2.2 The canonical contact structure Recall that a contact structure on a (2n + 1)−manifold N is a codimension-one distribution ξ ⊂ T N which is given, at least locally, as the kernel of a one-form α satisfying that the top form α ∧ (dα)n vanishes nowhere. We give two examples that will be relevant later. Example 2.4 For any manifold M, its unit cotangent bundle π̃ : ST ∗ M → M has a canonical contact structure (see, for instance, [9, Lem. 1.2.3]). A point ω ∈ ST ∗ M may be regarded as a linear form ω̃ ∈ Tπ̃∗(ω) M up to positive rescaling, which is determined by the hyperplane lω = ker ω̃ ⊂ Tπ̃ (ω) M. The canonical contact distribution on ST ∗ M is ξω := (Tω π̃)−1 (lω ) . Example 2.5 The unit tangent bundle π : ST M → M of a Riemannian manifold (M, g), which we define as ST M := {u ∈ T M | g(u, u) = 1}, has a canonical contact structure coming from ξ for ST ∗ M as in Example 2.4. Regard g as a map T M → T ∗ M and consider the contact structure   χ := g −1 ∗ ξ. Namely, for u ∈ ST M we have        −1    χu = g −1 ∗ ξ u = Tg(u) g −1 ◦ Tg(u) π̃ l g(u)   = (Tu (g ◦ π̃))−1 ker g(u) = (Tu π)−1 ( u ⊥ ), where u ⊥ denotes the orthogonal subspace to u in Tπ(u) M with respect to g. 123 On the space of null geodesics... Page 5 of 22 15 For the smooth manifold N, there is a canonical contact structure defined in terms of the so-called skies of the spacetime. Definition 2.6 Let (M, g) be a spacetime and x ∈ M. The sky of x is Sx = {γ ∈ N | x ∈ γ ⊂ M}. Note that, for any x ∈ M, the sky Sx is in correspondence with the projectivization of the cone C x . Hence, if m = dim M, we have Sx ∼ = Sm−2 . Definition 2.7 The canonical contact structure on the manifold of null geodesics N is the codimension-one distribution H defined as follows. For γ ∈ H, let x, y ∈ γ such that they cannot be joined by a one-parameter family of geodesics. Then, Hγ = Tγ Sx ⊕ Tγ S y . Note that the existence of two such points x and y follows from the fact that the exponential map provides a local diffeomorphism for any point of M. Proposition 2.8 [2, Sect. 2.4] The distribution H on the manifold N is well defined and is indeed a contact structure. 3 The contact manifold of null geodesics of (S2 × S1 , gc ) 3.1 The space of null geodesics Let (S2 , g◦ ) be the two-sphere with the round metric. Consider M = S2 × S1 and let t be the angle coordinate on S1 . For x ∈ S2 , we will refer to the point (x, t) ∈ M. Define, for c, d ∈ N+ coprime, the Lorentzian metric d2 2 gc/d = g◦ − dt . c2 The pair (M, gc/d ) is a Lorentzian manifold in which S2 × {t} is a spacelike surface for any t ∈ S1 , whereas {x} × S1 is a timelike submanifold for any x ∈ S2 . The vector field (0, ∂t ) ∈ T S2 ⊕ T S1 as a choice of future turns M into a spacetime. Lemma 3.1 The space of null geodesics Nc/d of (M, gc/d ) is given by  c   Nc/d ∼ = μ(s), s | μ is a unit-speed great circle in S2 . d Proof In a product chart using the coordinate t for S1 , the Christoffel symbol ikj of the metric gc/d vanishes whenever i, j or k equals 3, and all the others are the Christoffel symbols of g◦ in the chart of S2 . Hence, the geodesic equation for a curve γ : I → M defined by γ (s) = μ(s), t(s) is given by t¨ = 0, that is, t(s) = a + bs for some a, b ∈ R, and the geodesic equation for μ in S2 . Let u(s) ∈ T S2 be the vector tangent to the curve μ(s). Since S2 × {t} is a spacelike  surface for all t ∈ S1 , we can suppose, by reparametrizing γ , that gc/d (u, 0), (u, 0) = g◦ (u, u) = 1. Then, b2 d 2 b2 d 2 gc/d (γ̇ , γ̇ ) = g◦ (u, u) − ∂ t , ∂ t = 1 − , c2 c2 123 15 Page 6 of 22 A. Marín-Salvador, R. Rubio so γ is a future-pointing null geodesic if and only if b = dc , as − dc would give a past-pointing geodesic. By uniqueness of the geodesics in a pseudo-Riemannian manifold, all the null geodesics of (M, gc/d ) modulus reparametrization are of the form  c  γ (s) = μ(s), a + s , d where μ is a unit-speed great circle in S2 . Since γ intersects S2 × {0} at least at one point, we can suppose a = 0.   Remark 3.2 We use the notation Nc := Nc/1 . We start with the case c = 1. The speeds at which a geodesic travels the time direction S1 and the great circle in S2 are 1 to d. For γ ∈ N1 , there is a unique x ∈ S2 such that (x, 0) ∈ γ . Indeed, γ intersects S2 × {0} for s ∈ 2πdZ, and μ(2πdZ) is the unique point x on S2 . So γ is completely determined by x ∈ S2 and the tangent vector of the projection πS2 (γ ) at x, which is unitary by Lemma 3.1. In addition, any u ∈ ST S2 defines a unique null geodesic γ ⊂ M, which is the lift of the great circle μ ⊂ S2 defined by u, meaning that πS2 (γ ) = μ. Proposition 3.3 Thus, we have N1/d ∼ = N1 ∼ = ST S2 . Let us now consider Nc/d with c > 1. For a geodesic, the ratio between the turns around the time direction S1 and the turns around a great circle in S2 is c to d. Since geodesics are travelled at constant speed, every γ ∈ Nc/d intersects S2 × {0} at c points, namely those in μ(2π dc Z) = μ( 2πc Z), where the equality follows from the fact that gcd(c, d) = 1. These points are equidistantly spread over the great circle μ := πS2 (γ ), see Fig. 1. Conversely, any u ∈ ST S2 defines a unique null geodesic, which is the lift of the great circle defined by u to M. In order to get a proper description of N, one ought to identify the different elements of ST S2 defining the same null geodesic. If γ is a null geodesic of M that intersects S2 × {0}  (x, at  0) with  tangent  vector (u, ∂t ) ∈ ST S2 ⊕ T S1 , then γ intersects S2 × {0} at (x j , 0) := 2πd j μ c , 0 for j = 0, . . . , c − 1, where μ ⊂ S2 is the great circle defined by u, and     with velocity (u j , ∂t ) := μ̇ 2πdc j , ∂t . Note that (x j , u j ) can be obtained by a rotation 2πd j of (x, u) of c radians about the axis x × u,   xj cos 2πd j sin 2πd j x = c c . uj − sin c cos c j 2πd j 2πd u Hence, we have shown, Proposition 3.4 Consider the Zc -action on ST S2 generated by   y cos 2πd c sin 2πd c y → . v − sin 2πd cos 2πd v c c Then, Nc/d ∼ = ST S2 /Zc , where ST S2 /Zc denotes the orbit space of ST S2 under the action of Zc . 123 On the space of null geodesics... Page 7 of 22 15 Fig. 1 Null geodesic in S2 × S1 with c = 4, d = 1. The grey surface represents S2 × {0} and the radial coordinate is the S1 direction. The four elements of ST S2 represent the same red null geodesic γ The next step is to obtain an explicit description of the spaces Nc/d that allows us to compute their canonical contact structure. We will prove the following result. Theorem 3.5 For any c, d ≥ 1 with gcd(c, d) = 1, Nc/d ∼ = L(2c, 1). 3.2 A quaternionic approach to STS2 and the Hopf fibration In order to prove Theorem 3.5, we develop a quaternionic approach to ST S2 and the lens spaces L(2c, 1). Let H denote the division algebra of quaternions and V the three-dimensional vector space of pure imaginary quaternions. The canonical identification of H with R4 defined by α + ai + bj + ck → (α, a, b, c) gives an identification V ∼ = R3 , which provides V with a cross product induced by that of R : for u, v ∈ V, 3 uv − vu u×v = . 2 Let ∗ : H → H be the conjugation on H, which is an antiautomorphism allowing us to define a norm |q|2 = qq ∗ for q ∈ H. The restriction of such norm on V induces, via the polarization identity, an inner product on V defined, for u, v ∈ V, by uv + vu u, v = − , 2 which coincides with the euclidean inner product in R3 . We can also identify S3 ∼ = SH := {q ∈ H | |q| = 1} and S2 ∼ = SV := {u ∈ V | u, u = −u 2 = 1}. Finally, one has ST S2 ∼ = ST (SV) := {(u, v) ∈ SV × SV | u, v = 0}. Lemma 3.6 For any w ∈ SV, there exists a Hopf-like fibration map τw : SH → SV q → q −1 wq, which provides SH with the structure of an S1 -bundle over SV. The fibre over p ∈ SV is given by {ewθ q | θ ∈ R}, for any q ∈ τw−1 ( p). 123 15 Page 8 of 22 A. Marín-Salvador, R. Rubio Proof Let w ∈ SV. Let us show first that τw maps onto SV. Indeed, for q ∈ SH, q −1 wq + (q −1 wq)∗ = q −1 wq − q −1 wq = 0, q −1 wq, q −1 wq = −(q −1 wq)(q −1 wq) = −q −1 w 2 q = q −1 q = 1. We show next that the map is surjective. Let p ∈ SV and assume it is not collinear with θ w. Let θ = arccos w, p ∈ (0, π) and η = | p×w| p×w ∈ SV. Define q := eη 2 = cos θ2 + η sin θ2 . We have q −1 wq = (cos θ )w + (sin θ )(w × η), a rotation of w of angle −θ around the axis π η. Hence, q −1 wq = p, as needed. If p and w are collinear, take q = 1 or q = eu 2 , with u ∈ SV perpendicular to w, depending on whether p = w or p = −w. Finally, let q ∈ τw−1 ( p) be arbitrary. Then, if θ ∈ R, τw (ewθ q) = q −1 e−wθ wewθ q = q −1 wq = τw (q). In addition, if τw (q1 ) = τw (q2 ), then w = (q2 q1−1 )−1 w(q2 q1−1 ), which implies that q2 q1−1 = e−wθ for some θ ∈ R.   Proposition 3.7 The map  : SH → ST (SV) q → (q −1 kq, q −1 jq) provides a surjective local diffeomorphism in such a way that the preimage of a point in ST (SV) consists of exactly two antipodal points in SH. Proof Surjectivity follows using the same ideas as in the proof of Lemma 3.6. Also, for any q ∈ SH, we have (q) = (−q). If q1 , q2 ∈ SH are such that their images under  coincide, then q2 q1−1 = ekθk = e jθ j for some θ j , θk ∈ [0, 2π), which can only happen if θk = θ j = 0 or θk = θ j = π.   Remark 3.8 Actually, in Proposition 3.7 it is possible to replace j and k by any u, v ∈ SV such that u, v = 0 and the result remains true. Corollary 3.9 Let Z2 act on SH via the antipodal map. There exist diffeomorphisms SH/Z2 ∼ = ST (SV), and hence S3 /Z2 ∼ = ST S2 , which we also denote by . We establish now the connection to lens spaces. Definition 3.10 Consider the 3-sphere S3 ⊂ C2 and p ∈ Z+ . Define the Z p −action on S3 generated by  2πi 2πi  (z 0 , z 1 ) → e p z 0 , e p z 1 . The lens space L( p, 1) is the smooth manifold L( p, 1) := S3 /Z p . Identifying R4 ∼= C2 with H via (z 0 , z 1 ) → z 0 + z 1 j, the Z p action in Definition 3.10 2πi becomes q → e p q. Then, L( p, 1) = ∼ SH/Z p . Since the Z2 -action on SH that defines the lens space L(2, 1) is precisely the given by the antipodal map, we have shown, Proposition 3.11 We have ST (SV) ∼ = SH/Z2 ∼ = L(2, 1), so ST S2 ∼ = L(2, 1). This quaternionic approach allows us to formalize and prove the following result, which will give us, together with Propositions 3.3 and 3.4, the proof of Theorem 3.5. 123 On the space of null geodesics... Page 9 of 22 15 Proposition 3.12 Let  : SH/Z2 → ST (SV) and c ≥ 2. Then, the Z2c -action on SH that generates the lens space L(2c, 1) descends to a Zc -action on SH/Z2 that, via , induces the Zc -action on ST (SV) generated by    u cos 2π sin 2π u → c c . v − sin 2πc cos 2π c v Hence,  induces a diffeomorphism between ST (SV)/Zc and L(2c, 1). πi Proof Let q ∈ SH. The Z2c -action on SH is generated by q → e c q. Then, πi  πi πi πi πi   2πi 2πi  (e c q) = q −1 e− c ke c q, q −1 e− c je c q = q −1 ke c q, q −1 je c q . Now, consider (q) = (q −1 kq, q −1 jq). The result follows from 2π −1 2π −1  2π 2π  2πi cos q kq + sin q jq = q −1 k cos + i sin q = q −1 ke c q, c c c c 2π −1 2π −1  2π 2π  2πi −1 − sin q kq + cos q jq = q j cos + i sin q = q −1 je c q. c c c c   Proposition 3.12 is enough to prove Theorem 3.5 for d = 1. For d > 1, we need the following observation. Lemma 3.13 Let c, d ∈ Z+ coprime. The group automorphism of Zc defined by m → d · m sends the Zc −action on ST S2 in Proposition 3.12 to the Zc −action given by Equation (3.4). Hence, the two orbit spaces are diffeomorphic. Proof of Theorem 3.5 The case d, c = 1 follows from Proposition 3.3 and Proposition 3.11. The rest of the cases with d = 1 follow from Proposition 3.4 and Proposition 3.12. For d > 1, we make use of Lemma 3.13.   3.3 The canonical contact structures We next compute the canonical contact structures on the spaces L(2c, 1), c ≥ 1, which were seen as spaces of null geodesics. Note that, since the construction of the canonical structure is completely local, we can assume, without loss of generality, that d = 1. We do this throughout. Theorem 3.14 The canonical contact structure H on ST S2 = ∼ N1/d is the canonical contact structure χ on ST S2 . In general, the canonical contact structure H on L(2c, 1) ∼ = Nc/d for c > 1 and gcd(c, d) = 1, is r∗ χ, where r : ST S2 → L(2c, 1) is the projection. We break the proof of Theorem 3.14 into two lemmas, proving first the case c = 1 and then the cases with c > 1. Lemma 3.15 The canonical contact structure H on ST S2 ∼ = N1 is the canonical contact structure χ on ST S2 . Proof Let γ ∈ N1 ∼ = ST S2 . Recall that γ is the lift of the great circle μ : R → S2 defined by the pair (x, u) ∈ ST S2 representing γ . We will show that Hγ = χ(x,u) . 123 15 Page 10 of 22 A. Marín-Salvador, R. Rubio Fig. 2 The projection onto S2 of the sky of γ (τ ) is a circle tangent to u ⊥ Take (x, 0) ∈ γ and γ (τ )  = (x, 0), with 0 < τ < π. Note that all geodesics in the sky S(x,0) intersect S2 × {0} at (x, 0), so its projection is π(S(x,0) ) = {x}. Hence, Tx π(Tγ S(x,0) ) = {0} ⊆ u ⊥ and Tγ S(x,0) ⊂ χ(x,u) .   Consider now γ (τ )  = (x, 0). Since c = 1, we know that γ (s) = μ(s), s and hence   πS2 γ (τ ) = μ(τ ) =: y. Let v ∈ STy S2 such that μ̇(τ ), v = 0. Since all geodesics in S2 × S1 are travelled at the same speed, the projection of the sky of γ (τ ) is parametrized by π(Sγ (τ ) )(s) = y cos τ + (μ̇(τ ) cos s + v sin s) sin τ , which is a circle on (S2 , g◦ ) of radius τ and centre y, see Fig. 2, and π(γ ) = π(Sγ (τ ) )(0). Hence,  d Tx π(Tγ Sγ (τ ) ) = (y cos τ + (μ̇(τ ) cos s + v sin s) sin τ ) ds s=0 ⊥ = v sin τ ⊂ u , which implies Tγ Sγ (τ ) ⊂ χ(x,u) .   Since the distributions H and χ on ST S2 both have rank 2, the Lemma is proved. Let r : ST S2 ∼= L(2, 1) → L(2c, 1) be the canonical projection for c > 1. We use the notation [u] := r (u) ∈ L(2c, 1), which is the class of u ∈ ST S2 under the action of Zc . Let [u] ∈ L(2c, 1) and U a neighbourhood of u for which r |U : U → r (U ) is a diffeomorphism. We will show that H[u] = (r |U )∗ χu . The following lemma concludes the proof of Theorem 3.14. Lemma 3.16 For c > 1, the canonical contact structure H on L(2c, 1) ∼ = Nc , as the space of null geodesics, is r∗ χ. Proof Let x := π(u). We know that [u] describes the geodesic γ in S2 × S1 that intersects π(U ) × {0} only at (x, 0). Take (x, 0) ∈ γ and consider its sky S(x,0) . It is clear that S(x,0) = {[v] | v ∈ STx S2 }, and thus  −1      π ◦ r |U T S ∗ [u] (x,0) =Tx π(v) | v ∈ STx S2 =Tx ({x}) = {0} ⊂ u ⊥ , −1   from which we deduce that (r |U )∗ T[u] S(x,0) ⊂ χu and T[u] S(x,0) ⊂ (r |U )∗ χu . 123 On the space of null geodesics... Page 11 of 22 15 Take now γ (τ )  = (x, 0) but close enough so that Sγ (τ ) ⊂ U . As discussed previously, −1 π ◦ r |U (Sγ (τ ) ) describes a circle φ(s) in π(U ) whose tangent vector at x is orthogonal to u. Hence,   −1   −1 (π ◦ r |U )∗ T[u] Sγ (τ ) =Tx π ◦ r |U (Sγ (τ ) ) ⊥ =Tx {φ(s) | s ∈ R} ⊂ u , −1   which implies that (r |U )∗ T[u] Sγ (τ ) ⊂ χu , and hence T[u] Sγ (τ ) ⊂ (r |U )∗ χu . Therefore, Hγ = T[u] S(x,0) ⊕ T[u] Sγ (τ ) ⊂ (r |U )∗ χu and, since r |U is a submersion, equality follows. If v ∈ ST S2 is such that r (v) = [u] = r (u) and V is a neighbourhood of v such that r |V is a diffeomorphism, following the same argument, we have (r |V )∗ χv = Hγ = (r |U )∗ χu , and r∗ χ is indeed well defined.   3.4 Generalization to a class of compact spacetimes We extend now the results of Sect. 3.3 to the class of spacetimes d2 2 {(Z × S1 , g Z − dt )}c,d∈N+ coprime , c2 where (Z , g Z ) is a Zoll manifold, that is, such that all of its geodesics are closed and of the same minimal period (which we normalize to 2π). By Hopf-Rinow theorem, every Zoll manifold is compact, since it is clearly bounded. 2 Proposition 3.17 The space of null geodesics Nc/d of (Z ×S1 , g Z − dc2 dt 2 ) with c, d coprime, is diffeomorphic to the manifold ST Z /Zc , where the generator of Zc identifies tangent vectors on the same geodesic after 1/c of a turn, with the pushforward of the canonical contact structure on ST Z . Proof The same argument as in the proof of Proposition 3.3 implies N1/d ∼ = ST Z . For the contact structure, we can assume d = 1. We follow the proof of Lemma 3.15, which is geometrically more intuitive. Given a geodesic γ and (x, 0) ∈ γ ⊂ Z × S1 , all geodesics in the sky S(x,0) intersect Z × {0} at (x, 0) and we analogously have Tγ S(x,0) ⊂ χ(x,u) . For a different point γ (τ ) in the geodesic (close enough to γ (0)), the sky Sγ (τ ) is a sphere consisting of centre y := π Z (γ (τ )) and radius the distance between x and y. Since u points from x towards the direction of y, we have in general Tx π Z (Tγ Sγ (τ ) ) ⊂ u ⊥ . Indeed, let w ∈ Ty Z such that x = exp y (w) and complete it to an orthogonal basis (w, w2 , . . . , wn ) of Ty Z . The √ sphere π Z (Sγ (τ ) ) is the image under exp y of the sphere in Ty Z of radius g Z (w, w). Now, note that Tw exp y (w) ∈ u , and Tx (π Z (Sγ (τ ) )) = Tw exp y (w2 ), . . . , Tw exp y (wn ) . By Gauss’s Lemma, Tw exp y (w), Tw exp y (wi ) = w, wi = 0 for all 2 ≤ i ≤ n, and the claim follows. For the case c ≥ 2 and d = 1 we have an analogue of the end of Sect. 3.1. The generator of Zc identifies tangent vectors on the same geodesic after 1/c of a turn. The action is then free (as only the identity element would fix a vector). Since Zc is a finite group, the action is proper and the space ST Z /Zc is a manifold. For d > 1, we have an analogue of Proposition 3.12 and the lemma right after ensuring that Nc/d ∼ = Nc . For the contact structure, the proof of Lemma 3.16 applies by taking γ (τ ) close enough to (x, 0) so that the action of the group is trivial and replacing the circle φ(s) by the sphere described above.   123 15 Page 12 of 22 A. Marín-Salvador, R. Rubio Note that the purpose of Sect. 3.2 is giving a concrete description of ST Z /Zc for c ≥ 2, which we cannot have in the generality of Proposition 3.17, in the case of Z = S2 . The main examples of Zoll manifolds are Zoll surfaces (which are always spheres with Zoll metrics) and compact symmetric spaces of rank one [5]. 4 Engel structures as a tool in retrievability 4.1 Engel geometry and prolongations We recall here the main definitions on Engel manifolds and present the Cartan and Lorentz prolongations of, respectively, a contact and a Lorentzian three-manifold [8, 21]. A rank-three distribution E ⊂ T Q on a four-manifold Q is said to be an even-contact structure if it is everywhere non-integrable, that is, if [E, E] = T Q. Definition 4.1 Let Q be a four-manifold. A rank-two distribution D ⊂ T Q on Q is an Engel structure if E := [D, D] is an even-contact structure on Q. An Engel structure D on a four-manifold Q defines a unique line field W ⊂ E by the relation [W, E] ⊆ E. The line field W is known as the kernel (or characteristic line field) of the distribution and it can be shown to lie in the two-distribution D, for which it completes a flag W ⊂ D ⊂ E ⊂ T Q. Example 4.2 [Cartan prolongation] Let (N , ξ ) be a contact three-manifold and consider the S1 −bundle πC : S(ξ ) → N , where S(ξ )x is the quotient of ξx \{0} by the relation v ∼ λv for all λ ∈ R+ . We regard points in S(ξ ) as pairs (x, R) with x ∈ N and R an oriented line in ξx . The canonical Engel structure on S(ξ ) is D(x,R) := (T(x,R) πC )−1 (R). Let V , Y = ξ be a local frame on an embedded ball B ⊂ N . Then, B × S1 ∼ = S(ξ )| B via (x, t) → (x, R := X := V cos t + Y sin t), where  denotes the oriented spanned line. If we let the dot denote derivation with respect to the coordinate on the fibre, then D = ∂t , X , E = ∂t , X , Ẋ = ∂t ⊕ ξ and, since [∂t , Ẍ ] = − Ẋ ∈ E, we have W = ∂t . Remark 4.3 Whenever the leaf space S(ξ )/W is a manifold, then N∼ = S(ξ )/ ∂t = S(ξ )/W. Also, if the projection p : S(ξ ) → S(ξ )/W is a submersion, we have ξ ∼ = p∗ E. Example 4.4 [Lorentz Prolongation] Let M be a Lorentzian three-manifold. The set of null vectors on M induces an S1 −bundle π L : PC → M, where PC is fibrewise the projectiviza- tion of the cone C. A point (x, l) ∈ PC consists of a point x ∈ M and a line l in C x . Define an Engel structure on PC at (x, l) by D(x,l) := (T(x,l) π L )−1 (l). Let (V , Y , T ) be an orthonormal frame of T M, with V , Y spacelike and T timelike, on an open ball B ⊂ M. Then, B×S1 ∼ = PC| B via (x, θ ) → (x, l := X := V cos θ +Y sin θ +T ), where the vector fields are on x. Letting the dot denote derivation with respect to the fibre coordinate, [∂θ , Ẋ ] ∈ / E = ∂θ , X , Ẋ , which implies that W is always transverse to ∂θ . 123 On the space of null geodesics... Page 13 of 22 15 Following the ideas of Remark 4.3, we make the following observation, which will be useful in Sect. 4.3. Remark 4.5 The family of skies {PC x }x∈M defines a circle foliation of PC whose leaf space is diffeomorphic to the manifold M. In addition, the bundle of null cones of M can be recovered via the pushforward  of the Engel distribution D under the projection map π L , that is, C x = (π L )∗ D|PC x for all x ∈ M. From now on, we denote elements of PC and S(ξ ) by the line or oriented line that they define, dropping the base point of the three-manifold M. 4.2 The space of null geodesics as a deprolongation We make use of the deprolongation procedure in Remark 4.3 to present the space of null geodesics as a Cartan deprolongation of the Lorentz prolongation. (PC, D) Ex.4.4 ? Def.2.2 (M, g) (N, H) We first recall a technical result. Proposition 4.6 [15] Any point of a pseudo-Riemannian three-manifold admits a local chart in which the metric is diagonal. Let (M,  g) be a three-dimensional  spacetime and consider x ∈ M. Let ϕ : (x1 , x2 , x3 ) ∈ V → ϕ (x1 , x2 , x3 ) ∈ U be local coordinates around x for which g is diagonal. The matrix representation of g in the chart (U , ϕ −1 ) is ⎛ ⎞   g11 (x1 , x2 , x3 ) 0 0 g ϕ(x1 , x2 , x3 ) = ⎝ 0 g22 (x1 , x2 , x3 ) 0 ⎠ (12) 0 0 g33 (x1 , x2 , x3 ) for some smooth functions g11 , g22 , g33 on V . Since the metric is non-degenerate at every point, we can assume g11 , g22 > 0 and g33 < 0. In addition, the coordinate vector fields u i := ϕ∗ ei give the eigendirections of the metric at every point. This discussion allows us to define local coordinates on PC via    : V × (0, 2π) →  V × (0, 2π)  cos θ sin θ 1 (x1 , x2 , x3 , θ ) → √ u 1 + √ u 2 + √ u 3 ⊂ Tϕ(x1 ,x2 ,x3 ) M. g11 g22 −g33  in PC by the vector that spans it, identifying For therest of this section, we denote a line ∼ cos θ sin θ 1 PC = √ u 1 + √ u 2 + √ u3 . Let us denote by ∂x1 , ∂x2 , ∂x3 , ∂θ the g11 g22 −g33 θ ∈S1 ,x∈M coordinate vector fields defined by . We compute next the kernel of the Lorentz prolongation of any Lorentzian three-manifold.  W of the Engel Proposition 4.7 In the notation above, the kernel  distribution on PC defined by the Lorentz prolongation is spanned, on  V × (0, 2π) , by the vector field cos θ sin θ 1 Z := √ ∂x1 + √ ∂x2 + √ ∂x + (F cos θ + G sin θ + H )∂θ , g11 g22 −g33 3 123 15 Page 14 of 22 A. Marín-Salvador, R. Rubio where we define ⎧  ⎪ 1 1 ∂ g11 sin θ ∂ g11 ⎪ ⎪ F := √ +√ , ⎪ ⎪ 2g11 g22 ∂ x2 −g33 ∂ x3 ⎪ ⎪  ⎨ 1 1 ∂ g22 cos θ ∂ g22 G := − √ +√ , ⎪ ⎪ 2g 22 g11 ∂ x 1 −g33 ∂ x3 ⎪ ⎪  ⎪ ⎪ 1 sin θ ∂ g33 cos θ ∂ g33 ⎪ ⎩ H := √ −√ . 2g33 g11 ∂ x1 g22 ∂ x2 Proof Recall, from Example 4.4, that the Engel structure D on PC is given by D(x1 ,x2 ,x3 ,θ ) = D √ cos θ sin θ √ √ 1 g11 u 1 + g22 u 2 + −g33 u 3  −1 cos θ sin θ 1 = (T π L ) √ u1 + √ u2 + √ u3 , g11 g22 −g33 where π L : PC → M is the canonical projection. Since T π L (∂xi ) = u i and T π L (∂θ ) = 0, the distribution D is given by  cos θ sin θ 1 D = X := √ ∂x1 + √ ∂x2 + √ ∂ x , ∂θ . g11 g22 −g33 3 sin θ cos θ Define Ẋ := [∂θ , X ] = − √ ∂x1 + √ ∂x2 , so the even-contact structure E on PC is g11 g22 E = X , Ẋ , ∂θ . We have [X , Ẋ ] = A∂x1 + B∂x2 + C∂x3 , which is  ∂ ∂x   ∂ ∂x  x x cos2 θ √ 1 , √ 2 − sin2 θ √ 2 , √ 1 g11 g22 g22 g11  ∂ ∂   ∂ ∂x  x x x − sin θ √ 3 , √ 1 + cos θ √ 3 , √ 2 . −g33 g11 −g33 g22 Since A is the ∂x1 -component, and analogously for B and C, we obtain ⎧ ⎪ 1 ∂ g11 sin θ ∂ g11 ⎪ ⎪ A= √ + √ , ⎪ ⎪ 2g11 g11 g22 ∂ x2 2g11 −g11 g33 ∂ x3 ⎪ ⎪ ⎨ 1 ∂ g22 cos θ ∂ g22 B= − √ − √ , ⎪ ⎪ 2g22 g11 g22 ∂ x 1 2g22 −g 22 g33 ∂ x3 ⎪ ⎪ ⎪ ⎪ sin θ ∂ g33 cos θ ∂ g33 ⎪ ⎩C = − √ + √ . 2g33 −g11 g33 ∂ x1 2g33 −g22 g33 ∂ x2 Since the kernel W lies in D and is not spanned by ∂θ (as [∂θ , Ẋ ] ∈ / E), there exists a smooth function μ on PC such that W = X + μ∂θ . We have [∂θ , X + μ∂θ ] ∈ E and [X , X + μ∂θ ] ∈ E, whereas  cos θ sin θ  [ Ẋ , X + μ∂θ ] = Ẋ (μ)∂θ − (A∂x1 + B∂x2 + C∂x3 ) + μ √ ∂x1 + √ ∂x2 . g11 g22 Since Ẋ (μ)∂θ ∈ E, it is enough to impose that the last two terms belong to E. This is the case if and only if  cos θ sin θ  √ −(A∂x1 + B∂x2 + C∂x3 ) + μ √ ∂x1 + √ ∂x2 + C −g33 X ∈ E. g11 g22 123 On the space of null geodesics... Page 15 of 22 15 This vector field reads    −g33   −g33  cos θ sin θ  − A − C cos θ ∂x1 + B − C sin θ ∂x2 + μ √ ∂x1 + √ ∂x2 , g11 g22 g11 g22 and for it to belong to E, it is enough to impose that it is a multiple of Ẋ . We take √ √ √ μ := A g11 cos θ + B g22 sin θ − C −g33 , √ √ so that the vector field becomes (A g11 sin θ − B g22 cos θ ) Ẋ . Hence, by defining F := √ √ √ A g11 , G := B g22 and H := −C −g33 , the kernel is W = X + μ∂θ = Z := X + (F cos θ + G sin θ + H )∂θ .   Theorem 4.8 Let M be a three-dimensional spacetime. Then, N∼ = PC/W. In addition, if N is a manifold and p : PC → PC/W ∼ = N is a submersion, the canonical contact structure on N is H∼ = p∗ E. Proof We divide the proof of Theorem 4.8 into two parts. Part I: N ∼= PC/W. It is enough to show that W computed in Proposition 4.7 is pointwise  flow X g . Let (V, ϕ) be a local chart of M making the metric proportional to the geodesic diagonal, and let γ (t) = ϕ x1 (t), x2 (t), x3 (t) be a null geodesic. Then, (x1 )2 g11 + (x2 )2 g22 = −(x3 )2 g33 (13) and the curve x1 u 1 + x2 u 2 + x3 u 3 ∈ Tϕ(x1 ,x2 ,x3 ) M is an integral line of X g . Under our cos θ sin θ 1 identification PC ∼= { √ u1 + √ u2 + √ u 3 }θ ∈S1 ,x∈M , this curve reads g11 g22 −g33   x1 g11 u 1 x2 g22 u 2 u3   √ +  √ +√ ∈ Tϕ(x1 ,x2 ,x3 ) M =  x1 , x2 , x3 , θ , x3 −g33 g11 x3 −g33 g22 −g33 x  x2  g11 g22 where θ is such that cos θ = x1 −g 33 and sin θ = x  −g33 . Note that this makes sense 3 3 because of Equation (13), and that x3 is nonzero if γ is nonconstant. Then,  the tangent  vector to this curve, which gives the expression of X g in the coordinate chart V × (0, 2π) on PC, is x1 ∂x1 + x2 ∂x2 + x3 ∂x3 + θ  ∂θ √  cos θ sin θ 1 θ  = x3 −g33 √ ∂x1 + √ ∂x2 + √ ∂x3 +  √ ∂θ . g11 g22 −g33 x3 −g33  Hence, our claim is equivalent to x  √θ−g = F cos θ +G sin θ + H . Using that −θ  sin θ =    3 33 x1 g11 x3 −g33 and that the x i satisfy the geodesic equation the result follows, after computing the Christoffel symbols for the metric in Equation (12). Part II: H ∼ = p∗ E. Firstly, as argued in [1, p. 246], the even-contact structure E is invariant under the flow of any vector field generating W. Therefore, the pushforward p∗ E is well defined. 123 15 Page 16 of 22 A. Marín-Salvador, R. Rubio Let γ ∈ N ∼ = PC/W. Then, γ is given by a curve μ : (−ε, ε) → M which is a null geodesic in M, and γ = p μ̇(s) for all s ∈ (−ε, ε). Let q0 := μ(0) and define a coordinate system ϕ : (x1 , x2 , x3 ) → ϕ(x1 , x2 , x3 ) around q0 given by Proposition 4.6. Let (x1 , x2 , x3 , θ ) be coordinates around μ̇(0) in PC as defined before Proposi- tion 4.7. Now, for all s ∈ [0, ε) small enough, the point qs := μ(s) lies in the image of ϕ and μ̇(s)  lies in the image of  , and we can consider functions xi , θ such that μ̇(s) =  x1 (s), x2 (s), x3 (s), θ (s) . If s > 0 is small enough, the points q0 and qs are not conjugate in M. The sky of q0 is given by    Sq0 = { p ◦  x1 (0), x2 (0), x3 (0), θ | θ ∈ θ (0) − π, θ (0) + π }, and similarly for Sqs . Therefore, Hγ = Tγ Sq0 ⊕ Tγ Sqs = Tμ̇(0) p( ∂θ ) ⊕ Tμ̇(s) p( ∂θ ). We ought to express the second addend as a pushforward of a line over μ̇(0). Take s > 0 small enough and let K be a neighbourhood of μ̇(s) for which the flow −sZ : K →  Z (K ) −s at time −s of the vector field Z = X + (F cos θ + G sin θ )∂θ is a diffeomorphism. Then, p ◦ −s Z = p, and since  Z μ̇(s) = μ̇(0), we can compute −s Tγ Sqs = Tμ̇(s) p(∂θ ) = (Tμ̇(0) p ◦ Tμ̇(s) −s Z )(∂θ ) and ! " ! " Hγ = Tμ̇(0) p ∂θ , Tμ̇(s) −s Z (∂θ ) = Tμ̇(0) p ∂θ , Tμ̇(s) −s Z (∂θ ) − ∂θ # $ Tμ̇(s) −s Z (∂ ) − ∂ θ θ = Tμ̇(0) p ∂θ , , s for all s > 0 small enough.  true if we take the limit s → 0. Thus,  Hence, the result is still Tμ̇(s) −s Z (∂ )−∂ θ θ we obtain Hγ = Tμ̇(0) p ∂θ , lim s = Tμ̇(0) p ( ∂θ , [∂θ , Z ] ). We compute s→0 [∂θ , Z ] = Ẋ + (−F sin θ + G cos θ )∂θ and therefore % & % & Hγ = Tμ̇(0) p ( ∂θ , [∂θ , Z ] ) = Tμ̇(0) p ∂θ , Ẋ = Tμ̇(0) p ∂θ , Ẋ , Z % & = Tμ̇(0) p ∂θ , Ẋ , X + (F cos θ + G sin θ )∂θ % & = Tμ̇(0) p ∂θ , Ẋ , X = Tμ̇(0) p(E). This concludes the proof that H ∼ = p∗ E and hence the theorem.   Example 4.9 The Lorentzian prolongation of the spacetime (S2 × S1 , g1 ) is diffeomorphic to PC ∼ = ST S2 × S1 . Letting t be the S1 -coordinate and θ the coordinate on the fibre of ST S , the Engel flag over (u, s) ∈ ST S2 × S1 is, seeing T S2 ⊂ T (ST S2 ) via the Levi-Civita 2 connection on S2 , ⊥ u + ∂t ⊂ u + ∂t , ∂θ ⊂ u ⊕ u + ∂t , ∂θ ⊂ T(u,s) (ST S2 × S1 ). = ST S2 ∼ Hence, (ST S2 × S1 )/W = (ST S2 × S1 )/ u + ∂t ∼ = N1 , and letting p : ST S2 × ∼ ∼ S → ST S be the natural projection, p∗ E = χ = H. 1 2 123 On the space of null geodesics... Page 17 of 22 15 4.3 Retrieving the spacetime We investigate how Theorem 4.8 allows us to characterize the three-dimensional contact manifolds that are spaces of null geodesics of a spacetime. This provides a procedure to, given a contact manifold (N , ξ ) satisfying the necessary conditions, find a spacetime whose space of null geodesics is precisely (N , ξ ). The main idea is to Cartan-prolong the contact manifold and make use of Remark 4.5 to Lorentz-deprolong it. ∼ ? (PC, D) (S(ξ ), D) Ex.4.4 Ex.4.2 (5) ? Thm.4.8 (M, g) (N , ξ ) Def.2.2 We have the following result. Theorem 4.10 A three-dimensional contact manifold (N , ξ ) is contactomorphic to the space of null geodesics of a spacetime if and only if there exists an Engel manifold (Q, D) with Engel flag W ⊂ D ⊂ E ⊂ T Q such that N∼ = Q/W and ξ∼ = p∗ E, (14) for p : Q → Q/W the projection, and Q admits an oriented foliation by circles F such that (i) for all S ∈ F and x ∈ S, we have Tx S ⊕ Wx = Dx , (ii) the space of leaves M := Q/F is a manifold and the projection q : Q → M is a submersion, (iii) for every S ∈ F, the image q∗ D| S is a cone in the vector space Tq(S) M and the map x ∈ S → q∗ Dx is injective. In addition, if the above conditions are satisfied, (N , ξ ) is contactomorphic to the space of null geodesics of (Q/F, g), where g is a metric on Q/F with bundle of cones q∗ D. Proof Let (N , ξ ) be the space of null geodesics of a spacetime (L, h). The existence of Q satisfying (14) follows from Theorem 4.8 taking (Q, D) as the Lorentz prolongation of (L, h). Then, we can take F := {PC x }x∈L , which is a foliation by circles and is oriented because L is a spacetime. By definition of the Lorentz prolongation, T F ⊂ D, and by Example 4.4, the kernel W is transverse to T F. Hence i) is satisfied. Also, by Remark 4.5, Q/F ∼ = L, and therefore it is a manifold, and q is the projection π L : Q = PC → L, which is a submersion. Hence, ii) follows. Also by definition of the Lorentz prolongation, for all u ∈ Q, q∗ Du = u, where on the right hand side we regard u as a vector on a cone of L. Therefore, q∗ D|PC x = C x and iii) also follows. Finally, by Theorem 4.8 and (14), the last claim is also satisfied, as any other metric on L with the same bundle of cones is conformal to g, and therefore has the same space of null geodesics. For the converse, let (N , ξ ) be a contact manifold such that N = Q/W and ξ = p∗ E for an Engel manifold Q with flag W ⊂ D ⊂ E ⊂ T Q. Assume Q admits an oriented foliation F satisfying i), ii), iii) above. Let M := Q/F, which is a manifold by hypothesis, and q : Q → M the projection map. Now, since T F ⊂ D, the pushforward q∗ Dx is a line in Tq(x) M for all u ∈ PC and by iii), these create a cone in Tq(S) M when traveling the leaf S ∈ F containing x. Therefore, we obtain a smooth bundle of cones on T M, and hence there exists a metric g on M with such bundle of cones. In addition, any two metrics with the same 123 15 Page 18 of 22 A. Marín-Salvador, R. Rubio bundle of null cones are conformal, and hence produce the same contact manifold of null geodesics. Since F is oriented, we can assign a consistent orientation to each cone in the bundle and therefore (M, g) is a spacetime. ∼ ∼ ∼ ∼ Let now (PC, D) be the Lorentz prolongation of (M, g), with flag W ⊂ D ⊂ E ⊂ T (PC), and define the projection π M : PC → M. Consider the map  : Q → PC , x → q∗ D x which is a diffeomorphism by the definition of g and the hypothesis that x ∈ S → p∗ Dx is injective for any S ∈ F. In addition,   π L ∗ ◦ ∗ (Dx ) = (π M ◦ )∗ (Dx ) = q∗ Dx = (x), ∼ ∼ which implies ∗ D ⊂ D. Since  is a submersion, we find that ∗ D =D, and so  is an ∼ Engel-morphism. Let us define r : PC → PC/ W the projection. Then, by Theorem 4.8, we find that the space of null geodesics (N, H) of (M, g) is ∼ N∼ = PC/ W ∼ = Q/W ∼ =M and ∼ = r∗ E ∼ H∼ = p∗ E ∼ = ξ, where we make use of Remark 4.3.   Remark 4.11 We have later found out that [24, Rk. 1.7] suggests the necessity of a foliation F like the one in Theorem 4.10. Since the Cartan prolongation of (N , ξ ) satisfies (14), we obtain: Corollary 4.12 A three-dimensional contact manifold (N , ξ ) is contactomorphic to the con- tact manifold of null geodesics of a spacetime if the Cartan prolongation (S(ξ ), D) of (N , ξ ) admits an oriented foliation F by circles such that (i) for all S ∈ F and u ∈ S, we have Tu S ⊕ Wu = Du , where W denotes the kernel of (S(ξ ), D), (ii) the space of leaves M := S(ξ )/F is a manifold and the projection p : S(ξ ) → M is a submersion, (iii) for every S ∈ F, the image p∗ D| S is a cone in the vector space T p(S) M and the map u ∈ S → p∗ Du is injective. In addition, if i), ii), iii) are satisfied, (N , ξ ) is contactomorphic to the space of null geodesics of (M, g), where g is any metric on M with bundle of null cones p∗ D. We continue our discussion by exploring the relation between the Cartan prolongation of (N , ξ ) and the Engel manifold Q in Theorem 4.10, whenever it exists. The following proposition is an adaptation of [19, Prop. 5.4]. Proposition 4.13 Let (Q, D) be an Engel manifold with flag W ⊂ D ⊂ E ⊂ T Q. Assume Q/W is a manifold, which then can be endowed with a contact structure ξ := p∗ E for p : Q → Q/W the projection, provided it is a submersion. Then, there exists a local diffeomorphism  : Q → S(ξ ) to the Cartan prolongation which is compatible with the Engel structure. 123 On the space of null geodesics... Page 19 of 22 15 Proof Let x ∈ X . Since Wx ⊂ Dx , the pushforward p∗ Dx is a line in ξ p(x) . Hence, the map  : Q → S(ξ ) x → p∗ D x is well defined, and smooth. Let x ∈ Q and consider a neighbourhood U ⊂ Q of x in which we can trivialize D|U = Z , Y , for Z ∈ X(U ) spanning W. Since  is a bundle map over Q/W, it is enough to show that Ty  is surjective when restricted to W y for every y ∈ U . Therefore, by linearity, it is enough to show ∗ Z  = 0. By definition of , this is equivalent to [Z , Y ]  = 0, which holds because D is Engel. It is only left to show that  preserves the Engel structure. Let u ∈ Dx and define πC : S(ξ ) → X /W the projection. Then,   πC ∗ ◦ ∗ (Dx ) = (πC ◦ )∗ (Dx ) = p∗ D = (x). Since  is a submersion, the Proposition follows.   Corollary 4.14 If (N , ξ ) is a three-dimensional contact manifold contactomorphic to the space of null geodesics of a spacetime, the Engel manifold (Q, D) described in Theorem 4.12 comes with a canonical local diffeomorphism  : Q → S(ξ ) compatible with the Engel structures. Diagram (5) above is thus completed to ∼ Cor.4.14 (PC, D) (S(ξ ), D) Ex. 4.4 Ex. 4.2 Thm.4.10 Thm.4.8 (M, g) (N , ξ ). Space of null geodesics We believe that this approach can be useful in order to answer the open question of whether the contact structure of the space of null of geodesics can be overtwisted, but describing or just dealing with the foliation in Theorem 4.10 will require further work. We finally look at two illustrative examples where the manifolds involved can be described explicitly and the subtleties of the main results can be appreciated. Example 4.15 Consider R3 with coordinates (x, y, t) and tangent vector fields u x , u y , u t . Then, the Lorentz prolongation of (R3 , d x 2 + dy 2 − dt 2 ) is diffeomorphic to R3 × S1 via (x, y, t, θ ) → cos θ u x + sin θ u y + u t ⊆ T(x,y,t) R3 . Under this identification, the Engel structure is D(x,y,t,θ ) = cos θ ∂x +sin θ ∂ y +∂t , ∂θ , which implies W = cos θ ∂x +sin θ ∂ y + ∂t and E = D ⊕ − sin θ ∂x + cos θ ∂ y . The foliation F by circles is given by {{k} × S1 }k∈R3 . We also have R3 × S1 /W ∼ = R2 × S1 with projection map p : R3 × S1 → R2 × S1 (x, y, t, θ ) → (x − t cos θ, y − t sin θ, θ ). Take coordinates (u, v, θ ) on R2 × S1 with tangent vectors wu , wv , wθ . Then, the induced contact structure on R2 × S1 is ξ p(x,y,t,θ ) := p∗ E(x,y,t,θ ) = − sin θ wu + cos θ wv , t sin θ wu − t cos θ wv + wθ = − sin θ wu + cos θ wv , wθ , 123 15 Page 20 of 22 A. Marín-Salvador, R. Rubio which is the canonical contact structure on R2 × S1 ∼ = ST R2 . The Cartan prolongation S(ξ ) of (R × S , ξ ) is diffeomorphic to R × S × S via 2 1 2 1 1 (u, v, θ, ω) →  cos ω(− sin θ wu + cos θ wv ) + sin ωwθ  ∈ S(ξ )(u,v,θ ) . Under this identification, the Engel distribution reads ∼ D(u,v,θ,ω) = cos ω(− sin θ ∂u + cos θ ∂v ) + sin ω∂θ , ∂ω ⊂ T(u,v,θ,w) S(ξ ). The local diffeomorphism  defined in Proposition 4.13 is (x, y, t, θ ) → p∗ ∂θ = t sin θ wu − t cos θ wv + wθ  =  √ t 2 sin θ wu − √ t 2 cos θ wv + √ 1 2 wθ . Hence, 1+t 1+t 1+t using the identification above,  : R3 × S1 → R2 × S1 × S1 (x, y, t, θ ) → (x − t cos θ, y − t sin θ, θ, arccos √ t ), 1+t 2 where we take arccos √ t ∈ (0, π). Then,  is a global diffeomorphism onto its image 1+t 2 (R3 × S1 ) = R2 × S × (0, π). Note that the 1 image of F under  is {(x, y, t, θ ) | θ ∈ S1 }(x,y,t)∈R3 . This is still a foliation by circles on the image of  satisfying i) and ii) above with projection q : R2 × S1 × (0, π) → R2 × S1 × (0, π)/(F) ∼ = R2 × (0, π) ω ω (u, v, θ, ω) → (u + cos θ cos sin ω , v + sin θ cos sin ω , ω). Again, taking (u, v, ω) as coordinates on R2 × (0, π), we can compute  ∼ cos θ sin θ q∗ D= q∗ ∂ω = ∂ω − ∂ u − ∂ v =: V (ω, θ ), sin2 ω sin2 ω which indeed defines a bundle of cones with injective map θ ∈ S1 → V (ω, θ ) for a given ω ∈ (0, π). Hence, iii) is also satisfied. The bundle of cones obtained on R2 × (0, π) is the induced by the metric g = du 2 + dv − sin4 ωdω2 , and therefore we obtain a spacetime isometric to (R3 , d x 2 + dy 2 − dt 2 ) 2 via (x, y, t) → (x, y, arccos √ t 2 ). 1+t Example 4.16 Let (M, g) = (S2 × S1 , g◦ − dt 2 ). As argued in Example 4.9, PC ∼ = ST S2 with flag ⊥ u + ∂t ⊂ u + ∂t , ∂θ ⊂ u ⊕ u + ∂t , ∂θ ⊂ T(u,s) (ST S2 × S1 ), and (N, H) ∼ = (ST S2 , χ). Note that, since the S1 -bundle ST S2 → S2 is oriented, the vector field ∂θ is well defined everywhere. In addition, for u ∈ STx S2 , we can define u ⊥ ∈ STx S2 orthogonal to u with respect to g◦ and so that (u, u ⊥ , x) is a positive basis for S2 . Seeing T S2 ⊂ T (ST S2 ) via the Levi-Civita connection on (S2 , g◦ ), we can see u ⊥ ∈ Tu (ST S2 ), and we obtain a well-defined vector field P on ST S2 given pointwise by Pu = u ⊥ . Then, the contact structure on ST S2 is χ = ∂θ , P and, in particular, it is trivial as a vector bundle. Therefore the Cartan prolongation S(χ) of (ST S2 , χ) is S(χ) ∼ = ST S2 × S1 via (u, s) →  cos s P + sin s∂θ  ∈ Tu ST S2 . Now, the local diffeomorphism  : ST S2 × S1 → ST S2 × S1 is given by (u, s) = p∗ Du,s = p∗ ∂θ , 123 On the space of null geodesics... Page 21 of 22 15 which can be described as follows. The element in ST S2 of (u, s) is simply p(u, s), which recall can be described as taking the great circle μ ⊂ S2 defined by u, and parallel transporting u over μ an angle θ backwards. We get (u, s) =  cos s∂θ − sin s p(u, s)⊥  ∈ T p(u,s) ST S2 ,   that is, (u, s) = p(u, s), s + π2 . This is a global diffeomorphism  : ST S2 → ST S2 preserving the Engel structures. The foliation F = {STx S1 × {t}}(x,t)∈S2 ×S1 on PC ∼ = ST S2 × S1 gets sent under  to a foliation by circles F which can be described as follows. The foliation splits in a family of S1 −foliations of ST S2 . On ST S2 × {t}, F is formed by all circles on S2 with radius t and vectors pointing towards the centre of such circle. Hence, F satisfies i). Now, fixing t, every circle of F on ST S2 × {t} has a unique centre, and every point of S2 defines one such circle. Therefore, ST S2 /F ∼ = S2 × S1 , and the projection q : ST S2 → S2 × S1 is a submersion, since it is on every slice q| ST S2 ×{t} : ST S2 × {t} → S2 × {t}, as this is only parallel transporting a tangent vector an angle t over its great circle, and taking the basepoint. Therefore, ii) is also satisfied. The bundle of cones defined on S2 ×{t} is that of a strictly positive constant multiple of gc , and this constant varies smoothly with respect to t. Hence, we obtain a spacetime isometric to (S2 × S1 , gc × dt 2 ). Acknowledgements We are indebted to Francisco Presas for proposing the ideas that led to this work and for his invaluable comments. We also thank Miguel Sánchez Caja for his interest in this project. Funding Open Access Funding provided by Universitat Autonoma de Barcelona. Data availability There is no data associated to this work. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Adachi, J.: Engel structures with trivial characteristic foliations. Algebr. Geom. Topol. 2, 239–255 (2002) 2. 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