Invariants and Infinitesimal Transformations for Contact Sub-Lorentzian Structures on 3-Dimensional Manifolds

In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact 3 manifolds. Next we characterize vector fields which generate isometric and conformal symmetries in general sub-Lorentzian manifolds. We then focus attention back to the case where the underlying manifold is a contact 3 manifold and more specifically when the manifold is also a Lie group and the structure is left-invariant.


Basic notions and motivation.
For all details and facts concerning sub-Lorentzian geometry, the reader is referred to [8] and the references therein (see also [15,13]). In this section we only present the notions that are required for the formulation of the main results of the paper.
Let M be a smooth manifold. A sub-Lorentzian structure on M is a pair (H, g) where H is a bracket generating distribution of constant rank on M , and g is a Lorentzian metric on H. A triple (M, H, g), where (H, g) is a sub-Lorentzian structure on M , will be called a sub-Lorentzian manifold.
For any q ∈ M , a vector v ∈ H q will be called horizontal. A vector field X on M is horizontal if it takes values in H. We will denote the set of all local horizontal vector fields by Γ(H). To be more precise, X ∈ Γ(H) if and only if X is a horizontal vector field defined on some open subset U ⊂ M .
If (H, g) is a sub-Lorentzian metric on M then, as shown in [10] , H can be represented as a direct sum H = H − ⊕H + of subdistributions such that rank H − = 1 and the restriction of g to H − (resp. to H + ) is negative (resp. positive) definite. This type of decomposition will be called a causal decomposition of H . Now we say that (M, H, g) is time (resp. space) orientable if the vector bundle H − −→ M (resp. H + −→ M ) is orientable. It should be noted that although causal decompositions are not unique, the definition of time and space orientation is independent of a particular choice of a causal decomposition. Since a line bundle is orientable if and only if it is trivial, time orientability of (M, H, g) is equivalent to the existence of a continuous timelike vector field on M . A choice of such a timelike field is called a time orientation of (M, H, g).
Suppose that (M, H, g) is time oriented by a vector field X. A nonspacelike v ∈ H q will be called future (resp. past ) directed if g(v, X(q)) < 0 (resp. g(v, X(q)) > 0). A horizontal curve γ : [a, b] −→ M is called timelike future (past) directed iḟ γ(t) is timelike future (past) directed a.e. Similar classifications can be made for other types of curves, e.g. nonspacelike future directed etc. If q 0 ∈ M is a point and U is a neighborhood of q 0 , then by the future timelike (nonspacelike, null) reachable set from q 0 relative to U we mean the set of endpoints of all timelike (nonspacelike, null) future directed curves that start from q 0 and are contained in U .
Now we define a very important notion that will play a crucial role in the sequel. As it is known [8], any sub-Lorentzian structure (H, g) determines the so-called geodesic Hamiltonian which is defined as follows. The existence of the structure (H, g) is equivalent to the existence of the fiber bundle morphism G : T * M −→ H covering the identity such that if v, w are any horizontal vectors, then g(v, w) = ξ, Gη = η, Gξ whenever ξ ∈ G −1 (v), η ∈ G −1 (w). The geodesic Hamiltonian is the map h : T * M −→ R defined by h(λ) = 1 2 λ, Gλ .
If X 1 , ..., X k is an orthonormal basis for (H, g) with a time orientation X 1 , then A horizontal curve γ : [a, b] −→ M is said to be a Hamiltonian geodesic if there exists Γ : [a, b] −→ T * M such thatΓ = − → h and π(Γ(t)) = γ(t) on [a, b]; by π : T * M −→ M we denote the canonical projections, and − → h is the Hamiltonian vector field corresponding to h.
Let γ : [a, b] −→ M be a nonspacelike curve. The non-negative number L(γ) = b a |g(γ(t),γ(t))| 1/2 dt is called the sub-Lorentzian length of a curve γ. If U ⊂ M is an open subset, then the (local) sub-Lorentzian distance relative to U is the function d[U ] : U × U −→ [0, +∞] defined as follows: For q 1 , q 2 ∈ U , let Ω nspc q1,q2 (U ) denote the set of all nonspacelike future directed curves contained in U which join q 1 to q 2 , then It can be proved (see [8]) that every sufficiently small subarc of every nonspacelike future directed Hamiltonian geodesic is a U -maximizers for suitably chosen U .
Suppose now that we are given two sub-Lorentzian manifolds (M i , H i , g i ), i = 1, 2. A diffeomorphism ϕ : M 1 −→ M 2 is said to be a sub-Lorentzian isometry , if dϕ(H 1 ) ⊂ H 2 and for each q ∈ M 1 , the mapping dϕ q : (H 1 ) q −→ (H 2 ) ϕ(q) is a linear isometry, i.e., for every v 1 , v 2 ∈ (H 1 ) q it follows that Of course, any isometry maps timelike curves from M 1 to timelike curves on M 2 . The same for spacelike and null curves. Moreover isometries preserve the sub-Lorentzian length of nonspacelike curves. If (M i , H i , g i ), i = 1, 2, are both timeand space-oriented, then we can distinguish among all isometries those that preserve both orientations. More precisely, suppose that ϕ : is again a causal decomposition and we say that ϕ preserves time (resp. space) orientation if the vector bundle morphism dϕ |H − is orientation preserving. An isometry that preserves time and space orientation will be called a ts-isometry. It is clear that any ts-isometry preserves Hamiltonian geodesics, maximizers, and local sub-Lorentzian distance functions. Notice furthermore that the set of all isometries (M, H, g) −→ (M, H, g) is a Lie group and the set of all ts -isometries forms a connected component containing the identity.
A sub-Lorentzian manifold (M, H, g) is called a contact sub-Lorentzian manifold, if H is a contact distribution on M . Among sub-Lorentzian manifolds, those which are contact seem to be the easiest to study and hence well known. Contact sub-Lorentzian manifolds are studied for instance in papers [6,7,9,12,13,14,15,16]. The investigations go in two directions. The first addresses global aspects in the group case, e.g., in [9,7] the Heisenberg sub-Lorentzian metric is treated. More precisely, the future timelike, nonspacelike and null reachable sets from a point are computed, and a certain estimate on the distance function is given. Moreover, it is shown that the future timelike conjugate locus of the origin is zero, while the future null conjugate locus equals the union of the two null future directed Hamiltonian geodesics starting from the origin. In turn, in [15] and [9] it is proved that the set reachable from the origin by future directed timelike Hamiltonian geodesics coincides with the future timelike reachable set from the origin. In [15] the authors also study the set reachable by spacelike Hamiltonian geodesics and prove the uniqueness of geodesics in the Heisenberg case. Next, in the papers [15,16] the so-called H-type groups (i.e. higher dimensional analogues of the 3D Heisenberg group) with suitable sub-Lorentzian metrics are studied, and the main emphasis is put on the problem of connectivity by geodesics, i.e. given two points q 1, q 2 , figure out how many geodesics joining q 1 to q 2 exist. A similar problem is also dealt with in [14]. On the other hand, in [13] the group SL(2, R) with the sub-Lorentzian metric is studied. As it will become clear below, the cases of the Heisenberg group and that of SL(2, R) are especially interesting for us because these are exactly the cases that arise when the invarianth (defined below) vanishes.
As one can see, problems connected with isometric and conformal symmetry have not been examined in an explicit sense although in broader contexts such as parabolic geometry and Cartans's equivalence, there are applicable results. The aim of this paper is to embark on filling this gap. More precisely, first we construct invariants for contact sub-Lorentzian manifolds (M, H, g) with dim M = 3, more or less in the way as it is done in the contact sub-Riemannian case -cf. [4]. Our invariants are: a (1, 1)-tensorh on H and a smooth function κ on M . Then, we consider in some detail the case that M is a 3-dimensional Lie group such that h = 0. It turns out that in such a case M is locally either the Heisenberg group or the universal cover of SL(2, R). In these two cases we describe infinitesimal isometries and more generally infinitesimal conformal transformations.
1.2. The content of the paper. In section 2 we construct invariants for tsoriented contact sub-Lorentzian metrics on 3D manifolds. The construction follows the ideas of [4], however the full analogy does not exist due to the special character of indefinite case. Our main invariants for a manifold (M, H, g) are: a smooth (1, 1)tensorh on H and a smooth function κ on M . These invariants provide necessary conditions for two contact sub-Lorentzian manifolds to be locally ts-isometric. We also consider another invariant χ arising from the eigenvalues ofh which to a lesser extent also distinguishes the structure.
In section 3 we define and prove basic properties of infinitesimal sub-Lorentzian isometries and conformal transformations. Then we notice that the invarianth can be expressed in terms of the restricted Lie derivative of the metric g in the direction of the Reeb vector field. The immediate consequence of this latter fact is that the Reeb vector field X 0 is an infinitesimal isometry if and only ifh vanishes identically. Section 4 covers some other implications of certain combinations of the invariants vanishing. In particular we demonstrate (see proposition 4.5) that without any assumptions on orientation, the condition χ = 0 andh = 0 implies the existence of line sub-bundle L → M of H on which the metric g is equal to zero. We then begin to focus on the conditionh = 0 where κ comes to the fore. For example, when M is a simply connected Lie group, we show thath = 0 and κ = 0 implies that M is the Heisenberg group -cf. corollary 4.1, andh = 0 and κ = 0 implies that M is the universal cover of SL(2, R) -see corollary 4.2 . This contrasts with the sub-Riemannian case where a third group, namely SU (2), also appears. Section 6 is devoted to computing infinitesimal isometries and infinitesimal conformal transformations using Cartan's equivalence method and Section 7 presents an example of an isometrically rigid sub-Lorentzian structure. Finally, the appendix presents possible applications of our invariants to a noncontact case.

2.1.
Preliminaries. Let (M, H, g) be a contact sub-Lorentzian manifold, dim M = 3, which is supposed to be both time and space oriented or ts-oriented for short. Since H is of rank 2, any causal decomposition H = H − ⊕ H + splits H into a direct sum of line bundles. So in this case a space orientation is just a continuous spacelike vector field, and consequently H admits a global basis. Let us fix an orthonormal basis X 1 , X 2 for (H, g), i.e. g(X 1 , X 1 ) = −1, g(X 1 , X 2 ) = 0, g(X 2 , X 2 ) = 1, where X 1 (resp. X 2 ) is a time (resp. space) orientation. From now on we will work with ts-invariants, i.e. with invariants relative to ts-isometries. However, when reading the text the reader will see that the space orientation is only an auxiliary notion here and most of the results do not depend on it (some of them do not depend on an orientation at all).
Let ω be a contact 1-form such that H = ker ω. Without loss of generality we may assume that ω is normalized so that Next, denote by X 0 the so-called Reeb vector field on M which is defined by It is seen that X 0 is uniquely determined by ts-oriented sub-Lorentzian structure. Using (2.2) it is seen that the action of ad X0 preserves the horizontality of vector fields, i.e.

Induced bilinear form and linear operator.
In the introduction we defined the geodesic Hamiltonian h which can be written as We also consider the function h 0 (λ) = λ, X 0 and observe that by definition both h and h 0 are invariant with respect to ts-oriented structure (H, g). Therefore, it is the same with their Poisson bracket {h, h 0 } which, when evaluated at q ∈ M , gives a symmetric bilinear form , and then use (2.4).
In the assertion of lemma 2.1 and in many other places below we write c i jk for c i jk (q).
Let us recall (mutually inverse) musical isomorphisms determined by the metric g; these are ♯ : This permits us to define a bilinear symmetric formh q : Finally we define a linear mappingh q : H q −→ H q by the following formula: Using (2.7), it is seen that the matrix of the operatorh q in the basis {X 1 (q), X 2 (q)} is equal to 2.3. The ts-invariants. By our construction, the eigenvalues and determinant ofh q as well ash q itself, are all invariants for the ts-oriented structure (H, g).
view of (2.6), the trace ofh q is equal to 0, the eigenvalues ofh q are equal to as a functional ts-invariant for our structure. In analogy with the sub-Riemannian case ( [1], [3] and [4]), we consider the functional ts-invariant defined as follows: Unlike the sub-Riemannian case where χ and κ play the crucial role, it ish and κ that play the crucial role in the sub-Lorentzian setting.

2.3.1.
Proof of proposition 2.1. Let X 1 , X 2 is an orthonormal basis with a time orientation X 1 and a space orientation X 2 , and let c i jk be structures functions determined by this basis. Next, let θ = θ(q) be a smooth functions and consider an orthonormal basis Y 1 , Y 2 given by Then Y 1 (Y 2 ) is a time (space) orientation, and of course Let d i jk be the structure functions determined by the basis In order to prove proposition 2.1 we need the following lemma.
Lemma 2.2. The following formulas hold true: Proof. All formulas are proved by direct calculations. For instance, using (2.10) we write Then using (2.10) and (2.4) we arrive at from which the fifth and sixth equations in (2.12) follow. Now, using lemma 2.2, we see that Finally, we compute Y 2 (d 1 12 ) + Y 1 (d 2 12 ). To this end let us write . Combining (2.13), (2.14) and (2.15) completes the proof of Proposition 2.1.
In summary, our basic ts-invariants are: a smooth function κ on M and a (1, 1) tensorh on H.
In this section (M, H, g) is a fixed sub-semi-Riemannian manifold, rank H and dim M are arbitrary.
for every q ∈ M and every v, w ∈ H q . Of course, if ρ = 1 then f is an isometry of (M, H, g).
Along with conformal transformations and isometries we consider their infinitesimal variants. Let us note a simple lemma.
Lemma 3.1. Let Z be a vector field on M and denote by h t its flow. Then the following conditions are equivalent: Proof. Although the result is known, we give a proof for the sake of completeness.
(a) ⇒ (b) Following [19], we fix a point q and consider a basis X 1 , ..., X k of H defined on a neighborhood U of q. By our assumption, there exist smooth functions α ij , i, j = 1, ..., k , such that ad Z X i = k j=1 α ij X j on U and it follows where β ij (t) = (α ij • h −t ) (q). For any covector λ ∈ T * q M which annihilates H q , i.e. λ, v = 0 for every v ∈ H q , we obtain a system of linear differential equations for the functions w i (t) = λ, v i (t) , i = 1, ..., k: Take a point q, then for every t such that |t| is sufficiently small, we have (h t * X) (q) ∈ H q and it follows that (ad Z X) Suppose now that f : M −→ M is a diffeomorphism such that df (H) = H and let T be a tensor of type (0, 2) on H. We define a pull-backf * : where X, Y ∈ Γ(H) (tilde indicates that we restrict to horizontal vector fields).
We can now reformulate the definition of conformal transformations in a manner consistent with semi-Riemannian geometry: f is a conformal transformation of (M, H, g) if and only if there exists a function Suppose that Z is a vector field on M such that ad Z : Γ(H) −→ Γ(H) and let h t denote the (local) flow of Z. Using lemma 3.1, again by analogy to the classical geometry, we can define a local operatorL Z T : Γ(H) × Γ(H) −→ C ∞ (M ) which will be called the restricted Lie derivative: It turns out that Proof. Remembering that we use only horizontal vector fields, the proof is the same as in the classical geometry. Again h t is the flow of Z. "⇒" By lemma 3.1 we know that (i) is satisfied. If ( h t ) * g = ρ t g, where for each t the function ρ t is smooth and positive, then it follows that On the other hand, we also have that and so we see thatL Z g = λg where λ =ρ t − X(ρ t ) ρ t (note that ρ 0 = 1).
"⇐" From lemma 3.1 we know that dh t preserves H. From (ii) and (3.1) we have d dt By direct calculation we obtain for every X, Y ∈ Γ(H), which gives the following two Corollaries:  Proof. Fix a point q ∈ M . Under the above notation, for any n ∈ N and sufficiently small |t| Using corollary 3.1 we can remove from (3.4) terms of order 0 and 1 with respect to t. What we obtain is for |t| sufficiently small, which gives (3.3).

Some properties of invariants
In this section we assume all sub-Lorentzian manifolds to be ts-oriented. Let us start from an obvious observation.   H, g). First of all let us notice how the invarianth can be expressed in terms of the restricted Lie derivative of the metric g in the direction of the Reeb field. Indeed, knowing (3.2) it is clear that for every q ∈ M and every v, w ∈ H q Such an approach allows to define higher order invariants, namely ones that correspond to the bilinear forms .. In this way, however, we will not obtain any formulas involving the structure functions c i 12 , so κ is unavoidable. Using (4.1) we are in a position to prove the propositions below.   Next we study the effect on the invariants when we dilate the structure. To this end suppose that we have a sub-Lorentzian ts-oriented structure (H, g) which is given by an orthonormal frame X 1 , X 2 with a time (resp. space) orientation X 1 (resp X 2 ). Let λ > 0 be a constant. Consider the sub-Lorentzian structure (H ′ , g ′ ) defined by assuming the frame X ′ 1 = λX 1 , X ′ 2 = λX 2 to be orthonormal with the time (resp. space) orientation X ′ 1 (resp. X ′ 2 ). The normalized one form ω ′ which defines H ′ is given by ω ′ = 1 λ 2 ω, i.e., dω ′ (X ′ 1 , X ′ 2 ) = ω ′ ([X ′ 2 , X ′ 1 ]) = 1. It follows that the Reeb field is now λ 2 X 0 . Then it is easy to see that (2.4) can be rewritten as where c i ′ jk = λc i jk . As a corollary we obtain Proposition 4.4. Let χ, κ,h (resp. χ ′ , κ ′ ,h ′ ) be the ts-invariants of the sub-Lorentzian structure defined by an orthonormal basis X 1 , X 2 (resp. by X ′ 1 = λX 1 , 4.1. The case χ = 0,h = 0. Next let us assume that χ(q) = 0 buth q = 0 (i.e. c 1 01 = 0) everywhere. As we shall see we are given an additional structure in this case. Indeed, the correspondence q −→ kerh q defines an invariantly given field of directions. We can distinguish two cases: (i) and kerh q is spanned by X 1 (q) − X 2 (q) for each q. In the second case the matrix ofh q is equal to c 1 01 −c 1 01 c 1 01 −c 1 01 and kerh q is spanned by X 1 (q) + X 2 (q). Thus in the considered case there exists a line sub-bundle L −→ M of H on which g is equal to zero. Of course this result is trivial under assumption on ts-orientation because then H admits a global orthonormal basis X 1 , X 2 and we have in fact two such subbundles, namely Span{X 1 + X 2 } and Span{X 1 − X 2 }. What is interesting here is that the condition χ = 0,h = 0 does not depend on the assumption on orientation. Indeed, notice that if we change a time (resp. space) orientation keeping space (resp. time) one thenh is multiplied by −1 (because so is X 0 ). Moreover the condition χ = 0 , h = 0 means thath is a non-zero map with vanishing eigenvalues, the fact being independent of possible multiplication by −1. Therefore the condition χ = 0,h = 0 makes sense even for an unoriented contact sub-Lorentzian structures. In this way we are led to the following proposition. Proof. Fix an arbitrary point q ∈ M . Let Y 1 , Y 2 be an orthonormal basis for (H, g) defined on a neighborhood U of q, where Y 1 is timelike and Y 2 is spacelike. Supposing Y 1 (resp. Y 2 ) to be a time (resp. space) orientation we can apply the above construction of ts -invariants obtaining the corresponding objects χ U and h U . By our assumption and the above remark χ U = 0,h U = 0 on U , and we get an invariantly defined line sub-bundle L U −→ U : U ∋ q −→ ker(h U ) q =: L U (q). We repeat the same construction around any point q ∈ M , which results in the family {L U −→ U } U⊂M of line sub-bundles, indexed by elements U of an open covering of M . By construction L U (q) = L U ′ (q) for any q ∈ U ∩ U ′ .
Let us note that if M is simply connected, then the assertion of proposition 4.5 holds true no matter the values of χ andh are, because in this case the metric (H, g) admits a global orthonormal frame, see [10].
Proof. This is clear because X 0 is an infinitesimal isometry.
Rewriting as above (4.3) in terms of ν i 's we arrive at Lemma 4.1. The following identities hold −X 1 (c) − cc 2 12 + X 0 (c 1 12 ) = 0, X 2 (c) − cc 1 12 + X 0 (c 2 12 ) = 0. Proof. The lemma is obtained upon applying the exterior differential to both sides of the second and the third equation in (4.4).
Our next aim, which will be achieved in the next subsection, is to find a hyperbolic rotation of our frame X 1 , X 2 so that (4.3) significantly simplifies. More precisely we want to kill the terms c i 12 , i = 1, 2. To this end let us introduce the following 1-form (4.5) η = (κ + c)ν 0 + c 1 12 ν 1 − c 2 12 ν 2 whose significance will become evident below.
To end the proof we use lemma 4.1 and the definition of κ.

4.3.
The simply-connected Lie group case. Suppose that our contact sub-Lorentzian manifold (M, H, g) is such that M is a simply-connected Lie group and H, g are left-invariant; this means that left translations of M are sub-Lorentzian isometries (note that any left invariant bracket generating distribution on a 3dimensional Lie group is necessarily contact). In such a case, clearly, χ and κ are constant. We also remark that unlike the general situation, the assumption on ts-orientation is no longer restrictive since groups are parallelizable manifolds. As above, assume thath = 0 everywhere.
Recalling our aim formulated in the previous subsection we prove the following lemma.
Now we apply to our frame X 1 , X 2 , the hyperbolic rotation by the angle θ specified above. As a result, the frame Y 1 , Y 2 given by (2.10) satisfies Proof. It follows directly from facts proved in subsections 4.2, 4.3, from (4.3), and from lemma 2.2. When κ = 0 we have: If M is a simply-connected Lie group such thath vanishes and κ = 0, then it is isometric to a sub-Lorentzian structure on SL 2 (R) induced by the Killing form.
Before proving corollary 4.2 let us recall some basic facts about the Killig form and Cartan decompositions. For any Lie algebra the Killing form is the symmetric bilinear form defined by K(X, Y ) = Trace(ad X ad Y ). The Killing form has the following invariance properties: (1) K([X, Y ], Z) = K(X, [Y, Z]) and (2) K(T (X), T (Y )) = K(X, Y ) for all T ∈ Aut(g). If g is simple then any symmetric bilinear form satisfying the first invariance condition is a scalar multiple of the Killing form and Cartan's criterion states that a Lie algebra is semisimple if and only if the Killing form is non-degenerate.
A Cartan involution is any element Θ ∈ Aut(g) such that Θ 2 = I and X, Y Θ = −K(X, Θ(Y )) is positive definite. Corresponding with Θ we have a Cartan decomposition g = t ⊕ p where t and p are the eigenspaces corresponding with the eigenvalues 1 and −1 respectively. Since Θ is an automorphism, it follows that [t, t] ⊆ t, [t, p] ⊆ p and [p, p] ⊆ t. Moreover, the Killing form is negative definite on t and positive definite on p.
The standard Cartan involution on sl 2 is given by Θ(A) = −A T . In this case we have that t = span{f 1 } and p = span and the Lie brackets are If we set H e = span{n 1 , n 2 } and define B(n 1 , n 1 ) = −1, B(n 2 , n 2 ) = 1, B(n 1 , n 2 ) = 0, then the induced left invariant structure on SL 2 (R) is isometrically distinct from the cases above, indeedh = κ 1 0 0 −1 .

5.1.
Introduction. In this section we determine the conformal and isometry groups for the Heisenberg group and the universal cover of SL 2 . In particular we will see that in both cases the local infinitesimal conformal transformations are given by sl 3 .
In the context of this paper it would be natural to construct the vector fields using the criteria developed in section 3, however this leads to complicated systems of p.d.e. which we cannot provide explicit proof concerning solutions. Instead we apply Cartan's equivalence method which leads to the general solution without having to solve p.d.e..
In work in preparation with Alexandr Medvedev we explore further the application of the Cartan approach to sub-Lorentzian geometry. In particular the invariants discussed here appear in a much more systematic manner.

5.2.
Application of the equivalent method. The structure equations for the Cartan connection associated with the conformal symmetries of the sub-Lorentzian structure on the group given in proposition 4.8 are: and for the isometries the structure equations are: Consequently, for all κ, the conformal symmetries are given by SL 3 (R) and the isometries are always 4 dimensional.
The structure equations are obtained by staring with the ordered basis {ω 1 , ω 2 , ω 0 } such that an applying Cartan's equivalence method with structure group G given by the matrices: Note that G is the subgroup of GL 3 that leaves the sub-Lorentzian metric ω 2 2 − ω 2 1 conformally invariant modulo terms of the form η ⊙ ω 0 .
To begin we define ones forms Θ i on M × G by setting  with the following forms: The coefficients of the Θ i in Π j are determined by absorbing torsion and the α j are the Maurer-Cartan forms: In particular the coefficients B 1 , . . . , B 4 are undetermined parameters from absorption and so a prolongation is required. We write and consider the equivalence problem for the ordered basis with structure group G (1) consisting of matrices of the form The forms Θ 1 , Θ 2 , Θ 3 , Π 1 , Π 2 , Π 3 , Π 4 are now viewed as forms on the 11 dimensional manifold M × G × G (1) and again are augmented by forms {Ω 1 , Ω 2 , Ω 3 , Ω 4 } ⊂ T * (M × G × G (1) ). We get the following reductions of the structure group G (1) : and so M × G × G (1) becomes an 8 dimensional manifold and we only need Ω = Ω 1 to augment. Finally after absorption we arrive at the structure equations (5.1).
The structure equations for the isometries are obtained similarly but do not require prolongation.
The Killing form for the conformal structure equations is Since detK = 0 and K is indefinite with signature + + + + − − −, the Lie algebra must be sl 3 , i.e., the only 8 dimensional simple Lie algebras are sl 3 , su 3 and su 2,1 , however su 3 and su 2,1 are ruled out by the indefiniteness and signature. Alternatively one can simply compute the Lie brackets of the vector fields dual to the system of one forms and check that it isomorphic to sl 3 . The fact in that the universal cover of SL 2 (R) and the Heisenberg group both have SL 3 (R) as the conformal symmetry group implies that they are all conformally equivalent, see [17] proposition 2.3.2. and [5] section 2.5. We thus have the following Theorem conformal Darboux theorem.
Theorem 5.1. All left invariant sub-Lorentzian structures on the universal cover of SL 2 (R) such thath = 0 are locally conformally equivalent to the sub-Lorentzian Heisenberg group.

Rigid example.
As it is seen from the previous sections, in particular (5.2), the dimension of the algebra of infinitesimal isometries for left-invariant sub-Lorentzian structures is at least 3 and at most 4. The goal of this section is to show that this is not the case for general, not necessarily left-invariant, contact sub-Lorentzian structures. Since the computations based on considerations from Section 3 rely on solving systems of PDE's, even in the simplest cases, they are very complicated and do not lead to explicit solutions. We will use instead known facts from the geometry of second order ODE's.
Consider an equation where the right hand side is smooth. In terms of differential forms, equation (6.1) is encoded by by the coframe: defined on the first jet space J 1 = J 1 (R, R) with coordinates (x, u, p), where p = u ′ . More precisely, a curve γ(x) = (x, u(x), p(x)) in the space J 1 defines a solution to (6.1) if and only if γ * ω i = 0, i = 1, 2 (one can easily show that the vanishing of the two pull-backs is equivalent to u ′ (x) = p(x) and in turn to u ′′ (x) = Q(x, u(x), u ′ (x))). Consider a diffeomorphism Φ : R × R −→ R × R; Φ is called a point transformation or a point symmetry of (6.1) if and only if it maps graphs of solutions to (6.1) onto graphs of solutions to (6.1). It can be proved that a necessary and sufficient condition for Φ to be a point symmetry of (6.1) is the existence of smooth functions a i , i = 1, ..., 5, such that (6.2)Φ * ω 1 = a 1 ω 1 ,Φ * ω 2 = a 2 ω 1 + a 3 ω 2 ,Φ * ω 3 = a 4 ω 1 + a 5 ω 3 , whereΦ : J 1 −→ J 1 is the prolongation of Φ to the jet space J 1 . One of the most important problems in the geometric theory of ODE's is to classify ODE's with respect to point transformations. It is known that the group of point symmetries of a given second order ODE has dimension varying between 0 and 8. E.g. the flat equation u ′′ = 0 has its symmetry group of dimension 8 and the equation does not have (nontrivial) symmetries at all, see [18] page 182. Any equation (6.1) defines a conformal class of contact sub-Lorentzian metrics in terms of the forms (6.2). Indeed, let H = ker ω 1 , L 1 = ker ω 1 ∩ ker ω 2 , L 2 = ker ω 1 ∩ ker ω 3 . Clearly, H is a contact distribution that splits into the union of line bundles: H = L 1 ⊕ L 2 . Similarly as in the classical situation the splitting can be viewed as the field of null cones for a Lorentzian metric on H. Of course all such metrics are conformally equivalent. In particular it is seen that all solutions to (6.1) are determined by the trajectories of the null field ∂ ∂x + p ∂ ∂u + Q(x, u, p) ∂ ∂p spanning L 1 . Consider now any sub-Lorentzian structure (U, H, g), U being a neighborhood of 0 in R 3 , associated, in the just described manner, with the equation (6.3). It is easy to notice that (U, H, g) does not admit nontrivial infinitesimal isometries. Indeed, suppose that X is an infinitesimal isometry. Then its flow, say h t , preserves H and moreover (since it is isotopic with identity) h t (L i ) = L i , i = 1, 2. But then h t is induced by a point symmetry of the equation (6.3), therefore h t = id and, what follows, X = 0.

Appendix.
In this appendix we would like to draw the reader's attention to some possible applications of invariants in non-contact cases. Consider the simplest such case, namely the Martinet case. Martinet sub-Lorentzian structures (of Hamiltonian type) were studied in [11]. Let (M, H, g) be a sub-Lorentzian manifold where (H, g) is a Martinet sub-Lorentzian structure (or a metric). That is, there exists a hypersurface S, the so-called Martinet surface, with the following properties: (1) H is a contact structure on M \S, (2) dim(H q ∩ T q S) = 1 for every q ∈ M , (3) the field of directions L : S ∋ q −→ L q = H q ∩ T q S is timelike. It is a standard fact that trajectories of L are abnormal curves for the distribution H. Obviously our construction of the invariants can be carried out on the contact sub-Lorentzian manifold (M \S, H |M\S , g |M\S ). In this way we can produce necessary conditions for two Martinet sub-Lorentzian structures to be tsisometric. More precisely, let (M i , H i , g i ) be Martinet sub-Lorentzian manifolds such that (H i , g i ) are ts-oriented Martinet sub-Lorentzian metrics for i = 1, 2. Suppose that ϕ : (M 1 , H 1 , g 1 ) −→ (M 2 , H 2 , g 2 ) is a ts-isometry, then since abnormal curves are preserved by diffeomorphisms, ϕ(S 1 ) = S 2 where S i is the Martinet surface for H i , i = 1, 2. It follows that ϕ induces a ts-isometryφ = ϕ |M1\S1 : (M 1 \S 1 , H 1|M1\S1 , g 1|M1\S1 ) −→ (M 2 \S 2 , H 2|M2\S2 , g 2|M2\S2 ). Therefore, using results from section 4 we arrive at χ 1 =φ * χ 2 , κ 1 =φ * κ 2 , andh 1 =φ * h 2 , where χ i , κ i ,h i are the corresponding invariants for (M i \S i , H i|Mi\Si , g i|Mi\Si ), i = 1, 2.
As one might expect, the invariants become singular when one approaches the Martinet surface. Indeed, let us look at the following example.