New explicit Lorentzian Einstein-Weyl structures in 3-dimensions

On a 3D manifold, a Weyl geometry consists of pairs $(g, A) =$ (metric, 1-form) modulo gauge $\hat{g} = e^{2\phi} g$, $\hat{A} = A + d\phi$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(\mu\nu)} - \frac{1}{3} R g_{\mu\nu} = 0$ comes from an appropriate 3D leaf space quotient of a 7D connection bundle associated with a 3rd order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided 2 among 3 primary point invariants vanish: \[ Wunschmann(H) \,\equiv\, 0 \,\equiv\, Cartan(H). \] We find that point equivalence of a single PDE $z_y = F(x,y,z,z_x)$ with para-CR integrability $DF := F_x + z_x F_z \equiv 0$ leads to an isomorphic 7D Cartan bundle and connection. Then magically, the (complicated) equation $Wunschmann(H) \equiv 0$ becomes: \[ 0 \,\equiv\, Monge(F) \,:=\, 9\,F_{ppppp}\,F_{pp}^2 - 45\,F_{pppp}\,F_{ppp}\,F_{pp} + 40\,F_{ppp}^3 \ \ \ \ \ \ \ \ (p\,:=\,z_x), \] whose solutions are just conics in the $\{p, F\}$-plane. As an Ansatz, we take: \[ F(x,y,z,p) \,:=\, \frac{\alpha(y)(z-xp)^2+\beta(y)(z-xp)p+\gamma(y)(z-xp) + \delta(y)p^2+\varepsilon(y)p+\zeta(y)}{ \lambda(y)\,(z-xp) + \mu(y)\,p + \nu(y)}, \] with 9 arbitrary functions $\alpha, \dots, \nu$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv Monge(F)$, and we show that the condition $Cartan(H) \equiv 0 $ passes to a certain $K(F) \equiv 0$ which holds for any choice of $\alpha(y), \dots, \nu(y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $[ (g, A) ]$. These structures are nontrivial in the sense that $dA \not\equiv 0$ and $Cotton([g]) \not \equiv 0$.


INTRODUCTION
On an n-manifold M, a Weyl geometry is a pair (g, A) of a signature (k, n−k) pseudo-Riemannian metric modulo g = e 2ϕ g together with a 1-form A modulo A = A + dϕ, where ϕ : M → R is any function. As in Riemannian geometry, a symmetric Ricci tensor R (µν) with scalar curvature R can be defined (see [4,11,10] or Section 2).
Our main approach is to study point equivalences of a single PDE of the form: with unknown z = z(x, y). From para-CR geometry ( [16,12]), an integrability condition is required, namely: To exclude trivial PDEs, another point invariant condition must be assumed: In Theorem 5.1, we construct a 7-dimensional Cartan bundle connection P 7 −→ J 4 (x, y, z, p) canonically associated to point equivalences of such PDEs z y = F(x, y, z, z x ), we determine a canonical coframe θ 1 , θ 2 , θ 3 , θ 4 , Ω 1 , Ω 2 , Ω 3 on P 7 , and we find that its structure equations (4.4) incorporate exactly 3 primary invariants, named A 1 , B 1 , C 1 .
Quite unexpectedly, we realize that these structure equations are exactly the same as the structure equations of the canonical 7-dimensional Cartan bundle connection associated with point equivalences of 3 rd order ODEs y = H(x, y, y , y ). Furthermore, it is known that quite similarly, 3 primary differential invariants govern such geometries. Two among them are: the Wünschmann invariant W(H) and the Cartan invariant C(H). Since Cartan 1943, it is also known ( [4,13,9,11,10] This inspired us to try to work on the PDE side z y = F(x, y, z, z x ), instead of the ODE side. Then magically, W(H) ≡ 0 transforms into the much simpler classical invariant equation: When F pp = 0, it is known that M(F) ≡ 0 holds if and only if there exist functions A, B, C, K, L, M of (x, y, z) such that: Assuming A := 0, we then solve the problem completely.
At the end, we also present other families of functionally parametrized solutions, when A = 0.

WEYL GEOMETRY: A SUMMARY
In Einstein's theory, gravity is described in terms of a (pseudo-)riemannian metric g called the gravitational potential. In Maxwell's theory, the electromagnetic field is described in terms of a 1-form A called the Maxwell potential.
In his attempt Raum, Zeit, Materie [23] of unifying gravitation and electromagnetism, Weyl was inspired to introduce the synthetic geometric structure on any n-dimensional manifold M n which consists of classes of such pairs (g, A) under the equivalence relation: holding by definition if and only if there exists a function ϕ : M −→ R such that: (1) g = e 2 ϕ g; Clearly, the electromagnetic field F := dA depends only on the class. The signature (k, n − k) of g with p + q = n can be arbitrary. Einstein-conformal structures are a special class of Weyl structures, corresponding to the choice of a closed -hence locally exact -1-form A. In any (local) coframe ω µ , µ = 1, . . . , n, for the cotangent bundle T * M in which g = g µν ω µ ⊗ ω ν , the connection 1-forms Γ µ ν of D, or equivalently the Γ µν := g µρ Γ ρ ν , are indeed uniquely defined from the more explict conditions: Then the curvature of this Weyl connection identifies with the collection of n 2 curvature 2-forms: Ω µ ν := dΓ µ ν + Γ µ ρ ∧ Γ ρ ν , which produce the curvature tensor R µ νρσ by expanding in the given coframe ω µ : It turns out that R µ νρσ is a tensor density, which means in particular that its vanishing is independent of the choice of a representative (g, A), and hence as such, serves as a starting point for all invariants of a Weyl geometry M, [(g, A)] , produced by covariant differentiation.
Other invariant objects are: In particular, an appropriately contracted Bianchi identity shows that in 3-dimensions: In [4], Élie Cartan proposed dynamical Einstein equations for a Weyl geometry M, [(g, A)] postulating that the trace-free part of the symmetric Ricci tensor vanishes: where R := g µν R µν , with g µρ g ρν = δ µ ν and n = dim M. These equations (2.1) are called Einstein-Weyl equations, and a Weyl geometry satisfying (2.1) is called an Einstein-Weyl structure. The reason for this name is as follows.
Since a Weyl structure M, [g, A] with vanishing F = dA ≡ 0 is equivalent to a plain (pseudo-)conformal structure (M, [g]) and since the Weyl connection D then reduces to the Levi-Civita connection, these equations (2.1) are a natural generalization of Einstein's field equations. According to Weyl's approach, a gravity potential g is thereby coupled with an electromagnetic field F = dA.

CARTAN'S SOLUTION TO THE EINSTEIN-WEYL VACUUM EQUATIONS
In [3], Cartan gave a geometric description of all solutions to the Einstein-Weyl equations (2.1) in 3-dimensions. In particular, he showed that there is a one-to-one correspondence between 3 rd -order ODEs y = H(x, y, y , y ) considered modulo point transformations of variables which satisfy certain two point-invariant conditions: and 3-dimensional Einstein-Weyl structures with Lorentzian metrics g of signature (2, 1). Abbreviating p := y , q := y , in terms of the total differentiation operator: their explicit expressions are:  Some particular solutions are known, e.g.: or the 'horrible': They were all obtained by rather ad hoc methods. In fact, the main difficulty in getting a systematic approach to finding the solutions is an annoying nonlinearity of the Wünschmann condition W ≡ 0.

THIRD-ORDER ODES MODULO POINT TRANSFORMATIONS OF VARIABLES
It was Cartan ([3]) who solved the equivalence problem for 3 rd order ODEs considered modulo point transformations. Nowadays, the result may be stated more elegantly in terms of a certain Cartan connection ( [11,10]), as follows.
To any 3 rd order ODE: one associates a contact-like coframe on the space J 4 (x, y, p, q) of 2-jets of graphs x −→ y(x): It follows that if a 3 rd order ODE (4.1) undergoes a point transformation of variables: then the 1-forms ω 1 , ω 2 , ω 3 , ω 4 transform as: where the u i are certain functions on J 4 .
Actually, Cartan assures us that the entire equivalence problem for 3 rd order ODEs considered modulo point transformations of variables is the same as the equivalence problem for 1-forms (4.2), considered modulo transformations (4.3). There is a unique way of reducing these eight group parameters u i to only three u 3 , u 5 , u 7 , the other ones being expressed in terms of them. This is achieved by forcing the exterior differentials of the θ µ 's to satisfy the EDS (4.4) below.
Theorem 4.1. [3, 11, 10] A 3 rd order ODE y = H(x, y, y , y ) with its associated 1-forms: uniquely defines a 7-dimensional fiber bundle P 7 −→ J 4 over the space of second jets J 4 (x, y, p, q) and a unique coframe θ 1 , θ 2 , θ 3 , θ 4 , Ω 1 , Ω 2 , Ω 3 on P 7 enjoying structure equations of the shape: Exactly 3 (boxed) invariants are primary: A 1 , B 1 , C 1 , while others express in terms of them and their covariant derivatives. Point equivalence to y = 0 is characterized by 0 ≡ A 1 ≡ B 1 ≡ C 1 . Two relevant explicit expressions are: The seven 1-forms θ 1 , θ 2 , θ 3 , θ 4 , Ω 1 , Ω 2 , Ω 3 set up a Cartan connection ω on P 7 via: , and the structure equations (4.4) are just the equations for the curvature K of this connection: d ω + ω ∧ ω =: K. Now, the structure equations (4.4) guarantee that the bundle P 7 is foliated by a 4dimensional distribution annihilating the three 1-forms θ 1 , θ 3 , θ 4 , and that the leaf space M 3 of this foliation is equipped with a natural Weyl geometry, if and only if two among three primary invariants vanish identically: A representative (g, A) of the concerned Weyl class (g, A) on M 3 has then the signature (2, 1) symmetric bilinear form: which is obtained as the determinant of the lower-left 2 × 2 submatrix of the connection matrix ω, while the 1-form is defined as: It is thanks to the hypothesis A 1 ≡ 0 ≡ C 1 that g and A, originally defined on P 7 , descend on M 3 . Furthermore, it is the result of Cartan in [4] that any such Weyl geometry (g, A) defined on such a leaf space M 3 is automatically Einstein-Weyl! We stress that given H = H(x, y, p, q) satisfying A 1 ≡ 0 ≡ C 1 , or equivalently: one can in principle set up explicit formulae for the corresponding forms θ 1 , θ 3 , θ 4 , Ω 3 on P 7 , and this in turn can provide explicit formulae for (g, A) on M 3 . However, one substantial obstacle is the: In [12], it was shown that the equivalence problem for 3 rd -order ODEs considered modulo point transformations of variables can be embedded into an equivalence problem for 4-dimensional para-CR structures of type (1, 1, 2), cf. also [17]. This thus suggests us a new approach here to constructing Lorentzian Einstein-Weyl structures via para-CR structures of type (1, 1, 2).
Let us therefore associate a para-CR structure with PDEs on the plane of the form: for an unknown function z = z(x, y). Using the abbreviation z x =: p, we will consider such PDEs modulo point transformations of variables: (x, y, z) −→ (x, y, z) = x(x, y, z), y(x, y, z), z(x, y, z) .
This conducts to an equivalence problem for the four 1-forms: y, z, p) dy, ω 2 0 := dp, ω 3 0 := dx, ω 4 0 := dy, given up to transformations:  Within this coframe ω 1 0 , ω 2 0 , ω 3 0 , ω 4 0 , in terms of the two operators: D := ∂ x + p ∂ z and ∆ := ∂ y + F ∂ z , the exterior differential of any function F = F(x, y, z, p) rewrites as: The only nontrivial integrability condition for such an equivalence problem to constitute a true para-CR structure of type (1, 1, 2) comes from: From now on, we will only consider PDEs z y = F(x, y, z, z x ) satisfying DF ≡ 0. Furthermore, we will also assume that another point-invariant condition holds: Cartan's process conducts to choose more convenient representatives of these forms: and we will use this choice in the sequel. Using Cartan's method, it is then straightforward to solve the equivalence problem for point equivalence classes of such PDEs z y = F(x, y, z, z x ). The solution is summarized in the following Theorem 5.1. A PDE system z y = F(x, y, z, z x ) satisfying the two point-invariant conditions: with its associated 1-forms ω 1 , ω 2 , ω 3 , ω 4 as above, uniquely defines a 7-dimensional principal H 3 -bundle H 3 −→ P 7 −→ J 4 over the space of first jets J 4 (x, y, z, p) with the (reduced) structure group H 3 consisting of matrices: together with a unique coframe θ 1 , θ 2 , θ 3 , θ 4 , Ω 1 , Ω 2 , Ω 3 on P 7 where: such that the coframe enjoys precisely the structure equations (4.4). This time however, the curvature invariants A 1 , A 2 , A 3 , B 1 , B 2 , B 3 , B 4 , C 1 , C 2 , C 3 , E 1 , E 2 depend on F = F(x, y, z, p) and its derivatives up to order 6.
Two relevant explicit expressions are: where: Two equations z y = F(x, y, z, z x ) and z y = F x, y, z, z x satisfying DF = 0 = F zxzx and DF ≡ 0 = F z x z x are locally point equivalent if and only if there exists a bundle isomorphism Φ : P 7 ∼ −→ P 7 between the corresponding principal bundles H 3 −→ P 7 −→ J 4 and H 3 −→ P 7 −→ J 4 satisfying: This theorem shows that the geometry of 3 rd order ODEs y = H(x, y, y , y ) considered modulo point transformations of variables is the same as the geometry of PDEs z y = F(x, y, z, z x ) with DF ≡ 0 = F zxzx , also considered modulo point transformations. Thus provided that M(F) ≡ 0, there should exist a conformal Lorentzian metric on the leaf space of the integrable distribution in P 7 annihilated by θ 1 , θ 3 , θ 4 , and when moreover K(F) ≡ 0, all this should produce (new) Einstein-Weyl geometries. Actually, we gain the following Theorem 5.2. A PDE z y = F(x, y, z, z x ) defines a bilinear form g of signature (+, +, −, 0, 0, 0, 0) on the bundle P 7 (x, y, z, p, u 3 , u 5 , u 8 ): degenerate along the rank 4 integrable distribution D 4 which is the annihilator of θ 1 , θ 3 , θ 4 . The PDE z y = F(x, y, z, z x ) also defines the 1-form where: The degenerate bilinear form g descends to a Lorentzian conformal class When M(F) ≡ 0, the local coordinates on M 3 are (x, y, z) with the projection: x, y, z, p, u 3 , u 5 , u 8 −→ (x, y, z), and the conformal class [g] has a representative which is explicitly expressed in terms of dx, dy, dz, with coefficients depending only on (x, y, z). Next, Ω 3 descends to a 1-form denoted A given up to the differential of a function on M 3 (x, y, z), if and only if K(F) ≡ 0.
Moreover, the pair g, Ω 3 descends to a representative of a Einstein-Weyl structure (g, A) on M 3 , if and only if both M(F) ≡ 0 and K(F) ≡ 0.
Finally, this Weyl structure is actually Einstein-Weyl, namely it satisfies (2.1), and all Einstein-Weyl structures in 3-dimensions emerge from this construction.

TRANSFORMATION OF THE WÜNSCHMANN INVARIANT INTO THE MONGE INVARIANT
In particular, PDEs with A 1 ≡ 0 ≡ C 1 always define an Einstein-Weyl geometry on the leaf space M 3 of the integrable distribution in P 7 annihilated by θ 1 , θ 3 , θ 4 .
The advantage of looking at this Weyl geometry from the PDE point of view z y = F(x, y, z, z x ) rather than from the ODE side y = H(x, y, y , y ), is that now the Wünschmann invariant of the ODE becomes the much simpler and classical Monge invariant: Serendipitously, the identical vanishing M(F) ≡ 0 is well known to be equivalent to the condition that the graph of p −→ F(p) is contained in a conic of the (p, F)-plane, with parameters (x, y, z). More precisely: for some functions A, B, C, K, L, M depending only on (x, y, z).