New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 056, 16 pages
New Explicit Lorentzian Einstein–Weyl Structures
in 3-Dimensions
Joël MERKER † and Pawe l NUROWSKI ‡
† Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay,
91405 Orsay Cedex, France
E-mail: joel.merker@universite-paris-saclay.fr
URL: http://www.imo.universite-paris-saclay.fr/~merker/
‡ Centrum Fizyki Teoretycznej, Polska Akademia Nauk,
Al. Lotników 32/46, 02-668 Warszawa, Poland
E-mail: nurowski@cft.edu.pl
URL: http://www.fuw.edu.pl/~nurowski/
Received March 30, 2020, in final form June 08, 2020; Published online June 17, 2020
https://doi.org/10.3842/SIGMA.2020.056
Abstract. On a 3D manifold, a Weyl geometry consists of pairs (g,A) = (metric, 1-form)
modulo gauge ĝ = e2ϕg, Â = A + dϕ. In 1943, Cartan showed that every solution to
the Einstein–Weyl equations R(µν) − 1
3Rgµν = 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
Wünschmann(H) ≡ 0 ≡ Cartan(H).
We find that point equivalence of a single PDE zy = F (x, y, z, zx) with para-CR integrability
DF := Fx+zxFz ≡ 0 leads to a completely similar 7D Cartan bundle and connection. Then
magically, the (complicated) equation Wünschmann(H) ≡ 0 becomes
0 ≡ Monge(F ) := 9F 2
ppFppppp − 45FppFpppFpppp + 40F 3
ppp, p := zx,
whose solutions are just conics in the {p, F}-plane. As an ansatz, we take
F (x, y, z, p) :=
α(y)(z − xp)2+ β(y)(z − xp)p+ γ(y)(z − xp) + δ(y)p2+ ε(y)p+ ζ(y)
λ(y)(z − xp) + µ(y)p+ ν(y)
,
with 9 arbitrary functions α, . . . , ν of y. This F satisfies DF ≡ 0 ≡ Monge(F ), and we
show that the condition Cartan(H) ≡ 0 passes to a certain K(F ) ≡ 0 which holds for any
choice of α(y), . . . , ν(y). Descending to the leaf space quotient, we gain ∞-dimensional
functionally parametrized and explicit families of Einstein–Weyl structures
[
(g,A)
]
in 3D.
These structures are nontrivial in the sense that dA 6≡ 0 and Cotton([g]) 6≡ 0.
Key words: Einstein–Weyl structures; Lorentzian metrics; para-CR structures; third-order
ordinary differential equations; Monge invariant; Wünschmann invariant; Cartan’s method
of equivalence; exterior differential systems
2020 Mathematics Subject Classification: 83C15; 53C25; 83C20; 53C25; 53C10; 53C25;
53A30; 53A55; 34A26; 34C14; 58A15; 53-08
1 Introduction
On an n-manifoldM , a Weyl geometry is a pair (g,A) of a signature (k, n−k) pseudo-Riemannian
metric modulo ĝ = e2ϕg together with a 1-form A modulo  = A+ dϕ, where ϕ : M → R is any
function. As in Riemannian geometry, a symmetric Ricci tensor R(µν) with scalar curvature R
mailto:joel.merker@universite-paris-saclay.fr
http://www.imo.universite-paris-saclay.fr/~merker/
mailto:nurowski@cft.edu.pl
http://www.fuw.edu.pl/~nurowski/
https://doi.org/10.3842/SIGMA.2020.056
2 J. Merker and P. Nurowski
can be defined (see [3, 7, 8] or Section 2). The Einstein–Weyl equations in vacuum
R(µν) − 1
nRgµν = 0, 1 6 µ, ν 6 n, (1.1)
which depend only on the class [(g,A)], have raised interest, specially in dimension n = 3. We
find various functionally parametrized explicit families of solutions. On R3 3 (x, y, z), take for
instance 5 free arbitrary functions b, c, k, l, m of y with derivatives b′, k′.
Theorem 1.1. All pairs
(
g,A
)
such that
g := (k + bz)2dx2 + x2
(
l2 − cm
)
dy2 + x2b2dz2
+ 2x
(
ckz − blz + kl− bm
)
dxdy − 2xb
(
k + bz
)
dxdz − 2x2
(
ck− bl
)
dydz,
A :=
−ck + bl + b′k− bk′
x(ck2 − 2bkl + b2m)
(
xbdz − (k + bz)dx
)
+
bl2 − cbm− b′kl + bb′m + ckk′ − bk′l
ck2 − 2bkl + b2m
dy,
satisfy equations (1.1), hence define a Lorentzian Einstein–Weyl structure on R3.
Moreover, all such examples are generically conformally non-flat, and each of the 5 inde-
pendent components of the Cotton tensor of the underlying conformal structure (M, [g]) is not
identically zero.
We discover in fact even more general explicit families of solutions depending on 9 free
arbitrary functions of 1 variable y. Explicit examples of Einstein–Weyl structures in 3D were
known before [1, 3, 4, 5, 7, 8, 10, 11, 12, 17, 18, 19, 20].
According to [3], all Einstein–Weyl structures may be constructed by a certain quotient pro-
cess from a 7D Cartan bundle associated with equivalences of 3rd order ordinary differential
equations. Those, in turn, are known to be para-CR structures of type (1, 1, 2), cf. [9, Sec-
tion 5.1.3].
In the present paper, we explore the observation that PDEs on the plane (x, y) of the form
zy = F (x, y, z, zx), considered modulo point transformations, also happen to be (1, 1, 2) para-CR
structures, in certain circumstances. In Section 8, we show how equivalence classes of (1, 1, 2)
para-CR structures associated to PDEs zy = F are ‘embedded’ into the space of equivalence
classes of 3rd order ODEs. This distinguishes a certain class of 3rd order ODEs from which we
construct our explicit solutions to the Einstein–Weyl equations.
Thus, our main approach is to study point equivalences of a single PDE of the form (novelty)
zy = F (x, y, z, zx),
with unknown z = z(x, y). From para-CR geometry [9, 13], an integrability condition is required,
namely,
DF := Fx + zxFz ≡ 0.
To exclude trivial PDEs, another point invariant condition must be assumed:
Fpp 6= 0 (abbreviate p := zx).
In Theorem 5.2, we construct a 7-dimensional Cartan bundle
/
connection P7 −→ J4 3
(x, y, z, p) canonically associated to point equivalences of such PDEs zy = F (x, y, z, zx), we
determine a canonical coframe
{
θ1, θ2, θ3, θ4,Ω1,Ω2,Ω3
}
on P7, and we find that its structure
equations (4.4) incorporate exactly 3 primary invariants, named A1, B1, C1.
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 3
Quite unexpectedly, we realize that these structure equations have the same form as the
structure equations of the canonical 7-dimensional Cartan bundle
/
connection associated with
point equivalences of 3rd 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) [22] and the Cartan invariant C(H) [2, 3]. Since Cartan 1943, it
is also known [3, 6, 7, 8, 10] that all solutions to the Einstein–Weyl structure equations (1.1) can
be obtained from ODEs satisfying W(H) ≡ 0 ≡ C(H). Translating what is known for ODEs or
performing computations from scratch, we will set up and state Cartan’s construction from the
PDE side, see Theorem 5.3.
But from the ODE side unfortunately, it is quite difficult to solve Wünschmann’s nonlinear
equation incorporating 25 differential monomials
0 ≡W(H) := −18qHqHpq + 9pHyHqq + 18qHHpqq + 9qHpHqq − 18pHqHyq + 18pHHyqq
− 9HHqHqq + 18pqHypq + 18pHxyq + 18qHxpq + 9HxHqq + 18HHxqq
− 18HqHxq + 18HpHq + 9Hxxq − 27Hxp + 4H3
q + 9p2Hyyq − 27pHyp
+ 9qHyq + 9q2Hppq − 27qHpp − 18HHpq + 9H2Hqqq + 54Hy.
This inspired us to try to work on the PDE side zy = F (x, y, z, zx), instead of the ODE side.
Then magically, W(H) ≡ 0 transforms into the much simpler classical invariant of Monge [16]
0 ≡ Monge(F ) := 9F 2
ppFppppp − 45FppFpppFpppp + 40F 3
ppp,
When Fpp 6= 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
0 ≡ aF 2 + 2bFp+ cp2 + 2kF + 2lp+ m.
Assuming a := 0, we obtain the following
Proposition 1.2. The general solution F = F (x, y, z, p) to
0 ≡ Fx + pFz,
0 ≡ 0 + 2bFp+ cp2 + 2kF + 2lp+ m
is
F =
α(y)(z − xp)2 + β(y)(z − xp)p+ γ(y)(z − xp) + δ(y)p2 + ε(y)p+ ζ(y)
λ(y)(z − xp) + µ(y)p+ ν(y)
,
with 9 arbitrary functions α, β, γ, δ, ε, ζ, λ, µ, ν of y.
Of course, to the Cartan invariant C(H) from the ODE side there corresponds from the PDE
side a certain invariant we name K(F ): its expression appears in Theorem 5.2. Miraculously,
then, a direct calculation shows that no further constraint is imposed.
Proposition 1.3. For any choice of α(y), β(y), γ(y), δ(y), ε(y), ζ(y), λ(y), µ(y), ν(y), the
second condition
K
(
Fα,...,ν
)
≡ 0
for obtaining Weyl pairs [(g,A)] satisfying the Einstein–Weyl field equations (1.1) holds auto-
matically.
4 J. Merker and P. Nurowski
We then get – quite long – formulas for pairs
[
(g,A)
]
expressed explicitly in terms of α, β, γ,
δ, ε, ζ, λ, µ, ν. The subfamily for which β = 0, δ = 0, ε = 0, µ = 0 corresponds (with different
notations) to Theorem 1.1.
Theorem 1.4. Same conclusion as in Theorem 1.1 with
g := τ1τ2 + τ2τ1 + τ3τ3,
A := τ3
1
2Π
(
γλx− γµ+ xλν ′ + βλz + λµ′z − 2αµz
− λ′µz − µν ′ − xλ′ν − 2xαν + βν + µ′ν
)
,
with the coframe
τ1 := dx+
dy
xλ− µ
(
xβ − δ − x2α
)
,
τ2 :=
2dy
xλ− µ
Π,
τ3 := (−λz − ν) dx+
1
xλ− µ
dy
(
−εµ+ 2x2αν + xγµ− 2xβν − βµz + 2δλz + 2xαµz
+ xελ+ 2δν − x2γλ− xβλz
)
+ (xλ− µ) dz,
and the function
Π := x2ζλ2 + αµ2z2 + 2xαµνz + x2αν2 − βλµz2 − xβλνz + δλ2z2 + xελ2z − 2xζλµ
− βµνz − xβν2 + 2δλνz − ελµz + xελν − xγλµz − x2γλν + ζµ2 + δν2 − εµν
+ γµ2z + xγµν,
again with dA 6≡ 0 and Cotton([g]) 6≡ 0.
At the end, we also present other families of functionally parametrized solutions, when a 6= 0.
2 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 [21] of unifying gravitation and electromagnetism, Weyl
was inspired to introduce the synthetic geometric structure on any n-dimensional manifold Mn
which consists of classes of such pairs [(g,A)] under the equivalence relation
(g,A) ∼
(
ĝ, Â
)
holding by definition if and only if there exists a function ϕ : M −→ R such that
(1) ĝ = e2ϕg;
(2) Â = A+ dϕ.
Clearly, the electromagnetic field strength F := dA depends only on the class. The signature
(k, n − k) of g can be arbitrary. Conformally Einstein structures from ordinary conformal
geometry are a special class of Weyl structures, corresponding to the choice of a closed – hence
locally exact – 1-form A.
Inspired by Levi-Civita, Weyl established that to such a Weyl structure (M, [(g,A)]) is asso-
ciated a unique connection D on TM satisfying:
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 5
(A) D has no torsion;
(B) Dg = 2Ag for any representative (g,A) of the class [(g,A)].
In any (local) coframe ωµ, µ = 1, . . . , n, for the cotangent bundle T ∗M in which g = gµνω
µων ,
the connection 1-forms Γµν of D, or equivalently the Γµν := gµρΓ
ρ
ν , are indeed uniquely defined
from the more explicit conditions:
(A′) dωµ + Γµν ∧ ων = 0;
(B′) Dgµν := dgµν − Γµν − Γνµ = 2Agµν .
Then the curvature of this Weyl connection identifies with the collection of n2 curvature
2-forms
Ωµ
ν := dΓµν + Γµρ ∧ Γρν ,
which produce the curvature tensor Rµνρσ by expanding in the given coframe ωµ
Ωµ
ν = 1
2R
µ
νρσω
ρ ∧ ωσ.
It turns out that Rµνρσ 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:
• the (Weyl–)Ricci tensor Rµν := Rρµρν ;
• its symmetric part R(µν) := 1
2(Rµν +Rνµ);
• its antisymmetric part R[µν] := 1
2(Rµν −Rνµ).
In particular, an appropriately contracted Bianchi identity shows that in 3-dimensions
R[µν] = −3
2Fµν ,
where F = dA =: 1
2Fµνω
µ ∧ ων .
In [3], 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
R(µν) − 1
nRgµν = 0, (2.1)
where R := gµνRµν , with gµρgρν = δµν and n = dimM .
These equations (2.1) are called Einstein–Weyl equations, and a Weyl geometry satisfy-
ing (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.
6 J. Merker and P. Nurowski
3 Cartan’s solution to the Einstein–Weyl vacuum equations
In [2], 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
3rd-order ODEs y′′′ = H(x, y, y′, y′′) considered modulo point transformations of variables which
satisfy certain two point-invariant conditions
W(H) ≡ 0, (Wünschmann)
C(H) ≡ 0, (Cartan)
and 3-dimensional Einstein–Weyl structures with Lorentzian metrics g of signature (2, 1). Ab-
breviating p := y′, q := y′′, in terms of the total differentiation operator
D := ∂x + p∂y + q∂p +H∂q,
their explicit expressions are
W := 9DDHq − 27DHp − 18HqDHq + 18HqHp + 4H3
q + 54Hy, (3.1)
C := 18HqqDHq − 12HqqH
2
q − 54HqqHp + 36HpqHq − 108Hyq + 54Hpp. (3.2)
Although Cartan’s geometric arguments [3] offer, in the Lorentzian setting, a complete – but
abstract – understanding of the space of all solutions of the Einstein–Weyl equations (2.1), it is
quite difficult to find explicit solutions to the Wünschmann-Cartan equations 0 ≡W(H) ≡ C(H),
which would provide workable formulas for such Einstein–Weyl structures.
Some particular solutions are known, e.g.,
H =
3q2
2p
, H =
3q2p
p2 + 1
, H = q3/2, H = α
(
2qy − p2
)3/2
y2
, α ∈ R,
or the ‘horrible’
H =
pq
(
−12 + 3pq − 8
√
1− pq
)
+ 8
(
1 +
√
1− pq
)
p3
.
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.
4 Third-order ODEs modulo point transformations of variables
It was Cartan [2] who solved the equivalence problem for 3rd order ODEs considered modulo
point transformations. Nowadays, the result may be stated more elegantly in terms of a certain
Cartan connection [7, 8], as follows.
To any 3rd order ODE
y′′′ = H
(
x, y, y′, y′′
)
, (4.1)
one associates a contact-like coframe on the space J4 3 (x, y, p, q) of 2-jets of graphs x 7−→ y(x):
ω1 := dy − pdx, ω2 := dx, ω3 := dp− q dx, ω4 := dq −H(x, y, p, q) dx. (4.2)
It follows that if a 3rd order ODE (4.1) undergoes a point transformation of variables
(x, y) 7−→
(
x, y
)
=
(
x(x, y), y(x, y)
)
,
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 7
then the 1-forms
(
ω1, ω2, ω3, ω4
)
transform as
ω1
ω2
ω3
ω4
7−→
u1 0 0 0
u2 u3 0 0
u4 0 u5 0
u6 0 u7 u8
ω1
ω2
ω3
ω4
=:
θ1
θ2
θ3
θ4
, (4.3)
where the ui are certain functions on J4.
Actually, Cartan assures us that the entire equivalence problem for 3rd order ODEs con-
sidered 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 ui to only three u3, u5, u7, the other ones being expressed in terms of
them. This is achieved by forcing the exterior differentials of the θµ’s to satisfy the EDS (4.4)
below.
Theorem 4.1 ([2, 7, 8]). A 3rd order ODE y′′′ = H(x, y, y′, y′′) with its associated 1-forms
ω1 = dy − p dx, ω2 = dx, ω3 = dp− q dx, ω4 = dq −H(x, y, p, q) dx,
uniquely defines a 7-dimensional fiber bundle P7 −→ J4 over the space of second jets J4 3
(x, y, p, q) and a unique coframe
{
θ1, θ2, θ3, θ4,Ω1,Ω2,Ω3
}
on P7 enjoying structure equations of
the shape
dθ1 = Ω1 ∧ θ1 − θ2 ∧ θ3,
dθ2 = (Ω1 − Ω3) ∧ θ2 + B1 θ
1 ∧ θ3 − B2θ
1 ∧ θ4,
dθ3 = Ω2 ∧ θ1 + Ω3 ∧ θ3 + θ2 ∧ θ4,
dθ4 = (2Ω3 − Ω1) ∧ θ4 − Ω2 ∧ θ3 − A1 θ
1 ∧ θ2,
dΩ1 = Ω2 ∧ θ2 + (A2 − 2C1)θ
1∧ θ2 + (3B3 + E1)θ
1∧ θ3 + (2B1 − 3B4)θ
1 ∧ θ4 + B2θ
3 ∧ θ4,
dΩ2 = Ω2 ∧ (Ω1 − Ω3)− A3θ
1 ∧ θ2 + E2θ
1 ∧ θ3 − (B3 + E1)θ
1 ∧ θ4 + C1 θ
2 ∧ θ3
+ (B1 − 2B4)θ
3 ∧ θ4,
dΩ3 = −C1θ
1 ∧ θ2 + (2B3 + E1)θ
1 ∧ θ3 + 2(B1 − B4)θ
1 ∧ θ4 + 2B2θ
3 ∧ θ4. (4.4)
Moreover, two equations y′′′ = H(x, y, y′, y′′) and y′′′ = H
(
x, y, y′, y′
)
are locally point equiva-
lent if and only if there exists a local bundle isomorphism Φ: P7
∼−→ P 7 between the corresponding
bundles P7 −→ J4 and P 7 −→ J4 satisfying
Φ∗θ
µ
= θµ and Φ∗Ωi = Ωi, µ = 1, 2, 3, 4, i = 1, 2, 3.
Exactly 3 (boxed) invariants are primary: A1, B1, C1, while others express in terms of them
and their covariant derivatives. Point equivalence to y′′′ = 0 is characterized by 0 ≡ A1 ≡ B1 ≡
C1. Two relevant explicit expressions are
A1 =
1
54
u33
u31
W, (W in (3.1))
C1 =
1
54
u3
u21
(
C +
1
27
Wq
)
. (C in (3.2))
The seven 1-forms
(
θ1, θ2, θ3, θ4,Ω1,Ω2,Ω3
)
set up a Cartan connection ω̂ on P7 via
ω̂ :=
1
2Ω1
1
2Ω2 0 0
−θ2 Ω3 − 1
2Ω1 0 0
θ3 −θ4 1
2Ω1 − Ω3 −1
2Ω2
2θ1 θ3 θ2 −1
2Ω1
,
8 J. Merker and P. Nurowski
and the structure equations (4.4) are just the equations for the curvature K̂ of this connection
dω̂ + ω̂ ∧ ω̂ =: K̂.
Now, the structure equations (4.4) guarantee that the bundle P7 is foliated by a 4-dimensional
distribution annihilating the three 1-forms
(
θ1, θ3, θ4
)
, and that the leaf space M3 of this foliation
is equipped with a natural Weyl geometry, if and only if two among three primary invariants
vanish identically
0 ≡ A1(H) ≡ C1(H).
A representative (g,A) of the concerned Weyl class [(g,A)] on M3 has then the signature (2, 1)
symmetric bilinear form
g := θ3θ3 + θ1θ4 + θ4θ1,
which is obtained as the determinant of the lower-left 2×2 submatrix of the connection matrix ω̂,
while the 1-form is defined as
A := Ω3.
It is thanks to the hypothesis A1 ≡ 0 ≡ C1 that g and A, originally defined on P7, descend
on M3.
Furthermore, according to a result of Cartan in [3], any such Weyl geometry [(g,A)] defined
on such a leaf space M3 is automatically Einstein–Weyl!
We stress that given H = H(x, y, p, q) satisfying A1 ≡ 0 ≡ C1, or equivalently
W(H) ≡ 0 ≡ C(H),
one can in principle set up explicit formulas for the corresponding forms θ1, θ3, θ4, Ω3 on P7, and
this in turn can provide explicit formulas for (g,A) on M3. However, one substantial obstacle is
Question 4.2. How to solve W(H) ≡ 0 ≡ C(H)?
5 PDE on the plane zy = F (x, y, z, zx)
modulo point transformations
We recall that in [9], it was shown that the equivalence problem for 3rd-order ODEs considered
modulo point transformations of variables is in one-to-one correspondence with the equivalence
problem for 4-dimensional para-CR structures of type (1, 1, 2), cf. also [14, 15]. This thus suggests
a new approach for constructing Lorentzian Einstein–Weyl structures via para-CR structures
of type (1, 1, 2). Instead of working with general para-CR structures of type (1, 1, 2), we will
concentrate on a subclass determined in the following way.
We start with a class of PDEs of the form
zy = F (x, y, z, zx),
considered modulo point transformations, for an unknown function z = z(x, y). We then ask
when this class defines a para-CR structure of type (1, 1, 2).
To answer this (in Proposition 5.1), we need a little preparation. Using the abbreviation
zx =: p, we indeed consider such PDEs modulo point transformations of variables
(x, y, z) 7−→ (x, y, z) =
(
x(x, y, z), y(x, y, z), z(x, y, z)
)
.
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 9
This leads to an equivalence problem for the four 1-forms
ω1
0 := dz − p dx− F (x, y, z, p) dy, ω2
0 := dp, ω3
0 := dx, ω4
0 := dy,
given up to transformations
ω1
0
ω2
0
ω3
0
ω4
0
7−→
u1 0 0 0
u2 u3 0 0
u4 0 u5 u6
u7 0 u8 u9
ω1
0
ω2
0
ω3
0
ω4
0
. (5.1)
Within this coframe
{
ω1
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) can be rewritten as
dF = Fzω
1
0 + Fpω
2
0 +DFω3
0 + ∆Fω4
0.
Proposition 5.1. The coframe of 1-forms
{
ω1
0, ω
2
0, ω
3
0, ω
4
0
}
modulo transformations (5.1) defines
a para-CR structure of type (1, 1, 2) if and only if
0 ≡ DF = Fx + pFz.
Proof. The only nontrivial integrability condition required to constitute a true para-CR struc-
ture comes from
0 = dω1
0 ∧ ω1
0 ∧ ω2
0 = −DFω1
0 ∧ ω2
0 ∧ ω3
0 ∧ ω4
0. �
We will now show that for this class of para-CR structures there is an amazing coincidence
between its main invariant, which will happen to be the Monge invariant with respect to p, and
the classical Wünschmann invariant of the corresponding class of 3rd order ODEs modulo point
transformations.
From now on, we will only consider PDEs zy = F (x, y, z, zx) satisfying DF ≡ 0. Furthermore,
we will also assume that another point-invariant condition holds
0 6= Fpp (everywhere).
Cartan’s process leads one to choose more convenient representatives of these forms
ω1 := ω1
0,
ω2 := ω2
0 −
∆FpppFpp −∆FppFppp + 3FpFppFzpp − 3F 2
ppFzp − 2FpFpppFzp
6F 3
pp
ω1
0,
ω3 := ω3
0 + Fpω
4
0 −
1
3
Fppp
Fpp
ω1
0,
ω4 := Fppω
4
0 +
4F 2
ppp − 3FppFpppp
18F 2
pp
ω1
0,
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 zy = F (x, y, z, zx). The solution is summarized in the following
10 J. Merker and P. Nurowski
Theorem 5.2. A PDE system zy = F (x, y, z, zx) satisfying the two point-invariant conditions
DF ≡ 0 6= Fzxzx ,
with its associated 1-forms ω1, ω2, ω3, ω4 as above, uniquely defines a 7-dimensional principal
H3-bundle H3 −→ P7 −→ J4 over the space of first jets J4 3 (x, y, z, p) with the (reduced)
structure group H3 consisting of matrices
u3u5 0 0 0
0 u3 0 0
−u3u8 0 u5 0
−u3u
2
8
2u5
0 u8
u5
u3
, u3 ∈ R∗, u5 ∈ R∗, u8 ∈ R,
together with a unique coframe
{
θ1, θ2, θ3, θ4,Ω1,Ω2,Ω3
}
on P7 where
θ1
θ2
θ3
θ4
:=
u3u5 0 0 0
0 u3 0 0
−u3u8 0 u5 0
−u3u
2
8
2u5
0 u8
u5
u3
ω1
ω2
ω3
ω4
,
such that the coframe enjoys precisely the structure equations (4.4). This time however, the
curvature invariants A1, A2, A3, B1, B2, B3, B4, C1, C2, C3, E1, E2 depend on F = F (x, y, z, p)
and its derivatives up to order 6.
Two relevant explicit expressions are
A1 = − 1
54
1
u33
M
F 3
pp
, C1 =
1
3
1
u23u5
K
F 5
pp
,
where
M := 9FpppppF
2
pp − 45FppppFpppFpp + 40F 3
ppp,
K := ∆FpppppF
3
pp − 5∆FppppF
2
ppFppp + 12∆FpppFppF
2
ppp − 12∆FppF
3
ppp − 4∆FpppF
2
ppFpppp
+ 9∆FppFppFpppFpppp −∆FppF
2
ppFppppp + 5FpF
3
ppFppppz + 6F 4
ppFpppz
− 20FpF
2
ppFpppFpppz − 12F 3
ppFpppFppz + 36FpFppF
2
pppFppz − 12FpF
2
ppFppppFppz
+ 8F 2
ppF
2
pppFpz − 24FpF
3
pppFpz − 3F 3
ppFppppFpz + 18FpFppFpppFppppFpz
− 2FpF
2
ppFpppppFpz.
Two equations zy = F (x, y, z, zx) and zy = F
(
x, y, z, zx
)
satisfying DF = 0 6= Fzxzx and
DF ≡ 0 6= F zxzx are locally point equivalent if and only if there exists a bundle isomorphism
Φ: P7
∼−→ P 7 between the corresponding principal bundles H3 −→ P7 −→ J4 and H3 −→ P 7 −→
J4 satisfying
Φ∗θ
µ
= θµ and Φ∗Ωi = Ωi, µ = 1, 2, 3, 4, i = 1, 2, 3.
This theorem enables one to think about the geometry of a PDE zy = F (x, y, z, zx) with
DF ≡ 0 6= Fzxzx , considered modulo point transformations of variables, as the geometry of
a certain 3rd order ODE y′′′ = H(x, y, y′, y′′), also considered modulo point transformations. In
particular, one can ask how big is the subclass of point nonequivalent 3rd order ODEs which are
related to PDEs zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx .
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 11
We will not answer this question in this paper. Instead, we concentrate on the Einstein–Weyl
geometric aspect of the above observation.
Since the EDS staying behind the PDEs zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx is vi-
sibly the same as the EDS for 3rd order ODEs y′′′ = H(x, y, y′, y′′), one can look for PDEs
zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx , which in addition satisfy A1 = C1 = 0, and build
a corresponding Einstein Weyl geometry, not in terms of H(x, y, y′, y′′) satisfying W(H) ≡
C(H) ≡ 0, but in terms of the function F (x, y, z, zx) satisfying DF ≡ M(F ) ≡ K(F ) ≡ 0. If
only M(F ) ≡ 0, there exists a conformal Lorentzian metric on the leaf space of the integrable
distribution in P7 annihilated by
{
θ1, θ3, θ4
}
, and when moreover K(F ) ≡ 0, all this produces
Einstein–Weyl geometries. Actually, we gain the following
Theorem 5.3. A PDE zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx defines a bilinear form g̃ of
signature (+,+,−, 0, 0, 0, 0) on the bundle P7 3 (x, y, z, p, u3, u5, u8):
g̃ = θ3θ3 + θ1θ4 + θ4θ1 =
u25
9F 2
pp
{(
3Fpp[dx+ Fp dy]− Fppp[dz − p dx− F dy]
)2
+ (dz − p dx− F dy)
(
18F 3
ppdy + [4F 2
ppp − 3FppFpppp][dz − p dx− F dy]
)}
,
degenerate along the rank 4 integrable distribution D4 which is the annihilator of θ1, θ3, θ4.
The PDE zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx also defines the 1-form
Ω3 := rx dx+ ry dy + rz dz + 1
3d
[
log
(
u35Fpp
)]
,
where
rx =
1
3F 4
pp
{
∆FpppF
2
pp −∆FppFppFppp + 3FpF
2
ppFppz − F 3
ppFpz − 2FpFppFpppFpz
−∆FppppF
2
ppp+ 3∆FpppFppFpppp− 3∆FppF
2
pppp+ ∆FppFppFppppp
− 4FpF
2
ppFpppzp− 2F 3
ppFppzp+ 9FpFppFpppFppzp+ F 2
ppFpppFpzp
− 6FpF
2
pppFpzp+ 2FpFppFppppFpzp
}
,
ry =
1
3F 4
pp
{
−∆FppppFF
2
pp + ∆FpppFpF
2
pp −∆FppF
3
pp + 3∆FpppFFppFppp
−∆FppFpFppFppp − 3∆FppFF
2
ppp + ∆FppFFppFpppp − 4FFpF
2
ppFpppz
+ 3F 2
pF
2
ppFppz − 2FF 3
ppFppz + 9FFpFppFpppFppz − 3FpF
3
ppFpz
− 2F 2
pFppFpppFpz + FF 2
ppFpppFpz − 6FFpF
2
pppFpz + 2FFpFppFppppFpz + 3F 4
ppFz
}
,
rz =
1
3F 4
pp
{
∆FppppF
2
pp − 3∆FpppFppFppp + 3∆FppF
2
ppp −∆FppFppFpppp + 4FpF
2
ppFpppz
+ 2F 3
ppFppz − 9FpFppFpppFppz − F 2
ppFpppFpz + 6FpF
2
pppFpz − 2FpFppFppppFpz
}
.
The degenerate bilinear form g̃ descends to a Lorentzian conformal class [g] on the leaf
space M3 of the distribution D4, if and only if the Monge invariant M(F ) ≡ 0 vanishes identi-
cally.
When M(F ) ≡ 0, the local coordinates on M3 are (x, y, z) with the projection
P7 −→M3,(
x, y, z, p, u3, u5, u8
)
7−→ (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).
12 J. Merker and P. Nurowski
Next, Ω3 descends to a 1-form denoted A given up to the differential of a function on M3 3
(x, y, z), if and only if K(F ) ≡ 0.
Moreover, the pair
(
g̃,Ω3
)
descends to a representative of a Weyl structure [(g,A)] on M3, 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).
6 Transformation of the Wünschmann invariant
into the Monge invariant
As we now know, PDEs zy = F (x, y, z, zx) with DF ≡ 0 6= Fzxzx satisfying A1 ≡ 0 ≡ C1 always
define an Einstein–Weyl geometry on the leaf space M3 of the integrable distribution in P7
annihilated by
{
θ1, θ3, θ4
}
.
The advantage of looking at a Weyl geometry from the PDE zy = F (x, y, z, zx) point of view
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
A1(H) ∼M(F ) = 9F 2
ppFppppp − 45FppFppppFppp + 40F 3
ppp.
Serendipitously, the identical vanishing M(F ) ≡ 0 is well known to be equivalent to the
condition that the graph of p 7−→ F (p) is contained in a conic of the (p, F )-plane, with parameters
(x, y, z). More precisely,
0 ≡M(F ) ⇐⇒ aF 2 + 2bFp+ cp2 + 2kF + 2lp+ m ≡ 0, (6.1)
for some functions a, b, c, k, l, m depending only on (x, y, z).
Thus, passing from the formulation of Einstein–Weyl’s equations in terms of a 3rd order ODE
y′′′ = H(x, y, y′, y′′) to the formulation in terms of a PDE zy = F (x, y, z, zx), we are able to find
a rather large class of solutions to the equation
W(H) ≡ 0.
Indeed, by replacing W(H) ; M(F ), the solution (6.1) is just conical!
7 How to construct new explicit Lorentzian Einstein–Weyl
metrics?
But remember we also have to assure that
0 ≡ DF = ∂xF + p∂zF.
The simultaneous conditionsDF ≡ 0 ≡M(F ) can be achieved for instance by taking F satisfying
aF 2 + 2bF (z − px) + c(z − px)2 + 2kF + 2l(z − px) + m ≡ 0,
with a, b, c, k, l, m being now functions of y only!
From now on, we will analyze this special solution for M(F ) ≡ 0 ≡ DF . The simplest case
occurs when avoiding square root by choosing
a := 0,
so that
F :=
−c(z − xp)2 − 2l(z − xp)−m
2b(z − xp) + 2k . (7.1)
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 13
Here
b = b(y), c = c(y), k = k(y), l = l(y), m = m(y)
are free arbitrary differentiable functions of one variable y.
A direct check shows that remarkably this solution (7.1) also satisfies K(F ) ≡ 0!
Proposition 7.1. All such
F :=
−c(z − xp)2 − 2l(z − xp)−m
2b(z − xp) + 2k
with any functions b, c, k, l, m of y, lead to Einstein–Weyl structures in 3-dimensions.
Performing the Cartan procedure to determine the coframe
{
θ1, θ2, θ3, θ4,Ω1,Ω2,Ω3
}
, pro-
jecting both θ3θ3 + θ1θ4 + θ4θ1 and Ω3 to the leaf space of the annihilator M3 of
{
θ1, θ3, θ4
}
,
equipping M3 ≡ R3 with coordinates (x, y, z), we therefore obtain functionally parameterized
Einstein–Weyl structures
(
g,A
)
on R3 3 (x, y, z) represented by the signature (2, 1) Lorentzian
metric
g := (k + bz)2dx2 + x2
(
l2 − cm
)
dy2 + x2b2dz2
+ 2x(ckz − blz + kl− bm)dx dy − 2xb(k + bz)dx dz − 2x2(ck− bl)dy dz,
together with the differential 1-form
A :=
−ck + bl + b′k− bk′
x
(
ck2 − 2bkl + b2m
)(xb dz − (k + bz) dx
)
+
bl2 − cbm− b′kl + bb′m + ckk′ − bk′l
ck2 − 2bkl + b2m
dy.
An independent direct check confirms that equations (1.1) are indeed identically fulfilled.
As regards the Cotton tensor, we compute its 5 components, and find that they are not
identically zero. Hence the obtained Einstein–Weyl structures are generically conformally non-
flat. Thus, Theorem 1.1 is established. The story for Theorem 1.4 is quite similar.
Next, without assuming a ≡ 0 in (6.1), let us now make the ansatz that
aF 2 + 2bF (z − xp) + c(z − xp)2 + 2kF + 2l(z − xp) + m ≡ 0,
for some arbitrary functions a, b, c, k, l, m of y. The (two) solutions F automatically satisfy
DF ≡ 0 ≡M(F ).
Since the solutions to Monge’s equation are conics in the (p, F )-plane, we can rewrite in
a hyperbolic setting(
aF + b(z − xp) + c
)2 − (kF + l(z − xp) + m
)2 ≡ 1,
with changed functions a, b, c, k, l, m of y. To avoid transcendental functions in computations,
we parametrize cosh t = 1+q2
2q and sinh t = 1−q2
2q , and then, solving for F and for z− xp, we may
start from
F = a(y)
1 + q2
2q
+ b(y)
1− q2
2q
+ c(y),
z − xp = k(y)
1 + q2
2q
+ l(y)
1− q2
2q
+ m(y),
again with (changed) free functions a, b, c, k, l, m of y. Taking
ω1
0 := d(z − xp) + x dp− F dy, ω2
0 := dx, ω3
0 := dy, ω4
0 := dp,
and performing para-CR Cartan reduction to an {e}-structure/connection, we obtain
14 J. Merker and P. Nurowski
Proposition 7.2. The second invariant condition K(F ) ≡ 0 holds precisely in the following two
cases:
(1) k = l;
(2) c = m′ and a = bl+kk′−ll′
k .
In case (1), we obtain Einstein–Weyl structures for all free functions a, b, c, l, m of y given
by
g := 2τ1τ2 +
(
τ3
)2
, A := − 2(a + b)
x(a− b)lτ
2 − c−m′
x(a− b)
τ3,
where
τ1 := x(a + b) dy − 2l dx, τ2 := −1
2x(a− b) dy, τ3 := xc dy − x dz + (z −m) dx.
We verify that these Einstein–Weyl structures have nontrivial F = dA 6≡ 0 and nontrivial
Cotton([g]) 6≡ 0.
In case (2), we obtain Einstein–Weyl structures given by
g := 2τ1τ2 +
(
τ3
)2
, A := d
[
log
(
x2e
)]
,
where
τ1 := (k + l)k dx+ x(bk− bl + kk′ − ll′) dy,
τ2 := 1
2(k− l)k dx+ 1
2x(bk− bl + kk′ − ll′) dy,
τ3 := −(z −m)k dx− xkm dy + xk dz.
But this structure, which depends on 3 functions b, k, l of y, is flat
dA ≡ 0 ≡ Cotton([g]).
Finally, without replacing p by z − xp, let us make the ansatz that
aF 2 + 2bFp+ cp2 + 2kF + 2lp+ m ≡ 0.
Dealing similarly with the hyperbolic case,
F = a(y)
1 + q2
2q
+ b(y)
1− q2
2q
+ c(y),
p = k(y)
1 + q2
2q
+ l(y)
1− q2
2q
+ m(y),
we obtain nontrivial Einstein–Weyl structures. For instance, when k = l as in (1) above
g := 2τ1τ2 +
(
τ3
)2
, A := − m′
(a− b)lτ
3,
where
τ1 := 2l dx+ (a + b) dy, τ2 := −1
2(a− b) dy, τ3 := dz −m dx+ (a + b) dy.
Note that this is again nontrivial
dA 6≡ 0 6≡ Cotton([g]).
and note that we do not have x, z dependence here.
New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions 15
8 Transforming zy = F (zx) into w′′′ = w′′H(t)
We end up by exploring a link between our PDE systems and 3rd order ODEs. For simplicity,
we will assume that F = F (zx) depends only on p = zx.
To avoid notational confusion, 3rd-order ODEs will now be denoted as w′′′ = H(t, w,w′, w′′),
and the fundamental 1-forms as
ω1 := dz − p dx− F (p) dy, θ1 := dw − w1 dt,
ω2 := dp, θ2 := dt,
ω3 := dx, θ3 := dw1 − w2 dt,
ω4 := dy, θ4 := dw2 −H(t, w,w1, w2) dt.
We ask what equivalence class of 3rd-order ODE’s corresponds to the equivalence class of PDEs
zy = F (zx), still with Fpp 6= 0, and under the G-structures of Sections 4 and 5.
For this, since ω1 and θ1 are both defined up to plain dilations ω1 ∼ uω1 and θ1 ∼ uθ1, we
transform ω1 in order to make the shape of θ1 appear, using that F depends only on p
ω1 = d(z − xp− yF (p))− (−x− yFp(p)) dp =: dw − w1 dt
with t := p, w := z − xp − yF (p), w1 := −x − yFp(p). With this, ω2 = dp = dt = θ2. Next,
using ω3 ∼ −ω3 − udy, it comes
ω3 = dx = −
[
d(−x− yFp(p)) + yFpp(p) dp+ Fp(p) dy
]
∼
[
dw1 + yFpp(p) dp+ Fp(p) dy
]
− Fp(p) dy = dw1 − (−yFpp(p)) dp,
whence w2 := −yFpp(p).
A last computation using ω4 ∼ uω4
ω4 = dy = − 1
Fpp(p)
[
d(−yFpp(p)) + yFppp(p) dp
]
∼ dw2 −
(
−yFppp(p) dp
)
,
shows that the right-hand side function H = H(t, w2) of the associated ODE w′′′ = H is
independent of w,w1 as it must be
H := −yFppp(p) = w2
Fttt(t)
Ftt(t)
.
Hence
w′′′ = w′′
Fttt(t)
Ftt(t)
is the 3rd order ODE associated to the para-CR structure given by zy = F (zx). Observe that
zy = 1
2z
2
x becomes w′′′ = 0, leading to the flat Einstein–Weyl structure.
Assertion 8.1. The Wünschmann invariant for ODEs w′′′ = w′′ Fttt(t)
Ftt(t)
, where F (t) with Ftt 6= 0
is an arbitrary function of one variable, corresponds to the Monge invariant of the PDE zy =
F (zx):
Wünschmann
(
w2
Fttt(t)
Ftt(t)
)
=
9Ftt(t)Fttttt(t)− 45FttFtttFtttt + 40F 3
ttt
F 3
tt
=
Monge(F )
F 3
tt
.
Proof. Among the 25 terms of Wünschmann’s invariant shown in the Introduction, only 7
remain thanks to 0 ≡ Hw ≡ Hw1 :
Wünschmann
(
H(t, w2)
)
= −9HHw2Hw2w2 − 9HtHw2w2 + 18HHtw2w2 − 18Hw2Htw2
+ 9Httw2 + 4H3
w2
+ 9H2Hw2w2w2 ,
and a direct substitution of H := w2
Fttt(t)
Ftt(t)
leads to the result. �
16 J. Merker and P. Nurowski
Acknowledgements
Insights of the anonymous referees are gratefully acknowledged. This collaboration is supported
by the National Science Center, Poland, grant number 2018/29/B/ST1/02583.
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1 Introduction
2 Weyl geometry: a summary
3 Cartan's solution to the Einstein–Weyl vacuum equations
4 Third-order ODEs modulo point transformations of variables
5 PDE on the plane zy = F(x,y,z,zx) modulo point transformations
6 Transformation of the Wünschmann invariant into the Monge invariant
7 How to construct new explicit Lorentzian Einstein–Weyl metrics?
8 Transforming zy=F(zx) into w''' = w'' H(t)
References
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| id | nasplib_isofts_kiev_ua-123456789-210694 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:30Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Merker, Joël Nurowski, Paweł 2025-12-15T15:20:54Z 2020 New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions. Joël Merker and Paweł Nurowski. SIGMA 16 (2020), 056, 16 pages 1815-0659 2020 Mathematics Subject Classification: 83C15; 53C25; 83C20; 53C25; 53C10; 53C25; 53A30; 53A55; 34A26; 34C14; 58A15; 53-08 arXiv:1906.10880 https://nasplib.isofts.kiev.ua/handle/123456789/210694 https://doi.org/10.3842/SIGMA.2020.056 Insights of the anonymous referees are gratefully acknowledged. This collaboration is supported by the National Science Center, Poland, grant number 2018/29/B/ST1/02583. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions Article published earlier |
| spellingShingle | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions Merker, Joël Nurowski, Paweł |
| title | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions |
| title_full | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions |
| title_fullStr | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions |
| title_full_unstemmed | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions |
| title_short | New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions |
| title_sort | new explicit lorentzian einstein-weyl structures in 3-dimensions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210694 |
| work_keys_str_mv | AT merkerjoel newexplicitlorentzianeinsteinweylstructuresin3dimensions AT nurowskipaweł newexplicitlorentzianeinsteinweylstructuresin3dimensions |