New Explicit Lorentzian Einstein–Weyl Structures in 3-Dimensions

New Explicit Lorentzian Einstein–Weyl Structures
in 3-Dimensions

Joël MERKER † and Paweł NUROWSKI ‡

J. Merker and P. Nurowski

† Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université
Paris-Saclay,
† 91405 Orsay Cedex, France<EMAIL_ADDRESS>http://www.imo.universite-paris-saclay.fr/~merker/

‡ Centrum Fizyki Teoretycznej, Polska Akademia Nauk,
‡ Al. Lotników 32/46, 02-668 Warszawa, Poland<EMAIL_ADDRESS>http://www.fuw.edu.pl/~nurowski/

Received March 30, 2020, in final form June 08, 2020; Published online June
17, 2020

On a $3$D manifold, a Weyl geometry consists of pairs $(g,A)=$ (metric,
$1$-form) modulo gauge $\widehat{g}={\rm e}^{2\varphi}g$, $\widehat{A}=A+{\rm
d}\varphi$. In 1943, Cartan showed that every solution to the Einstein–Weyl
equations $R_{(\mu\nu)}-\frac{1}{3}Rg_{\mu\nu}=0$ comes from an appropriate
$3$D leaf space quotient of a $7$D connection bundle associated with a 3rd
order ODE $y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$ modulo
point transformations, provided $2$ among $3$ primary point invariants vanish

$\displaystyle\text{\sf W\"{u}nschmann}(H)\equiv 0\equiv\text{\sf Cartan}(H).$

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 a completely
similar $7$D Cartan bundle and connection. Then magically, the (complicated)
equation $\text{\sf W\"{u}nschmann}(H)\equiv 0$ becomes

$\displaystyle 0\equiv\text{\sf
Monge}(F):=9F_{pp}^{2}F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^{3},\qquad
p:=z_{x},$

whose solutions are just conics in the $\\{p,F\\}$-plane. As an ansatz, we
take

$\displaystyle
F(x,y,z,p):=\frac{\alpha(y)(z-xp)^{2}\\!+\beta(y)(z-xp)p+\gamma(y)(z-xp)+\delta(y)p^{2}\\!+\varepsilon(y)p+\zeta(y)}{\lambda(y)(z-xp)+\mu(y)p+\nu(y)},\\!$

with $9$ arbitrary functions $\alpha,\dots,\nu$ of $y$. This $F$ satisfies
$DF\equiv 0\equiv\text{\sf Monge}(F)$, and we show that the condition
$\text{\sf Cartan}(H)\equiv 0$ passes to a certain ${\text{K}}(F)\equiv 0$
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 $\big{[}(g,A)\big{]}$ in $3$D.
These structures are nontrivial in the sense that ${\rm d}A\not\equiv 0$ and
$\text{\sf Cotton}([g])\not\equiv 0$.

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

83C15; 53C25; 83C20; 53C25; 53C10; 53C25; 53A30; 53A55; 34A26; 34C14; 58A15;
53-08

##  1 Introduction

On an $n$-manifold $M$, a Weyl geometry is a pair $(g,A)$ of a signature
$(k,n-k)$ pseudo-Riemannian metric modulo $\widehat{g}={\rm e}^{2\varphi}g$
together with a $1$-form $A$ modulo $\widehat{A}=A+{\rm d}\varphi$, where
$\varphi\colon M\to\mathbb{R}$ is any function. As in Riemannian geometry, a
symmetric Ricci tensor $R_{(\mu\nu)}$ with scalar curvature $R$ can be defined
(see [3, 7, 8] or Section 2). The Einstein–Weyl equations in vacuum

$\displaystyle R_{(\mu\nu)}-\tfrac{1}{n}Rg_{\mu\nu}=0,\qquad
1\leqslant\mu,\nu\leqslant n,$ (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 $\mathbb{R}^{3}\ni(x,y,z)$, take for instance $5$ free
arbitrary functions b, c, k, l, m of $y$ with derivatives
${\text{b}}^{\prime}$, ${\text{k}}^{\prime}$.

######  Theorem 1.1.

All pairs $\big{(}g,A\big{)}$ such that

$\displaystyle g:=({\text{k}}+{\text{b}}z)^{2}{\rm
d}x^{2}+x^{2}\big{(}{\text{l}}^{2}-{\text{c}}{\text{m}}\big{)}{\rm
d}y^{2}+x^{2}{\text{b}}^{2}{\rm d}z^{2}$
$\displaystyle\hphantom{g:=}{}+2x\big{(}{\text{c}}{\text{k}}z-{\text{b}}{\text{l}}z+{\text{k}}{\text{l}}-{\text{b}}{\text{m}}\big{)}{\rm
d}x{\rm d}y-2x{\text{b}}\big{(}{\text{k}}+{\text{b}}z\big{)}{\rm d}x{\rm
d}z-2x^{2}\big{(}{\text{c}}{\text{k}}-{\text{b}}{\text{l}}\big{)}{\rm d}y{\rm
d}z,$ $\displaystyle
A:=\frac{-{\text{c}}{\text{k}}+{\text{b}}{\text{l}}+{\text{b}}^{\prime}{\text{k}}-{\text{b}}{\text{k}}^{\prime}}{x({\text{c}}{\text{k}}^{2}-2{\text{b}}{\text{k}}{\text{l}}+{\text{b}}^{2}{\text{m}})}\big{(}x{\text{b}}{\rm
d}z-({\text{k}}+{\text{b}}z){\rm d}x\big{)}$
$\displaystyle\hphantom{A:=}{}+\frac{{\text{b}}{\text{l}}^{2}-{\text{c}}{\text{b}}{\text{m}}-{\text{b}}^{\prime}{\text{k}}{\text{l}}+{\text{b}}{\text{b}}^{\prime}{\text{m}}+{\text{c}}{\text{k}}{\text{k}}^{\prime}-{\text{b}}{\text{k}}^{\prime}{\text{l}}}{{\text{c}}{\text{k}}^{2}-2{\text{b}}{\text{k}}{\text{l}}+{\text{b}}^{2}{\text{m}}}{\rm
d}y,$

satisfy equations (1.1), hence define a Lorentzian Einstein–Weyl structure on
$\mathbb{R}^{3}$.

Moreover, all such examples are generically conformally non-flat, and each of
the $5$ independent 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 $3$D 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 process from a $7$D 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, Section 5.1.3].

In the present paper, we explore the observation that PDEs on the plane
$(x,y)$ of the form $z_{y}=F(x,y,z,z_{x})$, 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 $z_{y}=F$ 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)

$z_{y}=F(x,y,z,z_{x}),$

with unknown $z=z(x,y)$. From para-CR geometry [9, 13], an integrability
condition is required, namely,

$DF:=F_{x}+z_{x}F_{z}\equiv 0.$

To exclude trivial PDEs, another point invariant condition must be assumed:

$F_{pp}\neq 0\qquad(\text{\rm abbreviate}\ p:=z_{x}).$

In Theorem 5.2, we construct a $7$-dimensional Cartan bundle/connection
$P_{7}\longrightarrow J_{4}\ni(x,y,z,p)$ canonically associated to point
equivalences of such PDEs $z_{y}=F(x,y,z,z_{x})$, we determine a canonical
coframe
$\big{\\{}\theta^{1},\theta^{2},\theta^{3},\theta^{4},\Omega_{1},\Omega_{2},\Omega_{3}\big{\\}}$
on $P_{7}$, and we find that its structure equations (4.4) incorporate exactly
$3$ primary invariants, named ${\text{A}}_{1}$, ${\text{B}}_{1}$,
${\text{C}}_{1}$.

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
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$. Furthermore, it
is known that quite similarly, $3$ primary differential invariants govern such
geometries. Two among them are: the Wünschmann invariant ${\text{W}}(H)$ [22]
and the Cartan invariant ${\text{C}}(H)$ [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 ${\text{W}}(H)\equiv
0\equiv{\text{C}}(H)$. 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

$\displaystyle
0\equiv{\text{W}}(H):=-18qH_{q}H_{pq}+9pH_{y}H_{qq}+18qHH_{pqq}+9qH_{p}H_{qq}-18pH_{q}H_{yq}+18pHH_{yqq}$
$\displaystyle\hphantom{0\equiv{\text{W}}(H):=}{}-9HH_{q}H_{qq}+18pqH_{ypq}+18pH_{xyq}+18qH_{xpq}+9H_{x}H_{qq}+18HH_{xqq}$
$\displaystyle\hphantom{0\equiv{\text{W}}(H):=}{}-18H_{q}H_{xq}+18H_{p}H_{q}+9H_{xxq}-27H_{xp}+4H_{q}^{3}+9p^{2}H_{yyq}-27pH_{yp}$
$\displaystyle\hphantom{0\equiv{\text{W}}(H):=}{}+9qH_{yq}+9q^{2}H_{ppq}-27qH_{pp}-18HH_{pq}+9H^{2}H_{qqq}+54H_{y}.$

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, ${\text{W}}(H)\equiv 0$ transforms
into the much simpler classical invariant of Monge [16]

$0\equiv\text{\sf
Monge}(F):=9F_{pp}^{2}F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^{3},$

When $F_{pp}\neq 0$, it is known that ${\text{M}}(F)\equiv 0$ holds if and
only if there exist functions a, b, c, k, l, m of $(x,y,z)$ such that

$0\equiv\text{\sc a}F^{2}+2\text{\sc b}Fp+\text{\sc c}p^{2}+2\text{\sc
k}F+2\text{\sc l}p+\text{\sc m}.$

Assuming $\text{\sc a}:=0$, we obtain the following

######  Proposition 1.2.

The general solution $F=F(x,y,z,p)$ to

$\displaystyle 0\equiv F_{x}+pF_{z},$ $\displaystyle 0\equiv 0+2\text{\sc
b}Fp+\text{\sc c}p^{2}+2\text{\sc k}F+2\text{\sc l}p+\text{\sc m}$

is

$F=\frac{\alpha(y)(z-xp)^{2}+\beta(y)(z-xp)p+\gamma(y)(z-xp)+\delta(y)p^{2}+\varepsilon(y)p+\zeta(y)}{\lambda(y)(z-xp)+\mu(y)p+\nu(y)},$

with $9$ arbitrary functions $\alpha$, $\beta$, $\gamma$, $\delta$,
$\varepsilon$, $\zeta$, $\lambda$, $\mu$, $\nu$ of $y$.

Of course, to the Cartan invariant ${\text{C}}(H)$ from the ODE side there
corresponds from the PDE side a certain invariant we name ${\text{K}}(F)$: 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 $\alpha(y)$, $\beta(y)$, $\gamma(y)$, $\delta(y)$,
$\varepsilon(y)$, $\zeta(y)$, $\lambda(y)$, $\mu(y)$, $\nu(y)$, the second
condition

${\text{K}}\big{(}F_{\alpha,\dots,\nu}\big{)}\equiv 0$

for obtaining Weyl pairs $[(g,A)]$ satisfying the Einstein–Weyl field
equations (1.1) holds automatically.

We then get – quite long – formulas for pairs $\big{[}(g,A)\big{]}$ expressed
explicitly in terms of $\alpha$, $\beta$, $\gamma$, $\delta$, $\varepsilon$,
$\zeta$, $\lambda$, $\mu$, $\nu$. The subfamily for which $\beta=0$,
$\delta=0$, $\varepsilon=0$, $\mu=0$ corresponds (with different notations) to
Theorem 1.1.

######  Theorem 1.4.

Same conclusion as in Theorem 1.1 with

$\displaystyle g:=\tau^{1}\tau^{2}+\tau^{2}\tau^{1}+\tau^{3}\tau^{3},$
$\displaystyle A:=\tau^{3}\frac{1}{2\Pi}\big{(}\gamma\lambda
x-\gamma\mu+x\lambda\nu^{\prime}+\beta\lambda
z+\lambda\mu^{\prime}z-2\alpha\mu z$
$\displaystyle\hphantom{A:=\tau^{3}\frac{1}{2\Pi}\big{(}}{}-\lambda^{\prime}\mu
z-\mu\nu^{\prime}-x\lambda^{\prime}\nu-2x\alpha\nu+\beta\nu+\mu^{\prime}\nu\big{)},$

with the coframe

$\displaystyle\tau^{1}:={\rm d}x+\frac{{\rm
d}y}{x\lambda-\mu}\big{(}x\beta-\delta-x^{2}\alpha\big{)},$
$\displaystyle\tau^{2}:=\frac{2{\rm d}y}{x\lambda-\mu}\Pi,$
$\displaystyle\tau^{3}:=(-\lambda z-\nu)\,{\rm
d}x+\frac{1}{x\lambda-\mu}\,{\rm
d}y\big{(}{-}\varepsilon\mu+2x^{2}\alpha\nu+x\gamma\mu-2x\beta\nu-\beta\mu
z+2\delta\lambda z+2x\alpha\mu z$
$\displaystyle\hphantom{\tau^{3}:=}{}+x\varepsilon\lambda+2\delta\nu-x^{2}\gamma\lambda-x\beta\lambda
z\big{)}+(x\lambda-\mu)\,{\rm d}z,$

and the function

$\displaystyle\Pi:=x^{2}\zeta\lambda^{2}+\alpha\mu^{2}z^{2}+2x\alpha\mu\nu
z+x^{2}\alpha\nu^{2}-\beta\lambda\mu z^{2}-x\beta\lambda\nu
z+\delta\lambda^{2}z^{2}+x\varepsilon\lambda^{2}z-2x\zeta\lambda\mu$
$\displaystyle\hphantom{\Pi:=}{}-\beta\mu\nu z-x\beta\nu^{2}+2\delta\lambda\nu
z-\varepsilon\lambda\mu z+x\varepsilon\lambda\nu-x\gamma\lambda\mu
z-x^{2}\gamma\lambda\nu+\zeta\mu^{2}+\delta\nu^{2}-\varepsilon\mu\nu$
$\displaystyle\hphantom{\Pi:=}{}+\gamma\mu^{2}z+x\gamma\mu\nu,$

again with ${\rm d}A\not\equiv 0$ and $\text{\sf Cotton}([g])\not\equiv 0$.

At the end, we also present other families of functionally parametrized
solutions, when $\text{\sc a}\neq 0$.

##  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 $M^{n}$ which consists of classes of
such pairs $[(g,A)]$ under the equivalence relation

$(g,A)\sim\big{(}\widehat{g},\widehat{A}\big{)}$

holding by definition if and only if there exists a function $\varphi\colon
M\longrightarrow\mathbb{R}$ such that

  1. (1)

$\widehat{g}={\rm e}^{2\varphi}g$;

  2. (2)

$\widehat{A}=A+{\rm d}\varphi$.

Clearly, the electromagnetic field strength $F:={\rm d}A$ 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 associated a unique connection D on $TM$ satisfying:

  1. (A)

D has no torsion;

  2. (B)

${\text{D}}g=2Ag$ for any representative $(g,A)$ of the class $[(g,A)]$.

In any (local) coframe $\omega^{\mu}$, $\mu=1,\dots,n$, for the cotangent
bundle $T^{\ast}M$ in which $g=g_{\mu\nu}\omega^{\mu}\omega^{\nu}$, the
connection $1$-forms ${\Gamma^{\mu}}_{\nu}$ of D, or equivalently the
$\Gamma_{\mu\nu}:=g_{\mu\rho}{\Gamma^{\rho}}_{\nu}$, are indeed uniquely
defined from the more explicit conditions:

  1. (A′)

${\rm d}\omega^{\mu}+{\Gamma^{\mu}}_{\nu}\wedge\omega^{\nu}=0$;

  2. (B′)

${\text{D}}g_{\mu\nu}:={\rm
d}g_{\mu\nu}-\Gamma_{\mu\nu}-\Gamma_{\nu\mu}=2Ag_{\mu\nu}$.

Then the curvature of this Weyl connection identifies with the collection of
$n^{2}$ curvature $2$-forms

${\Omega^{\mu}}_{\nu}:={\rm
d}{\Gamma^{\mu}}_{\nu}+{\Gamma^{\mu}}_{\rho}\wedge{\Gamma^{\rho}}_{\nu},$

which produce the curvature tensor ${R^{\mu}}_{\nu\rho\sigma}$ by expanding in
the given coframe $\omega^{\mu}$

${\Omega^{\mu}}_{\nu}=\tfrac{1}{2}{R^{\mu}}_{\nu\rho\sigma}\omega^{\rho}\wedge\omega^{\sigma}.$

It turns out that ${R^{\mu}}_{\nu\rho\sigma}$ 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 $R_{\mu\nu}:={R^{\rho}}_{\mu\rho\nu}$;

  * •

its symmetric part $R_{(\mu\nu)}:=\frac{1}{2}(R_{\mu\nu}+R_{\nu\mu})$;

  * •

its antisymmetric part $R_{[\mu\nu]}:=\frac{1}{2}(R_{\mu\nu}-R_{\nu\mu})$.

In particular, an appropriately contracted Bianchi identity shows that in
$3$-dimensions

$R_{[\mu\nu]}=-\tfrac{3}{2}F_{\mu\nu},$

where $F={\rm d}A=:\frac{1}{2}F_{\mu\nu}\omega^{\mu}\wedge\omega^{\nu}$.

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

$\displaystyle R_{(\mu\nu)}-\tfrac{1}{n}Rg_{\mu\nu}=0,$ (2.1)

where $R:=g^{\mu\nu}R_{\mu\nu}$, with
$g^{\mu\rho}g_{\rho\nu}={\delta^{\mu}}_{\nu}$ 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 $F={\rm d}A\equiv 0$ 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 $F={\rm d}A$.

##  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
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$ considered modulo
point transformations of variables which satisfy certain two point-invariant
conditions

$\displaystyle{\text{W}}(H)\equiv 0,$ (Wünschmann)
$\displaystyle{\text{C}}(H)\equiv 0,$ (Cartan)

and $3$-dimensional Einstein–Weyl structures with Lorentzian metrics $g$ of
signature $(2,1)$. Abbreviating $p:=y^{\prime}$, $q:=y^{\prime\prime}$, in
terms of the total differentiation operator

${\text{D}}:=\partial_{x}+p\partial_{y}+q\partial_{p}+H\partial_{q},$

their explicit expressions are

$\displaystyle{\text{W}}:=9DDH_{q}-27DH_{p}-18H_{q}DH_{q}+18H_{q}H_{p}+4H_{q}^{3}+54H_{y},$
(3.1)
$\displaystyle{\text{C}}:=18H_{qq}DH_{q}-12H_{qq}H_{q}^{2}-54H_{qq}H_{p}+36H_{pq}H_{q}-108H_{yq}+54H_{pp}.$
(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
$0\equiv{\text{W}}(H)\equiv{\text{C}}(H)$, which would provide workable
formulas for such Einstein–Weyl structures.

Some particular solutions are known, e.g.,

$H=\frac{3q^{2}}{2p},\qquad H=\frac{3q^{2}p}{p^{2}+1},\qquad H=q^{3/2},\qquad
H=\alpha\frac{\big{(}2qy-p^{2}\big{)}^{3/2}}{y^{2}},\qquad\alpha\in\mathbb{R},$

or the ‘horrible’

$H=\frac{pq\big{(}{-}12+3pq-8\sqrt{1-pq}\big{)}+8\big{(}1+\sqrt{1-pq}\big{)}}{p^{3}}.$

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
${\text{W}}\equiv 0$.

##  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

$\displaystyle
y^{\prime\prime\prime}=H\big{(}x,y,y^{\prime},y^{\prime\prime}\big{)},$ (4.1)

one associates a contact-like coframe on the space $J_{4}\ni(x,y,p,q)$ of
$2$-jets of graphs $x\longmapsto y(x)$:

$\displaystyle\omega^{1}:={\rm d}y-p\,{\rm d}x,\qquad\omega^{2}:={\rm
d}x,\qquad\omega^{3}:={\rm d}p-q\,{\rm d}x,\qquad\omega^{4}:={\rm
d}q-H(x,y,p,q)\,{\rm d}x.$ (4.2)

It follows that if a 3rd order ODE (4.1) undergoes a point transformation of
variables

$(x,y)\longmapsto\big{(}\overline{x},\overline{y}\big{)}=\big{(}\overline{x}(x,y),\overline{y}(x,y)\big{)},$

then the $1$-forms $\big{(}\omega^{1},\omega^{2},\omega^{3},\omega^{4}\big{)}$
transform as

$\displaystyle\left(\begin{matrix}\omega^{1}\\\ \omega^{2}\\\ \omega^{3}\\\
\omega^{4}\end{matrix}\right)\longmapsto\left(\begin{matrix}{\sf
u}_{1}&0&0&0\\\ {\sf u}_{2}&{\sf u}_{3}&0&0\\\ {\sf u}_{4}&0&{\sf u}_{5}&0\\\
{\sf u}_{6}&0&{\sf u}_{7}&{\sf
u}_{8}\end{matrix}\right)\left(\begin{matrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\\\ \omega^{4}\end{matrix}\right)=:\left(\begin{matrix}\theta^{1}\\\
\theta^{2}\\\ \theta^{3}\\\ \theta^{4}\end{matrix}\right),$ (4.3)

where the ${\sf u}_{i}$ are certain functions on $J_{4}$.

Actually, Cartan assures us that the entire equivalence problem for 3rd 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 ${\sf
u}_{i}$ to only three ${\sf u}_{3}$, ${\sf u}_{5}$, ${\sf u}_{7}$, the other
ones being expressed in terms of them. This is achieved by forcing the
exterior differentials of the $\theta^{\mu}$’s to satisfy the EDS (4.4) below.

######  Theorem 4.1 ([2, 7, 8]).

A $3$rd order ODE $y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$
with its associated $1$-forms

$\displaystyle\omega^{1}={\rm d}y-p\,{\rm d}x,\qquad\omega^{2}={\rm
d}x,\qquad\omega^{3}={\rm d}p-q\,{\rm d}x,\qquad\omega^{4}={\rm
d}q-H(x,y,p,q)\,{\rm d}x,$

uniquely defines a $7$-dimensional fiber bundle $P_{7}\longrightarrow J_{4}$
over the space of second jets $J_{4}\ni(x,y,p,q)$ and a unique coframe
$\big{\\{}\theta^{1},\theta^{2},\theta^{3},\theta^{4},\Omega_{1},\Omega_{2},\Omega_{3}\big{\\}}$
on $P_{7}$ enjoying structure equations of the shape

$\displaystyle{\rm
d}\theta^{1}=\Omega_{1}\wedge\theta^{1}-\theta^{2}\wedge\theta^{3},$
$\displaystyle
d\theta^{2}=(\Omega_{1}-\Omega_{3})\wedge\theta^{2}+\boxed{{\text{B}}_{1}\\!\\!}\,\theta^{1}\wedge\theta^{3}-{\text{B}}_{2}\theta^{1}\wedge\theta^{4},$
$\displaystyle{\rm
d}\theta^{3}=\Omega_{2}\wedge\theta^{1}+\Omega_{3}\wedge\theta^{3}+\theta^{2}\wedge\theta^{4},$
$\displaystyle{\rm
d}\theta^{4}=(2\Omega_{3}-\Omega_{1})\wedge\theta^{4}-\Omega_{2}\wedge\theta^{3}-\boxed{{\text{A}}_{1}\\!\\!}\,\theta^{1}\wedge\theta^{2},$
$\displaystyle{\rm
d}\Omega_{1}=\Omega_{2}\wedge\theta^{2}+({\text{A}}_{2}-2{\text{C}}_{1})\theta^{1}\\!\wedge\theta^{2}+(3{\text{B}}_{3}+{\text{E}}_{1})\theta^{1}\\!\wedge\theta^{3}+(2{\text{B}}_{1}-3{\text{B}}_{4})\theta^{1}\wedge\theta^{4}+{\text{B}}_{2}\theta^{3}\wedge\theta^{4},$
$\displaystyle{\rm
d}\Omega_{2}=\Omega_{2}\wedge(\Omega_{1}-\Omega_{3})-{\text{A}}_{3}\theta^{1}\wedge\theta^{2}+{\text{E}}_{2}\theta^{1}\wedge\theta^{3}-({\text{B}}_{3}+{\text{E}}_{1})\theta^{1}\wedge\theta^{4}+\boxed{{\text{C}}_{1}\\!\\!}\,\theta^{2}\wedge\theta^{3}$
$\displaystyle\hphantom{{\rm
d}\Omega_{2}=}{}+({\text{B}}_{1}-2{\text{B}}_{4})\theta^{3}\wedge\theta^{4},$
$\displaystyle{\rm
d}\Omega_{3}=-{\text{C}}_{1}\theta^{1}\wedge\theta^{2}+(2{\text{B}}_{3}+{\text{E}}_{1})\theta^{1}\wedge\theta^{3}+2({\text{B}}_{1}-{\text{B}}_{4})\theta^{1}\wedge\theta^{4}+2{\text{B}}_{2}\theta^{3}\wedge\theta^{4}.$
(4.4)

Moreover, two equations
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$ and
$\overline{y}^{\prime\prime\prime}=\overline{H}\big{(}\overline{x},\overline{y},\overline{y}^{\prime},\overline{y}^{\prime}\big{)}$
are locally point equivalent if and only if there exists a local bundle
isomorphism $\Phi\colon P_{7}\overset{\sim}{\longrightarrow}\overline{P}_{7}$
between the corresponding bundles $P_{7}\longrightarrow J_{4}$ and
$\overline{P}_{7}\longrightarrow\overline{J}_{4}$ satisfying

$\displaystyle\Phi^{\ast}\overline{\theta}^{\mu}=\theta^{\mu}\qquad\text{and}\qquad\Phi^{\ast}\overline{\Omega}_{i}=\Omega_{i},\qquad\mu=1,2,3,4,\quad
i=1,2,3.$

Exactly $3$ $($boxed$)$ invariants are primary: ${\text{A}}_{1}$,
${\text{B}}_{1}$, ${\text{C}}_{1}$, while others express in terms of them and
their covariant derivatives. Point equivalence to
$\overline{y}^{\prime\prime\prime}=0$ is characterized by
$0\equiv{\text{A}}_{1}\equiv{\text{B}}_{1}\equiv{\text{C}}_{1}$. Two relevant
explicit expressions are

$\displaystyle{\text{A}}_{1}=\frac{1}{54}\frac{{\sf u}_{3}^{3}}{{\sf
u}_{1}^{3}}{\text{W}},$ (W in (3.1))
$\displaystyle{\text{C}}_{1}=\frac{1}{54}\frac{{\sf u}_{3}}{{\sf
u}_{1}^{2}}\left({\text{C}}+\frac{1}{27}{\text{W}}_{q}\right).$ (C in (3.2))

The seven $1$-forms
$\big{(}\theta^{1},\theta^{2},\theta^{3},\theta^{4},\Omega_{1},\Omega_{2},\Omega_{3}\big{)}$
set up a Cartan connection $\widehat{\omega}$ on $P_{7}$ via

$\widehat{\omega}:=\left(\begin{matrix}\frac{1}{2}\Omega_{1}&\frac{1}{2}\Omega_{2}&0&0\vspace{1mm}\\\
-\theta^{2}&\Omega_{3}-\frac{1}{2}\Omega_{1}&0&0\vspace{1mm}\\\
\theta^{3}&-\theta^{4}&\frac{1}{2}\Omega_{1}-\Omega_{3}&-\frac{1}{2}\Omega_{2}\vspace{1mm}\\\
2\theta^{1}&\theta^{3}&\theta^{2}&-\frac{1}{2}\Omega_{1}\end{matrix}\right),$

and the structure equations (4.4) are just the equations for the curvature
$\widehat{K}$ of this connection

$\displaystyle{\rm
d}\widehat{\omega}+\widehat{\omega}\wedge\widehat{\omega}=:\widehat{K}.$

Now, the structure equations (4.4) guarantee that the bundle $P_{7}$ is
foliated by a $4$-dimensional distribution annihilating the three $1$-forms
$\big{(}\theta^{1},\theta^{3},\theta^{4}\big{)}$, 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

$0\equiv{\text{A}}_{1}(H)\equiv{\text{C}}_{1}(H).$

A representative $(g,A)$ of the concerned Weyl class $[(g,A)]$ on $M_{3}$ has
then the signature $(2,1)$ symmetric bilinear form

$g:=\theta^{3}\theta^{3}+\theta^{1}\theta^{4}+\theta^{4}\theta^{1},$

which is obtained as the determinant of the lower-left $2\times 2$ submatrix
of the connection matrix $\widehat{\omega}$, while the $1$-form is defined as

$A:=\Omega_{3}.$

It is thanks to the hypothesis ${\text{A}}_{1}\equiv 0\equiv{\text{C}}_{1}$
that $g$ and $A$, originally defined on $P_{7}$, descend on $M_{3}$.

Furthermore, according to a result of Cartan in [3], 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 ${\text{A}}_{1}\equiv
0\equiv{\text{C}}_{1}$, or equivalently

${\text{W}}(H)\equiv 0\equiv{\text{C}}(H),$

one can in principle set up explicit formulas for the corresponding forms
$\theta^{1}$, $\theta^{3}$, $\theta^{4}$, $\Omega_{3}$ on $P_{7}$, and this in
turn can provide explicit formulas for $(g,A)$ on $M_{3}$. However, one
substantial obstacle is

######  Question 4.2.

How to solve ${\text{W}}(H)\equiv 0\equiv{\text{C}}(H)$?

##  5 PDE on the plane $\boldsymbol{z_{y}=F(x,y,z,z_{x})}$
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

$z_{y}=F(x,y,z,z_{x}),$

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 $z_{x}=:p$, we indeed consider such PDEs modulo point
transformations of variables

$(x,y,z)\longmapsto(\overline{x},\overline{y},\overline{z})=\big{(}\overline{x}(x,y,z),\overline{y}(x,y,z),\overline{z}(x,y,z)\big{)}.$

This leads to an equivalence problem for the four $1$-forms

$\omega_{0}^{1}:={\rm d}z-p\,{\rm d}x-F(x,y,z,p)\,{\rm
d}y,\qquad\omega_{0}^{2}:={\rm d}p,\qquad\omega_{0}^{3}:={\rm
d}x,\qquad\omega_{0}^{4}:={\rm d}y,$

given up to transformations

$\displaystyle\left(\begin{matrix}\omega_{0}^{1}\\\ \omega_{0}^{2}\\\
\omega_{0}^{3}\\\
\omega_{0}^{4}\end{matrix}\right)\longmapsto\left(\begin{matrix}{\sf
u}_{1}&0&0&0\\\ {\sf u}_{2}&{\sf u}_{3}&0&0\\\ {\sf u}_{4}&0&{\sf u}_{5}&{\sf
u}_{6}\\\ {\sf u}_{7}&0&{\sf u}_{8}&{\sf
u}_{9}\end{matrix}\right)\left(\begin{matrix}\omega_{0}^{1}\\\
\omega_{0}^{2}\\\ \omega_{0}^{3}\\\ \omega_{0}^{4}\end{matrix}\right).$ (5.1)

Within this coframe
$\big{\\{}\omega_{0}^{1},\omega_{0}^{2},\omega_{0}^{3},\omega_{0}^{4}\big{\\}}$,
in terms of the two operators

$D:=\partial_{x}+p\partial_{z}\qquad\text{and}\qquad\Delta:=\partial_{y}+F\partial_{z},$

the exterior differential of any function $F=F(x,y,z,p)$ can be rewritten as

${\rm d}F=F_{z}\omega_{0}^{1}+F_{p}\omega_{0}^{2}+DF\omega_{0}^{3}+\Delta
F\omega_{0}^{4}.$

######  Proposition 5.1.

The coframe of $1$-forms
$\big{\\{}\omega_{0}^{1},\omega_{0}^{2},\omega_{0}^{3},\omega_{0}^{4}\big{\\}}$
modulo transformations (5.1) defines a para-CR structure of type $(1,1,2)$ if
and only if

$\displaystyle 0\equiv DF=F_{x}+pF_{z}.$

###### Proof.

The only nontrivial integrability condition required to constitute a true
para-CR structure comes from

$\displaystyle 0={\rm
d}\omega_{0}^{1}\wedge\omega_{0}^{1}\wedge\omega_{0}^{2}=-DF\omega_{0}^{1}\wedge\omega_{0}^{2}\wedge\omega_{0}^{3}\wedge\omega_{0}^{4}.$
∎

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 $z_{y}=F(x,y,z,z_{x})$ satisfying
$DF\equiv 0$. Furthermore, we will also assume that another point-invariant
condition holds

$0\neq F_{pp}\qquad(\text{\rm everywhere}).$

Cartan’s process leads one to choose more convenient representatives of these
forms

$\displaystyle\omega^{1}:=\omega_{0}^{1},$
$\displaystyle\omega^{2}:=\omega_{0}^{2}-\frac{\Delta F_{ppp}F_{pp}-\Delta
F_{pp}F_{ppp}+3F_{p}F_{pp}F_{zpp}-3F_{pp}^{2}F_{zp}-2F_{p}F_{ppp}F_{zp}}{6F_{pp}^{3}}\omega_{0}^{1},$
$\displaystyle\omega^{3}:=\omega_{0}^{3}+F_{p}\omega_{0}^{4}-\frac{1}{3}\frac{F_{ppp}}{F_{pp}}\omega_{0}^{1},$
$\displaystyle\omega^{4}:=F_{pp}\omega_{0}^{4}+\frac{4F_{ppp}^{2}-3F_{pp}F_{pppp}}{18F_{pp}^{2}}\omega_{0}^{1},$

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 5.2.

A PDE system $z_{y}=F(x,y,z,z_{x})$ satisfying the two point-invariant
conditions

$DF\equiv 0\neq F_{z_{x}z_{x}},$

with its associated $1$-forms $\omega^{1}$, $\omega^{2}$, $\omega^{3}$,
$\omega^{4}$ as above, uniquely defines a $7$-dimensional principal
$H_{3}$-bundle $H_{3}\longrightarrow P_{7}\longrightarrow J_{4}$ over the
space of first jets $J_{4}\ni(x,y,z,p)$ with the $($reduced$)$ structure group
$H_{3}$ consisting of matrices

$\left(\begin{matrix}{\sf u}_{3}{\sf u}_{5}&0&0&0\\\ 0&{\sf u}_{3}&0&0\\\
-{\sf u}_{3}{\sf u}_{8}&0&{\sf u}_{5}&0\\\ -\dfrac{{\sf u}_{3}{\sf
u}_{8}^{2}}{2{\sf u}_{5}}&0&{\sf u}_{8}&\dfrac{{\sf u}_{5}}{{\sf
u}_{3}}\end{matrix}\right),\qquad{\sf u}_{3}\in\mathbb{R}^{\ast},\quad{\sf
u}_{5}\in\mathbb{R}^{\ast},\quad{\sf u}_{8}\in\mathbb{R},$

together with a unique coframe
$\big{\\{}\theta^{1},\theta^{2},\theta^{3},\theta^{4},\Omega_{1},\Omega_{2},\Omega_{3}\big{\\}}$
on $P_{7}$ where

$\left(\begin{matrix}\theta^{1}\\\ \theta^{2}\\\ \theta^{3}\\\
\theta^{4}\end{matrix}\right):=\left(\begin{matrix}{\sf u}_{3}{\sf
u}_{5}&0&0&0\\\ 0&{\sf u}_{3}&0&0\\\ -{\sf u}_{3}{\sf u}_{8}&0&{\sf
u}_{5}&0\\\ -\dfrac{{\sf u}_{3}{\sf u}_{8}^{2}}{2{\sf u}_{5}}&0&{\sf
u}_{8}&\dfrac{{\sf u}_{5}}{{\sf
u}_{3}}\end{matrix}\right)\left(\begin{matrix}\omega^{1}\\\ \omega^{2}\\\
\omega^{3}\\\ \omega^{4}\end{matrix}\right),$

such that the coframe enjoys precisely the structure equations (4.4). This
time however, the curvature invariants ${\text{A}}_{1}$, ${\text{A}}_{2}$,
${\text{A}}_{3}$, ${\text{B}}_{1}$, ${\text{B}}_{2}$, ${\text{B}}_{3}$,
${\text{B}}_{4}$, ${\text{C}}_{1}$, ${\text{C}}_{2}$, ${\text{C}}_{3}$,
${\text{E}}_{1}$, ${\text{E}}_{2}$ depend on $F=F(x,y,z,p)$ and its
derivatives up to order $6$.

Two relevant explicit expressions are

${\text{A}}_{1}=-\frac{1}{54}\frac{1}{{\sf
u}_{3}^{3}}\frac{{\text{M}}}{F_{pp}^{3}},\qquad{\text{C}}_{1}=\frac{1}{3}\frac{1}{{\sf
u}_{3}^{2}{\sf u}_{5}}\frac{{\text{K}}}{F_{pp}^{5}},$

where

$\displaystyle{\text{M}}:=9F_{ppppp}F_{pp}^{2}-45F_{pppp}F_{ppp}F_{pp}+40F_{ppp}^{3},$
$\displaystyle{\text{K}}:=\Delta F_{ppppp}F_{pp}^{3}-5\Delta
F_{pppp}F_{pp}^{2}F_{ppp}+12\Delta F_{ppp}F_{pp}F_{ppp}^{2}-12\Delta
F_{pp}F_{ppp}^{3}-4\Delta F_{ppp}F_{pp}^{2}F_{pppp}$
$\displaystyle\hphantom{{\text{K}}:=}{}+9\Delta
F_{pp}F_{pp}F_{ppp}F_{pppp}-\Delta
F_{pp}F_{pp}^{2}F_{ppppp}+5F_{p}F_{pp}^{3}F_{ppppz}+6F_{pp}^{4}F_{pppz}$
$\displaystyle\hphantom{{\text{K}}:=}{}-20F_{p}F_{pp}^{2}F_{ppp}F_{pppz}-12F_{pp}^{3}F_{ppp}F_{ppz}+36F_{p}F_{pp}F_{ppp}^{2}F_{ppz}-12F_{p}F_{pp}^{2}F_{pppp}F_{ppz}$
$\displaystyle\hphantom{{\text{K}}:=}{}+8F_{pp}^{2}F_{ppp}^{2}F_{pz}-24F_{p}F_{ppp}^{3}F_{pz}-3F_{pp}^{3}F_{pppp}F_{pz}+18F_{p}F_{pp}F_{ppp}F_{pppp}F_{pz}$
$\displaystyle\hphantom{{\text{K}}:=}{}-2F_{p}F_{pp}^{2}F_{ppppp}F_{pz}.$

Two equations $z_{y}=F(x,y,z,z_{x})$ and
$\overline{z}_{\overline{y}}=\overline{F}\big{(}\overline{x},\overline{y},\overline{z},\overline{z}_{\overline{x}}\big{)}$
satisfying $DF=0\neq F_{z_{x}z_{x}}$ and $\overline{D}\overline{F}\equiv
0\neq\overline{F}_{\overline{z}_{\overline{x}}\overline{z}_{\overline{x}}}$
are locally point equivalent if and only if there exists a bundle isomorphism
$\Phi\colon P_{7}\overset{\sim}{\longrightarrow}\overline{P}_{7}$ between the
corresponding principal bundles $H_{3}\longrightarrow P_{7}\longrightarrow
J_{4}$ and
$\overline{H}_{3}\longrightarrow\overline{P}_{7}\longrightarrow\overline{J}_{4}$
satisfying

$\displaystyle\Phi^{\ast}\overline{\theta}^{\mu}=\theta^{\mu}\qquad\text{and}\qquad\Phi^{\ast}\overline{\Omega}_{i}=\Omega_{i},\qquad\mu=1,2,3,4,\quad
i=1,2,3.$

This theorem enables one to think about the geometry of a PDE
$z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq F_{z_{x}z_{x}}$, considered modulo
point transformations of variables, as the geometry of a certain 3rd order ODE
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$, 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
$z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq F_{z_{x}z_{x}}$.

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 $z_{y}=F(x,y,z,z_{x})$ with $DF\equiv
0\neq F_{z_{x}z_{x}}$ is visibly the same as the EDS for 3rd order ODEs
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$, one can look for
PDEs $z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq F_{z_{x}z_{x}}$, which in
addition satisfy $A_{1}=C_{1}=0$, and build a corresponding Einstein Weyl
geometry, not in terms of $H(x,y,y^{\prime},y^{\prime\prime})$ satisfying
${\text{W}}(H)\equiv{\text{C}}(H)\equiv 0$, but in terms of the function
$F(x,y,z,z_{x})$ satisfying $DF\equiv{\text{M}}(F)\equiv{\text{K}}(F)\equiv
0$. If only ${\text{M}}(F)\equiv 0$, there exists a conformal Lorentzian
metric on the leaf space of the integrable distribution in $P_{7}$ annihilated
by $\big{\\{}\theta^{1},\theta^{3},\theta^{4}\big{\\}}$, and when moreover
${\text{K}}(F)\equiv 0$, all this produces Einstein–Weyl geometries. Actually,
we gain the following

######  Theorem 5.3.

A PDE $z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq F_{z_{x}z_{x}}$ defines a
bilinear form $\widetilde{g}$ of signature $(+,+,-,0$, $0,0,0)$ on the bundle
$P_{7}\ni(x,y,z,p,u_{3},u_{5},u_{8})$:

$\displaystyle\widetilde{g}=\theta^{3}\theta^{3}+\theta^{1}\theta^{4}+\theta^{4}\theta^{1}=\frac{{\sf
u}_{5}^{2}}{9F_{pp}^{2}}\Big{\\{}\big{(}3F_{pp}[{\rm d}x+F_{p}\,{\rm
d}y]-F_{ppp}[{\rm d}z-p\,{\rm d}x-F\,{\rm d}y]\big{)}^{2}$
$\displaystyle\hphantom{\widetilde{g}=}{}+({\rm d}z-p\,{\rm d}x-F\,{\rm
d}y)\big{(}18F_{pp}^{3}{\rm d}y+[4F_{ppp}^{2}-3F_{pp}F_{pppp}][{\rm
d}z-p\,{\rm d}x-F\,{\rm d}y]\big{)}\Big{\\}},$

degenerate along the rank $4$ integrable distribution $\mathcal{D}_{4}$ which
is the annihilator of $\theta^{1}$, $\theta^{3}$, $\theta^{4}$.

The PDE $z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq F_{z_{x}z_{x}}$ also
defines the $1$-form

$\Omega_{3}:=r_{x}\,{\rm d}x+r_{y}\,{\rm d}y+r_{z}\,{\rm d}z+\tfrac{1}{3}{\rm
d}\big{[}\log\big{(}u_{5}^{3}F_{pp}\big{)}\big{]},$

where

$\displaystyle r_{x}=\frac{1}{3F_{pp}^{4}}\big{\\{}\Delta
F_{ppp}F_{pp}^{2}-\Delta
F_{pp}F_{pp}F_{ppp}+3F_{p}F_{pp}^{2}F_{ppz}-F_{pp}^{3}F_{pz}-2F_{p}F_{pp}F_{ppp}F_{pz}$
$\displaystyle\hphantom{r_{x}=}{}-\Delta F_{pppp}F_{pp}^{2}p+3\Delta
F_{ppp}F_{pp}F_{ppp}p-3\Delta F_{pp}F_{ppp}^{2}p+\Delta F_{pp}F_{pp}F_{pppp}p$
$\displaystyle\hphantom{r_{x}=}{}-4F_{p}F_{pp}^{2}F_{pppz}p-2F_{pp}^{3}F_{ppz}p+9F_{p}F_{pp}F_{ppp}F_{ppz}p+F_{pp}^{2}F_{ppp}F_{pz}p$
$\displaystyle\hphantom{r_{x}=}{}-6F_{p}F_{ppp}^{2}F_{pz}p+2F_{p}F_{pp}F_{pppp}F_{pz}p\big{\\}},$
$\displaystyle r_{y}=\frac{1}{3F_{pp}^{4}}\big{\\{}{-}\Delta
F_{pppp}FF_{pp}^{2}+\Delta F_{ppp}F_{p}F_{pp}^{2}-\Delta
F_{pp}F_{pp}^{3}+3\Delta F_{ppp}FF_{pp}F_{ppp}$
$\displaystyle\hphantom{r_{y}=}{}-\Delta F_{pp}F_{p}F_{pp}F_{ppp}-3\Delta
F_{pp}FF_{ppp}^{2}+\Delta F_{pp}FF_{pp}F_{pppp}-4FF_{p}F_{pp}^{2}F_{pppz}$
$\displaystyle\hphantom{r_{y}=}{}+3F_{p}^{2}F_{pp}^{2}F_{ppz}-2FF_{pp}^{3}F_{ppz}+9FF_{p}F_{pp}F_{ppp}F_{ppz}-3F_{p}F_{pp}^{3}F_{pz}$
$\displaystyle\hphantom{r_{y}=}{}-2F_{p}^{2}F_{pp}F_{ppp}F_{pz}+FF_{pp}^{2}F_{ppp}F_{pz}-6FF_{p}F_{ppp}^{2}F_{pz}+2FF_{p}F_{pp}F_{pppp}F_{pz}+3F_{pp}^{4}F_{z}\big{\\}},$
$\displaystyle r_{z}=\frac{1}{3F_{pp}^{4}}\big{\\{}\Delta
F_{pppp}F_{pp}^{2}-3\Delta F_{ppp}F_{pp}F_{ppp}+3\Delta
F_{pp}F_{ppp}^{2}-\Delta F_{pp}F_{pp}F_{pppp}+4F_{p}F_{pp}^{2}F_{pppz}$
$\displaystyle\hphantom{r_{z}=}{}+2F_{pp}^{3}F_{ppz}-9F_{p}F_{pp}F_{ppp}F_{ppz}-F_{pp}^{2}F_{ppp}F_{pz}+6F_{p}F_{ppp}^{2}F_{pz}-2F_{p}F_{pp}F_{pppp}F_{pz}\big{\\}}.$

The degenerate bilinear form $\widetilde{g}$ descends to a Lorentzian
conformal class $[g]$ on the leaf space $M_{3}$ of the distribution
$\mathcal{D}_{4}$, if and only if the Monge invariant ${\text{M}}(F)\equiv 0$
vanishes identically.

When ${\text{M}}(F)\equiv 0$, the local coordinates on $M_{3}$ are $(x,y,z)$
with the projection

$\displaystyle P_{7}$ $\displaystyle\longrightarrow M_{3},$
$\displaystyle\big{(}x,y,z,p,{\sf u}_{3},{\sf u}_{5},{\sf u}_{8}\big{)}$
$\displaystyle\longmapsto(x,y,z),$

and the conformal class $[g]$ has a representative which is explicitly
expressed in terms of ${\rm d}x$, ${\rm d}y$, ${\rm d}z$, with coefficients
depending only on $(x,y,z)$.

Next, $\Omega_{3}$ descends to a $1$-form denoted $A$ given up to the
differential of a function on $M_{3}\ni(x,y,z)$, if and only if
${\text{K}}(F)\equiv 0$.

Moreover, the pair $\big{(}\widetilde{g},\Omega_{3}\big{)}$ descends to a
representative of a Weyl structure $[(g,A)]$ on $M_{3}$, if and only if both
${\text{M}}(F)\equiv 0$ and ${\text{K}}(F)\equiv 0$.

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 $z_{y}=F(x,y,z,z_{x})$ with $DF\equiv 0\neq
F_{z_{x}z_{x}}$ satisfying ${\text{A}}_{1}\equiv 0\equiv{\text{C}}_{1}$ always
define an Einstein–Weyl geometry on the leaf space $M_{3}$ of the integrable
distribution in $P_{7}$ annihilated by
$\big{\\{}\theta^{1},\theta^{3},\theta^{4}\big{\\}}$.

The advantage of looking at a Weyl geometry from the PDE
$z_{y}=F(x,y,z,z_{x})$ point of view rather than from the ODE side
$y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$, is that now the
Wünschmann invariant of the ODE becomes the much simpler and classical Monge
invariant

${\text{A}}_{1}(H)\sim{\text{M}}(F)=9F_{pp}^{2}F_{ppppp}-45F_{pp}F_{pppp}F_{ppp}+40F_{ppp}^{3}.$

Serendipitously, the identical vanishing ${\text{M}}(F)\equiv 0$ is well known
to be equivalent to the condition that the graph of $p\longmapsto F(p)$ is
contained in a conic of the $(p,F)$-plane, with parameters $(x,y,z)$. More
precisely,

$\displaystyle 0\equiv{\text{M}}(F)\quad\Longleftrightarrow\quad\text{\sc
a}F^{2}+2\text{\sc b}Fp+\text{\sc c}p^{2}+2\text{\sc k}F+2\text{\sc
l}p+\text{\sc m}\equiv 0,$ (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 $y^{\prime\prime\prime}=H(x,y,y^{\prime},y^{\prime\prime})$ to
the formulation in terms of a PDE $z_{y}=F(x,y,z,z_{x})$, we are able to find
a rather large class of solutions to the equation

${\text{W}}(H)\equiv 0.$

Indeed, by replacing ${\text{W}}(H)\leadsto{\text{M}}(F)$, 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

$0\equiv DF=\partial_{x}F+p\partial_{z}F.$

The simultaneous conditions $DF\equiv 0\equiv{\text{M}}(F)$ can be achieved
for instance by taking $F$ satisfying

${\text{a}}F^{2}+2{\text{b}}F(z-px)+{\text{c}}(z-px)^{2}+2{\text{k}}F+2{\text{l}}(z-px)+{\text{m}}\equiv
0,$

with a, b, c, k, l, m being now functions of $y$ only!

From now on, we will analyze this special solution for ${\text{M}}(F)\equiv
0\equiv DF$. The simplest case occurs when avoiding square root by choosing

${\text{a}}:=0,$

so that

$\displaystyle
F:=\frac{-{\text{c}}(z-xp)^{2}-2{\text{l}}(z-xp)-{\text{m}}}{2{\text{b}}(z-xp)+2{\text{k}}}.$
(7.1)

Here

${\text{b}}={\text{b}}(y),\qquad{\text{c}}={\text{c}}(y),\qquad{\text{k}}={\text{k}}(y),\qquad{\text{l}}={\text{l}}(y),\qquad{\text{m}}={\text{m}}(y)$

are free arbitrary differentiable functions of one variable $y$.

A direct check shows that remarkably this solution (7.1) also satisfies
${\text{K}}(F)\equiv 0$!

######  Proposition 7.1.

All such

$F:=\frac{-{\text{c}}(z-xp)^{2}-2{\text{l}}(z-xp)-{\text{m}}}{2{\text{b}}(z-xp)+2{\text{k}}}$

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
$\big{\\{}\theta^{1},\theta^{2},\theta^{3},\theta^{4},\Omega_{1},\Omega_{2},\Omega_{3}\big{\\}}$,
projecting both
$\theta^{3}\theta^{3}+\theta^{1}\theta^{4}+\theta^{4}\theta^{1}$ and
$\Omega_{3}$ to the leaf space of the annihilator $M^{3}$ of
$\big{\\{}\theta^{1},\theta^{3},\theta^{4}\big{\\}}$, equipping
$M_{3}\equiv\mathbb{R}^{3}$ with coordinates $(x,y,z)$, we therefore obtain
functionally parameterized Einstein–Weyl structures $\big{(}g,A\big{)}$ on
$\mathbb{R}^{3}\ni(x,y,z)$ represented by the signature $(2,1)$ Lorentzian
metric

$\displaystyle g:=({\text{k}}+{\text{b}}z)^{2}{\rm
d}x^{2}+x^{2}\big{(}{\text{l}}^{2}-{\text{c}}{\text{m}}\big{)}{\rm
d}y^{2}+x^{2}{\text{b}}^{2}{\rm d}z^{2}$
$\displaystyle\hphantom{g:=}{}+2x({\text{c}}{\text{k}}z-{\text{b}}{\text{l}}z+{\text{k}}{\text{l}}-{\text{b}}{\text{m}}){\rm
d}x\,{\rm d}y-2x{\text{b}}({\text{k}}+{\text{b}}z){\rm d}x\,{\rm
d}z-2x^{2}({\text{c}}{\text{k}}-{\text{b}}{\text{l}}){\rm d}y\,{\rm d}z,$

together with the differential $1$-form

$\displaystyle
A:=\frac{-{\text{c}}{\text{k}}+{\text{b}}{\text{l}}+{\text{b}}^{\prime}{\text{k}}-{\text{b}}{\text{k}}^{\prime}}{x\big{(}{\text{c}}{\text{k}}^{2}-2{\text{b}}{\text{k}}{\text{l}}+{\text{b}}^{2}{\text{m}}\big{)}}\big{(}x{\text{b}}\,{\rm
d}z-({\text{k}}+{\text{b}}z)\,{\rm d}x\big{)}$
$\displaystyle\hphantom{A:=}{}+\frac{{\text{b}}{\text{l}}^{2}-{\text{c}}{\text{b}}{\text{m}}-{\text{b}}^{\prime}{\text{k}}{\text{l}}+{\text{b}}{\text{b}}^{\prime}{\text{m}}+{\text{c}}{\text{k}}{\text{k}}^{\prime}-{\text{b}}{\text{k}}^{\prime}{\text{l}}}{{\text{c}}{\text{k}}^{2}-2{\text{b}}{\text{k}}{\text{l}}+{\text{b}}^{2}{\text{m}}}\,{\rm
d}y.$

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 $\text{\sc a}\equiv 0$ in (6.1), let us now make the
ansatz that

$\displaystyle{\text{a}}F^{2}+2{\text{b}}F(z-xp)+{\text{c}}(z-xp)^{2}+2{\text{k}}F+2{\text{l}}(z-xp)+{\text{m}}\equiv
0,$

for some arbitrary functions a, b, c, k, l, m of $y$. The (two) solutions $F$
automatically satisfy $DF\equiv 0\equiv{\text{M}}(F)$.

Since the solutions to Monge’s equation are conics in the $(p,F)$-plane, we
can rewrite in a hyperbolic setting

$\big{(}{\text{a}}F+{\text{b}}(z-xp)+{\text{c}}\big{)}^{2}-\big{(}{\text{k}}F+{\text{l}}(z-xp)+{\text{m}}\big{)}^{2}\equiv
1,$

with changed functions a, b, c, k, l, m of $y$. To avoid transcendental
functions in computations, we parametrize $\cosh t=\frac{1+q^{2}}{2q}$ and
$\sinh t=\frac{1-q^{2}}{2q}$, and then, solving for $F$ and for $z-xp$, we may
start from

$\displaystyle
F={\text{a}}(y)\frac{1+q^{2}}{2q}+{\text{b}}(y)\frac{1-q^{2}}{2q}+{\text{c}}(y),$
$\displaystyle
z-xp={\text{k}}(y)\frac{1+q^{2}}{2q}+{\text{l}}(y)\frac{1-q^{2}}{2q}+{\text{m}}(y),$

again with (changed) free functions a, b, c, k, l, m of $y$. Taking

$\omega_{0}^{1}:={\rm d}(z-xp)+x\,{\rm d}p-F\,{\rm
d}y,\qquad\omega_{0}^{2}:={\rm d}x,\qquad\omega_{0}^{3}:={\rm
d}y,\qquad\omega_{0}^{4}:={\rm d}p,$

and performing para-CR Cartan reduction to an $\\{e\\}$-structure/connection,
we obtain

######  Proposition 7.2.

The second invariant condition ${\text{K}}(F)\equiv 0$ holds precisely in the
following two cases:

  1. $(1)$

${\text{k}}={\text{l}}$;

  2. $(2)$

${\text{c}}={\text{m}}^{\prime}$ and
${\text{a}}=\frac{{\text{b}}{\text{l}}+{\text{k}}{\text{k}}^{\prime}-{\text{l}}{\text{l}}^{\prime}}{{\text{k}}}$.

In case (1), we obtain Einstein–Weyl structures for all free functions a, b,
c, l, m of $y$ given by

$\displaystyle g:=2\tau^{1}\tau^{2}+\big{(}\tau^{3}\big{)}^{2},\qquad
A:=-\frac{2({\text{a}}+{\text{b}})}{x({\text{a}}-{\text{b}}){\text{l}}}\tau^{2}-\frac{{\text{c}}-{\text{m}}^{\prime}}{x({\text{a}}-{\text{b}})}\tau^{3},$

where

$\displaystyle\tau^{1}:=x({\text{a}}+{\text{b}})\,{\rm d}y-2{\text{l}}\,{\rm
d}x,\qquad\tau^{2}:=-\tfrac{1}{2}x({\text{a}}-{\text{b}})\,{\rm
d}y,\qquad\tau^{3}:=x{\text{c}}\,{\rm d}y-x\,{\rm d}z+(z-{\text{m}})\,{\rm
d}x.$

We verify that these Einstein–Weyl structures have nontrivial $F={\rm
d}A\not\equiv 0$ and nontrivial $\text{\sf Cotton}([g])\not\equiv 0$.

In case (2), we obtain Einstein–Weyl structures given by

$g:=2\tau^{1}\tau^{2}+\big{(}\tau^{3}\big{)}^{2},\qquad A:={\rm
d}\big{[}\log\big{(}x^{2}{\text{e}}\big{)}\big{]},$

where

$\displaystyle\tau^{1}:=({\text{k}}+{\text{l}}){\text{k}}\,{\rm
d}x+x({\text{b}}{\text{k}}-{\text{b}}{\text{l}}+{\text{k}}{\text{k}}^{\prime}-{\text{l}}{\text{l}}^{\prime})\,{\rm
d}y,$
$\displaystyle\tau^{2}:=\tfrac{1}{2}({\text{k}}-{\text{l}}){\text{k}}\,{\rm
d}x+\tfrac{1}{2}x({\text{b}}{\text{k}}-{\text{b}}{\text{l}}+{\text{k}}{\text{k}}^{\prime}-{\text{l}}{\text{l}}^{\prime})\,{\rm
d}y,$ $\displaystyle\tau^{3}:=-(z-{\text{m}}){\text{k}}\,{\rm
d}x-x{\text{k}}{\text{m}}\,{\rm d}y+x{\text{k}}\,{\rm d}z.$

But this structure, which depends on $3$ functions b, k, l of $y$, is flat

${\rm d}A\equiv 0\equiv\text{\sf Cotton}([g]).$

Finally, without replacing $p$ by $z-xp$, let us make the ansatz that

$\displaystyle{\text{a}}F^{2}+2{\text{b}}Fp+{\text{c}}p^{2}+2{\text{k}}F+2{\text{l}}p+{\text{m}}\equiv
0.$

Dealing similarly with the hyperbolic case,

$\displaystyle
F={\text{a}}(y)\frac{1+q^{2}}{2q}+{\text{b}}(y)\frac{1-q^{2}}{2q}+{\text{c}}(y),$
$\displaystyle
p={\text{k}}(y)\frac{1+q^{2}}{2q}+{\text{l}}(y)\frac{1-q^{2}}{2q}+{\text{m}}(y),$

we obtain nontrivial Einstein–Weyl structures. For instance, when
${\text{k}}={\text{l}}$ as in (1) above

$g:=2\tau^{1}\tau^{2}+\big{(}\tau^{3}\big{)}^{2},\qquad
A:=-\frac{{\text{m}}^{\prime}}{({\text{a}}-{\text{b}}){\text{l}}}\tau^{3},$

where

$\displaystyle\tau^{1}:=2{\text{l}}\,{\rm d}x+({\text{a}}+{\text{b}})\,{\rm
d}y,\qquad\tau^{2}:=-\tfrac{1}{2}({\text{a}}-{\text{b}})\,{\rm
d}y,\qquad\tau^{3}:={\rm d}z-{\text{m}}\,{\rm
d}x+({\text{a}}+{\text{b}})\,{\rm d}y.$

Note that this is again nontrivial

${\rm d}A\not\equiv 0\not\equiv\text{\sf Cotton}([g]).$

and note that we do not have $x$, $z$ dependence here.

##  8 Transforming $\boldsymbol{z_{y}=F(z_{x})}$ into
$\boldsymbol{w^{\prime\prime\prime}=w^{\prime\prime}H(t)}$

We end up by exploring a link between our PDE systems and 3rd order ODEs. For
simplicity, we will assume that $F=F(z_{x})$ depends only on $p=z_{x}$.

To avoid notational confusion, 3rd-order ODEs will now be denoted as
$w^{\prime\prime\prime}=H(t,w,w^{\prime},w^{\prime\prime})$, and the
fundamental $1$-forms as

$\displaystyle\omega^{1}:={\rm d}z-p\,{\rm d}x-F(p)\,{\rm d}y,\qquad$
$\displaystyle\theta^{1}:={\rm d}w-w_{1}\,{\rm d}t,$
$\displaystyle\omega^{2}:={\rm d}p,\qquad$ $\displaystyle\theta^{2}:={\rm
d}t,$ $\displaystyle\omega^{3}:={\rm d}x,\qquad$
$\displaystyle\theta^{3}:={\rm d}w_{1}-w_{2}\,{\rm d}t,$
$\displaystyle\omega^{4}:={\rm d}y,\qquad$ $\displaystyle\theta^{4}:={\rm
d}w_{2}-H(t,w,w_{1},w_{2})\,{\rm d}t.$

We ask what equivalence class of 3rd-order ODE’s corresponds to the
equivalence class of PDEs $z_{y}=F(z_{x})$, still with $F_{pp}\neq 0$, and
under the $G$-structures of Sections 4 and 5.

For this, since $\omega^{1}$ and $\theta^{1}$ are both defined up to plain
dilations $\omega^{1}\sim{\sf u}\omega^{1}$ and $\theta^{1}\sim{\sf
u}\theta^{1}$, we transform $\omega^{1}$ in order to make the shape of
$\theta^{1}$ appear, using that $F$ depends only on $p$

$\omega^{1}={\rm d}(z-xp-yF(p))-(-x-yF_{p}(p))\,{\rm d}p=:{\rm d}w-w_{1}\,{\rm
d}t$

with $t:=p$, $w:=z-xp-yF(p)$, $w_{1}:=-x-yF_{p}(p)$. With this,
$\omega^{2}={\rm d}p={\rm d}t=\theta^{2}$. Next, using
$\omega^{3}\sim-\omega^{3}-\underline{{\sf u}\,{\rm d}y}$, it comes

$\displaystyle\omega^{3}={\rm d}x=-\big{[}{\rm
d}(-x-yF_{p}(p))+yF_{pp}(p)\,{\rm d}p+F_{p}(p)\,{\rm d}y\big{]}$
$\displaystyle\hphantom{\omega^{3}={\rm d}x}{}\sim\big{[}{\rm
d}w_{1}+yF_{pp}(p)\,{\rm d}p+F_{p}(p)\,{\rm
d}y\big{]}-\underline{F_{p}(p)\,{\rm d}y}={\rm d}w_{1}-(-yF_{pp}(p))\,{\rm
d}p,$

whence $w_{2}:=-yF_{pp}(p)$.

A last computation using $\omega^{4}\sim{\sf u}\omega^{4}$

$\displaystyle\omega^{4}={\rm d}y=-\frac{1}{F_{pp}(p)}\big{[}{\rm
d}(-yF_{pp}(p))+yF_{ppp}(p)\,{\rm d}p\big{]}\sim{\rm
d}w_{2}-\big{(}{-}yF_{ppp}(p)\,{\rm d}p\big{)},$

shows that the right-hand side function $H=H(t,w_{2})$ of the associated ODE
$w^{\prime\prime\prime}=H$ is independent of $w,w_{1}$ as it must be

$H:=-yF_{ppp}(p)=w_{2}\frac{F_{ttt}(t)}{F_{tt}(t)}.$

Hence

$w^{\prime\prime\prime}=w^{\prime\prime}\frac{F_{ttt}(t)}{F_{tt}(t)}$

is the 3rd order ODE associated to the para-CR structure given by
$z_{y}=F(z_{x})$. Observe that $z_{y}=\frac{1}{2}z_{x}^{2}$ becomes
$w^{\prime\prime\prime}=0$, leading to the flat Einstein–Weyl structure.

######  Assertion 8.1.

The Wünschmann invariant for ODEs
$w^{\prime\prime\prime}=w^{\prime\prime}\frac{F_{ttt}(t)}{F_{tt}(t)}$, where
$F(t)$ with $F_{tt}\neq 0$ is an arbitrary function of one variable,
corresponds to the Monge invariant of the PDE $z_{y}=F(z_{x})$:

$\displaystyle\text{\sf
W\"{u}nschmann}\left(w_{2}\frac{F_{ttt}(t)}{F_{tt}(t)}\right)=\frac{9F_{tt}(t)F_{ttttt}(t)-45F_{tt}F_{ttt}F_{tttt}+40F_{ttt}^{3}}{F_{tt}^{3}}=\frac{\text{\sf
Monge}(F)}{F_{tt}^{3}}.$

###### Proof.

Among the $25$ terms of Wünschmann’s invariant shown in the Introduction, only
$7$ remain thanks to $0\equiv H_{w}\equiv H_{w_{1}}$:

$\displaystyle\text{\sf
W\"{u}nschmann}\big{(}H(t,w_{2})\big{)}=-9HH_{w_{2}}H_{w_{2}w_{2}}-9H_{t}H_{w_{2}w_{2}}+18HH_{tw_{2}w_{2}}-18H_{w_{2}}H_{tw_{2}}$
$\displaystyle\hphantom{\text{\sf
W\"{u}nschmann}\big{(}H(t,w_{2})\big{)}=}{}+9H_{ttw_{2}}+4H_{w_{2}}^{3}+9H^{2}H_{w_{2}w_{2}w_{2}},$

and a direct substitution of $H:=w_{2}\frac{F_{ttt}(t)}{F_{tt}(t)}$ leads to
the result. ∎

### 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.

## References

  * [1] Calderbank D.M.J., Pedersen H., Einstein–Weyl geometry, in Surveys in Differential Geometry: Essays on Einstein Manifolds, Surv. Differ. Geom., Vol. 6, Int. Press, Boston, MA, 1999, 387–423.
  * [2] Cartan E., La geometria de las ecuaciones diferenciales de tercer orden, Rev. Mat. Hispano-Amer. 4 (1941), 1–31.
  * [3] Cartan E., Sur une classe d’espaces de Weyl, Ann. Sci. École Norm. Sup. 60 (1943), 1–16.
  * [4] Dunajski M., Mason L.J., Tod P., Einstein–Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001), 63–93, arXiv:math.DG/0004031.
  * [5] Eastwood M.G., Tod K.P., Local constraints on Einstein–Weyl geometries: the 3-dimensional case, Ann. Global Anal. Geom. 18 (2000), 1–27.
  * [6] Frittelli S., Kozameh C., Newman E.T., Differential geometry from differential equations, Comm. Math. Phys. 223 (2001), 383–408, arXiv:gr-qc/0012058.
  * [7] Godlinski M., Geometry of third-order ordinary differential equations and its applications in general relativity, arXiv:0810.2234.
  * [8] Godlinski M., Nurowski P., Geometry of third order ODEs, arXiv:0902.4129.
  * [9] Hill C.D., Nurowski P., Differential equations and para-CR structures, Boll. Unione Mat. Ital. 3 (2010), 25–91, arXiv:0909.2458.
  * [10] Hitchin N.J., Complex manifolds and Einstein’s equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Math., Vol. 970, Springer, Berlin – New York, 1982, 73–99.
  * [11] Jones P.E., Tod K.P., Minitwistor spaces and Einstein–Weyl spaces, Classical Quantum Gravity 2 (1985), 565–577.
  * [12] LeBrun C., Mason L.J., The Einstein–Weyl equations, scattering maps, and holomorphic disks, Math. Res. Lett. 16 (2009), 291–301, arXiv:0806.3761.
  * [13] Merker J., Lie symmetries and CR geometry, J. Math. Sci. 154 (2008), 817–922, arXiv:math.CV/0703130.
  * [14] Merker J., Nurowski P., On degenerate para-CR structures: Cartan reduction and homogeneous models, arXiv:2003.08166.
  * [15] Merker J., Pocchiola S., Explicit absolute parallelism for 2-nondegenerate real hypersurfaces $M^{5}\subset{\mathbb{C}}^{3}$ of constant Levi rank 1, J. Geom. Anal. 30 (2020), 2689–2730.
  * [16] Monge G., Sur les équations différentielles des courbes du second degré, Corr. l’École Impériale Polytech. (1810), no. 2, 51–54; Bull. Soc. Philomat. Paris (1810), 87–88.
  * [17] Nurowski P., Differential equations and conformal structures, J. Geom. Phys. 55 (2005), 19–49, arXiv:math.DG/0406400.
  * [18] Pedersen H., Tod K.P., Three-dimensional Einstein–Weyl geometry, Adv. Math. 97 (1993), 74–109.
  * [19] Tod K.P., Compact $3$-dimensional Einstein–Weyl structures, J. London Math. Soc. 45 (1992), 341–351.
  * [20] Tod K.P., Einstein–Weyl spaces and third-order differential equations, J. Math. Phys. 41 (2000), 5572–5581.
  * [21] Weyl H., Raum, Zeit, Materie, Springer-Verlag, Berlin, 1919.
  * [22] Wünschmann K., Uber Berührungsbedingungen bei Integralkurven von Differentialgleichungen, Inaug. Dissert., Leipzig, Teubner, 1905.


The University of Minnesota,
Minneapolis, Minnesota, USA<EMAIL_ADDRESS>
# A Sub-Picosecond Digitally-Controlled Phase Delay

D. Dehmeshki  E. Frahm  R. Rusack  R. Saradhy  Y. TousiCorresponding author

###### Abstract

The use of precision timing measurements will be a major tool at the HL-LHC,
where it will be used to suppress pile-up and to search for long-lived
particles. To control a reference clock with sub-picosecond accuracy, we have
fabricated in the TSMC 65 nm process a digitally controlled phase shifter. It
is composed of a chain of 66 cells, each with a digitally controlled planar
wave guide with either a short or long delay. With this a reference clock’s
phase can be controlled to a precision of 200 fs with dynamic range of 12 ps.

Digital electronic circuits, Analogue electronic circuits, Instrumentation and
methods for time-of-flight (TOF) spectroscopy, Timing Detectors TWEPP 2021
20${}^{\text{th}}$ \- 24${}^{\text{th}}$ September 2021
Virtual

##  1 Introduction

Precision measurement and control of signal delay is instrumental in both
wireless and wireline communication systems. The ability to adjust the delay
of a reference clock with sub-picosecond accuracy is necessary to meet the
challenge in particle physics of measuring signal arrival times with a sub-
picosecond precision. To distribute a precisely synchronized reference clock
from a single source to different detector elements, which can be separated by
distances of up to 20m, requires that subtle variations, such as temperature
variations in optical fibers, can be detected and corrected in real time. A
core part of a calibration scheme is a high precision time delay that can
track and adjust for such variations. True time delays have been in use since
the early days of microwave systems for tuning a variety of components and
signals. The traditional method for delay control is based on physical
adjustment of a transmission line using a trombone-like mechanical knob [1].
Such a method can provide a reliable and accurate control over the delay of
phase across a broad frequency. While trombone type delay lines are widely
used in smaller and discrete component systems, they have limited use in more
complex integrated systems. This is mainly due to the relatively large size of
the structure limiting the number of tunable delays in the system, as well as
its slow and manual means of delay adjustment.

A large body of literature is dedicated to improving on these limitations by
replacing mechanical tuning with electrical switches or analog control. A
common method to replace mechanical adjustment of the length is to do so by
switching the signal path between two predesignated paths, each with its own
unit length [2, 3]. These approaches rely on artificial construction of a
true-time delay that is subsequently adjusted by switching in and out
different sections. A cascaded series of such switched delays with either
binary or uniform delay steps can ideally provide accurate tuning of the
effective length similar to the trombone structure. However, due to the need
for multiple switches in series to the signal path such systems require low
loss electrical switches to avoid significant signal degradation. Diode-based
switches have demonstrated the feasibly of such methods in off-chip
implementations. However, due to the higher loss in standard integrated
technologies, such as CMOS, such signal switching is not effective for on chip
implementation.

The objective of this work is to develop a true time delay line structure that
is fully integrated, while maintaining the benefit of traditional tunable
delays. The remainder of this paper will discuss the proposed structure,
followed by simulation results and prototype implementation and measurement
results.

##  2 Tunable transmission line

The relation between the input and output of a TEM mode transmission line
closely represents an ideal delay line. This is because it represents wave
propagation in a passive broadband structure that provides a dispersion-free
and broadband group delay. Although physical adjustment of the length has
proven to a be powerful delay tuning technique, an on-chip implementation of a
transmission line imposes a fixed overall length. Furthermore, multi-path
switched-based length adjustment has limited utility, due to their limited
accuracy and insertion loss. However, due to its linear dispersion relation,
the concept of using a transmission line is superior to multi-section phase
adjustment, and it is highly desirable to implement a tunable delay structure
than maintains the core benefit of a dispersion free transmission line.

The output of a loss-less and dispersion-free transmission line, shown in Fig.
1, can be related to its input by the following equation:

$v_{o}\left(t\right)=v_{i}\left(t-t_{d}\right),$ (1)

where, $t_{d}=L$/$V$ is the effective delay of this transmission line, $L$ is
the length of the line and $V$ is propagation velocity of the wave in the
line. This suggests that in addition to adjusting the length, the delay can
also be controlled by adjusting the propagation velocity, which depends on the
speed of light in the material, given by $V=c/\sqrt{\epsilon\mu}$. As it is
very difficult to adjust these material properties in an normal transmission
line, an artificial transmission line can be constructed as shown in Fig. 1
using lumped components, which can mimic the behavior of a transmission line
with a highly granular structure.

### Artificial transmission line

Given the linear nature of this delay line, one can divide it into small unit
delays represented by lumped circuit elements, such as inductors and
capacitors. This is theoretically possible for any linear electromagnetic
system as long as individual unit cells have a small phase delay compared to
the wavelength [4]. In such a representation, we can divide the transmission
line into $N$ unit delay sections, each consisting of an inductance $L$ and
capacitance $C$. Solving for the relation between the input and output voltage
in this equivalent circuit, the relationship between the continuous wave
parameters and the circuit values in this artificial construct can be found.
In each section the delay line the characteristic impedance is
$Z_{0}=\sqrt{L/C}$ and the delay is $\sqrt{LC}$. By combining $N$ cells a
transmission line with a total delay of $t_{d}=N\sqrt{LC}$ can be constructed.

Figure 1: Left: An tunable transmission line and its equivalent distributed
circuit with tunable values. Right: physical layout of the delay unit cell and
the two delay settings.

This representation is only valid when delay per section is small compared to
the signal period, that is:

$f\ll\frac{1}{\sqrt{LC}}.$ (2)

This criteria needs to be taken into account in the design of the delay and
the choice of the physical size of the unit cells, which will depend on the
maximum frequency of operation required for the delay line.

The overall delay is adjusted by controlling the the unit delay of each cell.
To do this we introduce switched inductors and capacitors to the delay cell.
As shown in Fig. 1, each cell consists of two switches. The L-switch adjusts
the ground return path of the transmission line: when the switch is on the
return path is close to the signal line resulting in a smaller magnetic flux
and inductance compared to the off mode. Similarly, the C-switch adds a shunt
capacitor to the unit structure. The combination of these two switches allows
control of both the delay, $\sqrt{LC}$, and $Z_{o}$. This allows the delay to
be adjusted without modifying the impedance of the line and affecting wave
propagation. This effectively results in a dispersion free transmission line
with a tunable propagation velocity. Furthermore, due to the granular nature
of this control, this method can provide a highly accurate control of the
delay by simply breaking down the transmission line into small unit cells.
Simulation results of the group delay for different settings is shown in in
Fig. 2.

Figure 2: Simulation results of the proposed delay line: left: S-parameters,
middle: pulse waveforms, right: pulsed input delay.

The dimensions of the parallel lines and the value of the characteristic
impedance are chosen based on electromagnetic extraction of the distributed
structure, followed by a Spice based simulation of the extracted structure,
along with the lumped capacitances and switches. In addition to minimal
dispersion, a delay step of significantly less than 1 ps can be achieved. Due
to the inherently discrete tuning method in this digitally-controlled delay
line, the size of the unit delay cell plays a critical role in the overall
resolution of the structure. As a result, we designed the unit cells with the
smallest possible dimensions, while allowing sufficient margin within the
fabrication design rules. Fig. 1 shows the dimensions of one of the 66 delay
cells used in this design.

In order to adjust the delay line, a serial interface provides access to the
L-bit and C-bit in each delay cell. To adjust the delay, the design employs a
thermometer coding scheme where the delay cells are switched in a sequential
manner, starting from all the cells being in the low delay mode and
subsequently switching them one-by-one to the high-delay mode. This results in
a gradual increase in the total delay, while minimizing abrupt transitions in
the characteristic impedance of the line. As a result, the delay steps are
monotonic and uniform.

### Chip implementation

The chip prototype is fabricated using the TSMC 65nm CMOS process. The chip
consists of two parallel delay lines for use in a differential signal
environment. As shown in the die micrograph, the 66 cells are placed in three
rows of 22 cells for an area efficient design. The 180-degree bend in the
structure required between adjacent rows is optimized to create equal delay
with minimal effect on the line dispersion. We use the top metal for both
ground and signal routing, while the switches are placed as close as
physically possible to the return path to ensure efficiency. We measured the
chip using the Keysight N5242A PNA-X Network Analyzer with the two sides of
the delay line connecting to the PNA through differential RF probes.

Figure 3: Chip level measurements. Left: chip microphoto, middle: phase vs.
frequency for minimum and maximum settings, group delay vs. digital setting.

Fig. 3 illustrates the measured phase S-parameters of the line from DC to
15GHz, showing the delay vs. frequency characteristic for the highest and
lowest delay settings of the circuit. The measured average group delay and
loss for all 66 digital codes demonstrates the linear delay vs. digital
relationship. The tunable delay line provides a total delay range of 9.4psec
with an average delay step of 140fs. The maximum phase error is less than 0.15
degrees and there is a close alignment between simulation and measured
S-parameter data.

##  3 Board-level testing

As a further test of the structure and as a demonstration of its use with
digital signals, we have tested the delay line with a digital clock at
different clock frequencies. For this we have used a circuit similar to the
one described in our previous presentation at TWEPP 2019 [5]. Our test setup
is sketched in Fig. 4, in it the output of an oven-controlled crystal
oscillator with a frequency of 40 MHz is input to Silicon Labs PLL (Si5344) to
generate a highly stable clock at selected frequencies. This clock is then
injected into the circuit and the phase is controlled using the serial
interface. The output of the ASIC is coupled to a Samtec Firefly 12TX
transmitter, which is coupled through a multimode fiber optic cable to a
Samtec Firefly 12RX receiver on the same PCB. The received signal is compared
with the original clock signal using a digital dual mean time difference
circuit.

The results obtained at different clock frequencies for different settings of
the phase delay are shown in Fig. 4, and the step size is measured to be
$\approx 200$fs.

Figure 4: Left: The test setup used to measure the changes of phase in the
time domain. Right: The phase changes measured with the DDMTD for digital
clock frequencies between 60 and 160 MHZ.

##  4 Summary

The proposed transmission-line based solution enables substantially better
delay resolution across a wide bandwidth, while ensuring a monotonic response.
Furthermore, due to the passive nature of the line, the proposed delay line is
more energy efficient compared to active delay tuning methods. As a result,
this chip demonstrates a robust tunable true-time delay with sub-picosecond
accuracy for precision timing applications.

We are grateful the US DOE Office of High Energy Physics for their support
under the award DE-SC0020185.

## References

  * [1] K. Jackson, S. Newton, B. Moslehi, M. Tur, C. Cutler, J. Goodman et al., _Optical fiber delay-line signal processing_ , _IEEE Transactions on Microwave Theory and Techniques_ 33 (1985) 193.
  * [2] F. Hu and K. Mouthaan, _A 1–20 ghz 400 ps true-time delay with small delay error in 0.13 µm cmos for broadband phased array antennas_ , in _2015 IEEE MTT-S International Microwave Symposium_ , pp. 1–3, 2015, DOI.
  * [3] M.H. Ghazizadeh and A. Medi, _Novel trombone topology for wideband true-time-delay implementation_ , _IEEE Transactions on Microwave Theory and Techniques_ 68 (2020) 1542.
  * [4] D.M. Pozar, _Microwave engineering_ , John wiley & sons (2011).
  * [5] R. Rusack, J. Brashear, E. Frahm, S. Jain and R. Saradhy, _A Precision Pure Clock Distribution System_ , _PoS_ TWEPP2019 (2020) 056.