Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 056, 16 pages      arXiv:1906.10880

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

Joël Merker a and Paweł Nurowski b
a) Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France
b) Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Al. Lotników 32/46, 02-668 Warszawa, Poland

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 $R_{(\mu\nu)} - \frac{1}{3} R g_{\mu\nu} = 0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3$^{\rm rd}$ order ODE $y''' = H(x,y,y',y'')$ modulo point transformations, provided $2$ among $3$ primary point invariants vanish \begin{gather*} \text{Wünschmann}(H) \equiv 0\equiv \text{Cartan}(H). \end{gather*}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 $\text{Wünschmann}(H) \equiv 0$ becomes \begin{gather*}0\equiv\text{Monge}(F):=9F_{pp}^2F_{ppppp}-45F_{pp}F_{ppp}F_{pppp}+40F_{ppp}^3,\qquad p:=z_x, \end{gather*} whose solutions are just conics in the $\{p, F\}$-plane. As an ansatz, we take \begin{gather*}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)}, \end{gather*} with $9$ arbitrary functions $\alpha, \dots, \nu$ of $y$. This $F$ satisfies $DF \equiv 0 \equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H) \equiv 0 $ passes to a certain $\boldsymbol{K}(F) \equiv 0$ 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 $\text{Cotton}([g]) \not \equiv 0$.

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.

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