New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

Abstract

On a $3$D manifold, a Weyl geometry consists of pairs $(g,A)=$ (metric, $1$-form) modulo gauge $\hat{g}={\mathrm{e}}^{2\phi }g$, $\hat{A}=A+\mathrm{d}\phi$. In 1943, Cartan showed that every solution to the Einstein-Weyl equations $R_{(\mu \nu )}-\frac{1}{3}R{g}_{\mu \nu }=0$ comes from an appropriate $3$D leaf space quotient of a $7$D connection bundle associated with a 3${}^{\mathrm{rd}}$ 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 $$\text{Wu¨nschmann}(H)\equiv 0\equiv \text{Cartan}(H).$$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 $\text{Wu¨nschmann}(H)\equiv 0$ becomes $$0\equiv \text{Monge}(F):=9F_{pp}^2{F}_{ppppp}-45F_{pp}{F}_{ppp}{F}_{pppp}+40F_{ppp}^3,p:=z_x,$$ whose solutions are just conics in the $\{p,F\}$-plane. As an ansatz, we take $$F(x,y,z,p):=\frac{\alpha (y)(z-xp{)}^2+\beta (y)(z-xp)p+\gamma (y)(z-xp)+\delta (y)p^2+\epsilon (y)p+\zeta (y)}{\lambda (y)(z-xp)+\mu (y)p+\nu (y)},$$ with $9$ arbitrary functions $\alpha ,\dots ,\nu$ of $y$. This $F$ satisfies $DF\equiv 0\equiv \text{Monge}(F)$, and we show that the condition $\text{Cartan}(H)\equiv 0$ passes to a certain $\mathbf{K}(F)\equiv 0$ which holds for any choice of $\alpha (y),\dots ,\nu (y)$. Descending to the leaf space quotient, we gain $\infty$-dimensional functionally parametrized and explicit families of Einstein-Weyl structures $[(g,A)]$ in $3$D. These structures are nontrivial in the sense that $\mathrm{d}A\not\equiv 0$ and $\text{Cotton}([g])\not\equiv 0$.