New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions
Abstract
On a 3D manifold, a Weyl geometry consists of pairs (g,A)= (metric, 1-form) modulo gauge gˆ=e2φg, Aˆ=A+dφ. In 1943, Cartan showed that every solution to the Einstein-Weyl equations R(μν)−13Rgμν=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):=9F2ppFppppp−45FppFpppFpppp+40F3ppp, 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≢0 and Cotton([g])≢0.
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