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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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https://hal-universite-paris-saclay.archives-ouvertes.fr/hal-03286283
Contributor : Joël Merker <>
Submitted on : Wednesday, July 14, 2021 - 9:56:17 AM
Last modification on : Sunday, July 18, 2021 - 3:29:06 AM

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Joël Merker, Paweł Nurowski. New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions. Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2020, 16, pp.056. ⟨10.3842/sigma.2020.056⟩. ⟨hal-03286283⟩

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