On Degenerate Para-CR Structures: Cartan Reduction and Homogeneous Models

Merker, Joël; Nurowski, Paweł

doi:10.1007/s00031-022-09746-4

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2023

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Cited by 3 publications

References 19 publications

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Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Merker,

Nurowski

2023

J Geom Anal

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Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Merker,

Nurowski

2023

J Geom Anal

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New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

Merker¹,

Nurowski²

2020

SIGMA

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On a 3D manifold, a Weyl geometry consists of pairs (g, A) = (metric, 1-form) modulo gauge g = e 2ϕ g, A = A + dϕ. In 1943, Cartan showed that every solution to the Einstein-Weyl equations R (µν) − 1 3 Rg µν = 0 comes from an appropriate 3D leaf space quotient of a 7D connection bundle associated with a 3 rd order ODE y = H(x, y, y , y ) modulo point transformations, provided 2 among 3 primary point invariants vanishWe 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 ≡ 0 leads to a completely similar 7D Cartan bundle and connection. Then magically, the (complicated) equation Wünschmann(H) ≡ 0 becomeswhose solutions are just conics in the {p, F }-plane. As an ansatz, we takewith 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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On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds

Foo¹,

Merker²,

Ta³

2020

Preprint

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Firstly, applying Lie's elementary theory for appropriate prolongations to jet spaces of orders 1 and 2, we show that any C ω hypersurface M 5 ⊂ C 3 in the class C 2,1 carries two sorts of Cartan-Moser chains, that are of orders 1 and 2.Secondly, integrating and straightening any given order 2 chain passing through any point p ∈ M to be the v-axis in coordinates (z, ζ, w = u + i v) centered at p, without setting up the formal theory in advance, we show that there exists a convergent change of complex coordinates (z, ζ, w) −→ (z , ζ , w ) fixing the origin in which γ is the v-axis and in which M has Poincaré-Moser reduced equation (suppressing primes):

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On Degenerate Para-CR Structures: Cartan Reduction and Homogeneous Models

Cited by 3 publications

References 19 publications

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

Homogeneous CR and Para-CR Structures in Dimensions 5 and 3

New Explicit Lorentzian Einstein-Weyl Structures in 3-Dimensions

On Convergent Poincaré-Moser Reduction for Levi Degenerate Embedded $5$-Dimensional CR Manifolds

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