On the Einstein-Weyl and conformal self-duality equations

Abstract

The equations governing anti-self-dual and Einstein-Weyl conformal geometries can be regarded as "master dispersionless systems" in four and three dimensions, respectively. Their integrability by twistor methods has been established by Penrose and Hitchin. In this note, we present, in specially adapted coordinate systems, explicit forms of the corresponding equations and their Lax pairs. In particular, we demonstrate that any Lorentzian Einstein-Weyl structure is locally given by a solution to the Manakov-Santini system, and we find a system of two coupled third-order scalar partial differential equations for a general anti-self-dual conformal structure in neutral signature. C 2015 AIP Publishing LLC. [http://dx

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

“…Differentiating each of these relations by a, b, p, q and λ, we obtain 30 relations which, in the non-degenerate case, can be uniquely resolved for all secondorder partial derivatives of P and Q, thus leading to a closed system. It can be verified directly that the resulting system is involutive if and only if the functions f and g satisfy integrability conditions (17). This finishes the proof of Theorem 2.…”

“…Remark 1. Second-order relations (31) governing linearly degenerate systems in 3D are not in involution, and their prolongation implies all third-order integrability conditions (17). This requires differentiating equation (31) three times and solving for the fifth-, fourth-and third-order partial derivatives of f and g in terms of the first-and second-order derivatives.…”

“…Introducing ω by formula (4) and calculating Einstein-Weyl conditions (3) we obtain expressions depending on third-order partial derivatives of u and v. These have to vanish identically modulo (8), in other words, on every solution. Using (8) and its differential consequences to eliminate all partial derivatives involving differentiation by t, we obtain differential expressions that are polynomial in the remaining second-and third-order partial derivatives of u and v. It can be verified directly that the vanishing of all coefficients of these polynomials is equivalent to integrability conditions (17). More precisely, the Einstein-Weyl conditions contain terms of two types: linear in third-order derivatives, and quadratic in second-order derivatives of u and v. Coefficients at third-order derivatives of u and v vanish identically due to the choice of ω, while coefficients at quadratic terms (105 coefficients altogether) give all of the 40 integrability conditions (17).…”

“…Thus, the moduli space of integrable systems depends on 30 − 24 = 6 essential parameters. To prove the above statement let us note that the submanifold in the space of 3-jets of f, g given by integrability conditions (17) projects diffeomorphically onto the space of 2-jets J 2 (R 4 , R 2 ) (which has dimension 30).…”

“…Note that it is sufficient to specify g and ω only, then the first set of equations uniquely determines D. The integrability of Einstein-Weyl equation (3) by twistor-theoretic methods was established by Hitchin [33]. It was shown in [17] that generic Einstein-Weyl structures are governed by the Manakov-Santini system introduced in [34] as a two-component integrable generalisation of the dKP equation. We will prove that for integrable systems (1) the conformal structure g defined by the characteristic variety must be Einstein-Weyl on every solution, with the covector ω expressed in terms of g by the universal formula…”