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On the Einstein-Weyl and conformal self-duality equations. (English) Zbl 1325.53058
Summary: 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.
©2015 American Institute of Physics

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C28 Twistor methods in differential geometry
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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