Einstein-Weyl geometry, the dKP equation and twistor theory

It is shown that Einstein-Weyl (EW) equations in 2+1 dimensions contain the dispersionless Kadomtsev-Petviashvili (dKP) equation as a special case: if an EW structure admits a constant-weighted vector then it is locally given by h = dy(2) - 4 dx dt - 4u dt(2), v = -4u(x) dt, where u = u (x, y, t) satisfies the dKP equation (u(t) - uu(x))(x) = u(yy). Linearised solutions to the dKP equation are shown to give rise to four-dimensional anti-self-dual conformal structures with symmetries. All four-dimensional hyper-Kahler metrics in signature (+ + - -) for which the self-dual part of the derivative of a Killing vector is null arise by this construction. Two new classes of examples of EW metrics which depend on one arbitrary function of one variable are given, and characterised. A Lax representation of the EW condition is found and used to show that all EW spaces arise as Symmetry reductions of hyper-Hermitian metrics in four dimensions. The EW equations are reformulated in terms of a simple and closed two-form on the CP1-bundle over a Weyl space. It is proved that complex solutions to the dKP equations, module a certain coordinate freedom, are in a one-to-one correspondence with mini-twistor spaces (two-dimensional complex manifolds Z containing a rational curve with normal bundle O(2)) that admit a section of kappa (-1/4), where kappa is the canonical bundle of Z. Real solutions are obtained if the mini-twister space also admits an anti-holomorphic involution with fixed points together with a rational curve and section of kappa (-1/4) that are invariant under the involution. (C) 2001 Elsevier Science B.V. All rights reserved. MSC: 58F07; 81R25.