COMPLEX PARACONFORMAL MANIFOLDS - THEIR DIFFERENTIAL GEOMETRY AND TWISTOR-THEORY

A complex paraconformal manifold is a pq-dimensional complex manifold (p, q greater-than-or-equal-to 2) whose tangent bundle factors as a tensor product of two bundles of ranks p and q. We also assume that we are given a fixed isomorphism of the highest exterior powers of the two bundles. Examples of such manifolds include 4-dimensional conformal manifolds (with spin structure) and complexified quaternionic, quaternionic Kahler and hyperKahler manifolds. We develop the differential geomety of these structures, which is formally very similar to that of the special case of four dimensional conformal structures [30]. The examples have the property that they have a rich twistor theory, which we discuss in a unified way in the paraconformal category. In particular, we consider the 'non-linear graviton' construction [29], and discuss the structure on the twistor space corresponding to quaternionic Kahler and hyperKahler metrics. We also define a family of special curves for these structures which in the 4-dimensional conformal case coincide with the conformal circles [34,2]. These curves have an intrinsic, naturally defined projective structure. In the particular case of complexified 4k-dimensional quaternionic structures, we obtain a distinguished 8k + 1 parameter family of special curves satisfying a third order ODE in local coordinates.