Exotic holonomy on moduli spaces of rational curves

Bryant [3] proved the existence of torsion free connections with exotic holonomy, i.e., with holonomy that does not occur on the classical List of Berger [1]. These connections occur on moduli spaces y of rational contact curves in a contact threefold W. Therefore, they are naturally contained in the moduli space Z of all rational curves in W. We construct a connection on Z whose restriction to y is torsion free. However, the connection on Z has torsion unless both y and Z are flat. This answers a question of Bryant as to whether the GL(2, C) x SL(2, C)structures which arise from such a moduli space Z always admit a torsion free connection in the negative. We also show the existence of a new exotic holonomy which is a certain six-dimensional representation of SL(2, C) x SL(2, C). We show that every regular H-3-connection(cf. [3]) is the restriction of a unique connection with this holonomy.