THE SPACE OF LEAVES OF A SHEAR-FREE CONGRUENCE, MULTIPOLE EXPANSIONS, AND ROBINSON THEOREM

What it means for (relative) sheaf cohomology classes to have a pole of a given order on a surface S in twistor space will be defined and that they can be described in terms of some formal neighborhood sheaves will be shown. In space-time, S corresponds to a foliation by alpha-surfaces and the filtration of cohomology gives a filtration on the fields that extends the idea of being algebraically special along the foliation. This idea is also used for the case of the "double-valued" congruence associated with a world line, in which case the filtration applied to soruced fields is essentially a multipole expansion. In the case of curved space-times, it will be shown that if a certain curvature condition holds, then the space of leaves of a foliation by alpha surfaces has an ambient twistor space defined to first order, and we relate this to an extended version of Robinson's theorem.