On higher dimensional Einstein spacetimes with a non-degenerate double Weyl aligned null direction

Abstract

We prove that higher dimensional Einstein spacetimes which possess a geodesic, non-degenerate double Weyl aligned null direction (WAND) must additionally possess a second double WAND (thus being of type D) if either: (a) the Weyl tensor obeys (, i.e. the Weyl type is II(abd)); (b) is twistfree. Some comments about an extension of the Goldberg–Sachs theorem to six dimensions are also made.