Null geodesic surfaces and Goldberg–Sachs theorem in complex Riemannian spaces

Abstract

An extension of the Goldberg–Sachs theorem for the case of a complex V4 is given with a simple proof. The interpretation of the theorem, however, no longer applies the concept of the geodesic and shearless congruence of null directions; instead, the existence of a geodesic 2-surface (complex), the tangent vectorial space to which (i) contains only null vectors, (ii) is parallelly propagated along the surface, is now essential.