The Riemannian Goldberg–Sachs Theorem

Abstract

The paper contains a description of compact Hermitian complex surfaces whose Riemannian Ricci tensor is of type (1,1). This in turn comes as a consequence of a Riemannian version of the well-known (generalized) Goldberg–Sachs theorem of the General Relativity. A complete proof of the Riemannian version is given in the framework of "classical" Hermitian geometry. The paper includes some more results also pertaining to "Riemannian Goldberg–Sachs theory", as well as a "dual theory" involving the Penrose operator.