The Riemannian Goldberg-Sachs theorem

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.