  In this brief survey, we will remark the interaction among the Hessian tensor
on a semi-Riemannian manifold and some of the several questions in Lorentzian
(and also in semi-Riemannian) geometry where this 2-covariant tensor is
involved. In particular, we deal with the characterization of Killing vector
fields and the study of a set of consequences of energy conditions in the
framework of standard static space-times.
