2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/214133In 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.12 pages, changed title and contentDifferential GeometryGeneral Relativity and Quantum CosmologyMathematical Physics53C21; 53C50; 53C80text