On the class of pseudo-Riemannian geometries which can not be locally described using curvature scalars solely

Abstract

A classic problem with intriguing implications at the level of both applied differential geometry and theoretical physics is dealt with in this short work: Is there any criterion in order to decide whether a pseudo-Riemannian space can be locally described using curvature scalars solely? Surprisingly enough, this question is susceptible of a very simple and elegant answer. In order to avoid unnecessary complexity, the analysis is restricted to local rather than global considerations, without any loss of not only the generality but also the insights to the initial problem.