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

TL;DR

This paper investigates when pseudo-Riemannian geometries can be fully characterized by curvature scalars, providing a simple criterion to distinguish such spaces, with implications for differential geometry and physics.

Contribution

It offers a straightforward criterion to determine if a pseudo-Riemannian space can be locally described solely by curvature scalars.

Findings

Identifies a class of geometries not fully captured by curvature scalars

Provides an elegant criterion for local scalar description

Focuses on local rather than global geometric properties

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.