On the essential constants in Riemannian geometries

TL;DR

This paper provides a covariant criterion to distinguish essential constants in Riemannian metrics, reducing the problem to solving first-order PDEs, with a focus on local, smooth metric tensors.

Contribution

It introduces a necessary and sufficient covariant condition to identify whether a constant in a Riemannian metric is essential or spurious.

Findings

The criterion is expressed as a system of first-order PDEs.

The analysis is purely local and applies to smooth metric tensors.

The method simplifies the identification of essential constants in Riemannian geometries.

Abstract

In the present work the problem of distinguishing between essential and spurious (i.e., absorbable) constants contained in a metric tensor field in a Riemannian geometry is considered. The contribution of the study is the presentation of a sufficient and necessary criterion, in terms of a covariant statement, which enables one to determine whether a constant is essential or not. It turns out that the problem of characterization is reduced to that of solving a system of partial differential equations of the first order. In any case, the metric tensor field is assumed to be smooth with respect to the constant to be tested. It should be stressed that the entire analysis is purely of local character.