The definition of scalar and vector concomitant of a metric

Question

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: Link

In the section 4, the scalar concomitant of the metric tensor g is defined in the sense of Schouten, 1954. And I found it's a book called Tensor Analysis for Physicists and it's very hard to locate that particular definition in the book. Also, I checked online but did not find very clear and straightforward definition of a scalar(vector) concomitant of the metric. I would like to ask if someone who could give a lucid and precise definition of that term.

Thanks.

Answer 1

Schouten's "Tensor Analysis for Physicists" is a kind of gentle introduction, and he does not use the word "concomitant" there. The reference in Defrise-Carter's paper is to the English translation (1954) of Schouten's "Ricci Kalkül" (1924).

The term "concomitant" comes from the ancient (XIX century) invariant theory. It has many uses and varying senses. See the semantics in this Wikipedia disambiguation article. It has further links to an invariant theory meaning and an algebraic-geometric meaning, both being the reflections of their respective authors' understanding. Those definitions, albeit correct, are not fully relevant to the article, which you are reading.

In modern differential geometry, there is a tendency to use the term "invariant" instead of antiquated "concomitants", "covariants" etc, however they are still frequently encountered in the literature.

A very clear general definition of a concomitant in differential geometry is given in the thesis of J.B. Pitts "General Covariance, Artificial Gauge Freedom and Empirical Equivalence" (2008), on p. 86:

A geometric object $\psi$ is called a concomitant of another geometric object $\phi$ if the components of $\phi$ determine those of $\psi$ and do so in the same way in every coordinate system.

The words "the same way" mean that $\psi$ is given by a universal formula. The requirement "in every coordinate system" means that the quantity $\psi$ is natural, that is it must commute with the pullbacks of diffeomorphisms.

Just to give an example, the scalar curvature in Riemannian geometry is a scalar concomitant of the Riemannian metric (the given geometric structure) in the above sense. Further examples are all the curvature invariants.

It is useful to compare this definition with the point of view of classical invariant theory, see e.g. p.69 in Dolgachev's "Lectures on Invariant Theory".