TITLE: Classification of natural invariants of Riemannian structures
QUESTION [8 upvotes]: Before I formulate my question, let me remind P. B. Gilkey's characterization of Pontryagin forms,following the paper "On the heat equation and the index theorem" by Atiyah, Bott, Patodi.
By definition, a $q$-form valued invariant of Riemannian structures is a function $\omega$ which attaches to any smooth Riemannian manifold $(M,g)$ a $q$-form on it such that for any open imbedding of another smooth manifold $f\colon M'\to M$ one has
$$\omega(M',f^*g)=f^*\omega(M,g).$$
Such an $\omega$ is called regular if the following conditions hold: in any local coordinate system the components of $\omega(M,g)$ are given by polynomials in:
1) the components $g_{ij}$ of the metric tensor;
2) a finite number of derivatives $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$;
3) the inverse of $\det g=\det(g_{ij})$.
Furthermore such an $\omega$ is called homogeneous of weight $k$ if for any $\lambda>0$
$$\omega(M,\lambda^2g)=\lambda^k\omega(M,g).$$
Theorem (Gilkey). The only regular $q$-form valued invariants of structures of non-negative weights are polynomials in the Pontryagin forms (thus must have weight 0).
QUESTION. Is there anything known about $q$-form invariants of negative weights, or on other tensor-valued invariants (not forms)? I am particularly curious in the case of functions, i.e. 0-forms of negative weight.
Below are some examples of such regular $0$-form (function) valued invariants of negative weight.
Example 1. The scalar curvature of the metric. It has weight -2.
Example 2. More generally, let us denote by $R_{ijkl}$ the Riemann curvature tensor, and by $R^{ij}_{kl}$ this tensor with the first two indices lifted up using the metric.
For any even number $e\in [0,n],\, n:=\dim M$ let us define the following functions on $M$:
\begin{eqnarray*}
H_e=\sum_{\alpha,\beta}sgn(\alpha,\beta)R^{\beta_1\beta_2}_{\alpha_1\alpha_2}\dots
R^{\beta_{e-1}\beta_e}_{\alpha_{e-1}\alpha_e},
\end{eqnarray*}
where the sum runs over all sequences of elements in the set $\{1,2,\dots,n\}$ such that all elements in $\alpha$ are distinct, all elements in $\beta$ are also distinct, and elements of $\beta$ are just permutation of $\alpha$. The sign of this permutation is denoted above by $sgn(\alpha,\beta)$.
Remark. If in Example 2 one takes $e=2$, one recovers the scalar curvature from EXample 1. Expressions from Example 2 have interesting geometric meaning: these are coefficients in the H. Weyl's formula for volume of an $\varepsilon$-neighborhood of a submanifold in a Eucliden space.

REPLY [4 votes]: The objects you are after are sometimes called invariant tensors or curvature tensors. The first observation that simplifies their classification is that, by the invariance (or covariance, if you wish) criterion, the dependence on $\frac{\partial^\alpha}{\partial x^\alpha}g_{ij}$ always factors through the Riemann curvature tensor and its covariant derivatives. This result is sometimes known as the Thomas replacement theorem (see arXiv:gr-qc/9404030 by Anderson & Torre for a proof).
In certain branches of mathematical physics, computing and classifying invariant tensors once constituted a minor industry. See for instance Fulling et al. CQG 9 1151, as well as citations therein and thereof. There, the classification is not exactly by weight, but by rank (number of tensor indices), order (of the highest derivative of the metric) and degree (number of factors of the Riemann tensor or one of its derivatives). But I think you should be able to reconstruct the weight from that information.