Local differential geometry and invariant theory

Question

Can someone please give me pointers to the literature for local differential differential geometry according to invariant theory in the following sense, provided such a literature exists?

Start with the observation that any given diffeomorphism $\rho:M\rightarrow M$ of a Riemannian manifold $M$ allows for pulling the metric on $M$ back along the map $\rho$.

Now consider just those Riemannian metrics on a neighborhoods of $0$ in ${\Bbb R}^n$ given by analytic functions $m$ on ${\Bbb R}^n$. $m$ takes values, of course, in real, symmetric, positive-definite $n\times n$ matrices. Assume also (without serious loss of generality) that $m(0)$ equals the identity matrix.

Locally, one can get a new (still analytic) metric by pulling such an $m$ back along any analytic injection $\iota$ from a neighborhood of $0$ into ${\Bbb R}^n$ that fixes $0$. If, furthermore, $\iota$ has its Jacobian at $0$ sitting in the orthogonal group, the new metric at $0$ will still get represented by the identity matrix.

Viewed only up to order $j+1$, these analytic injections $\iota$ will form a finite-dimensional Lie group. This group acts on metrics $m$ viewed up to order $j$ (as one takes derivatives to form the Jacobians needed to pull back metrics).

Now metrics themselves don't form a linear space (taking differences can kill positive definiteness). But (germs of) functions $m - m(0)$ do comprise a linear space, and when viewed only up to order $j$, a finite-dimensional linear space.

So if I said everything right, I have a finite-dimensional Lie group acting on a finite dimensional linear space, and invariant theory should have something to say about classifying the orbits. Metrics in a common orbit will then share all local geometric invariants up to order $k$.

Answer 1

This question was answered decisively by Weyl and Cartan. The essential point is that there is a canonical way to reduce this nonlinear action to the linear action of $\mathrm{O}(n)$ on a finite dimensional vector space: Use normal coordinates.

The point is that a metric $g$ can be written in $p$-centered coordinates $x=(x^i)$ in the form $$ g = g_{ij}(x)\,\mathrm{d}x^i\,\mathrm{d}x^j $$ where $g_{ij}(x)$ satisfies the system of $n$ (affine) linear equations $$ g_{ij}(x)x^j = x^i. $$ Such coordinates are normal coordinates for $g$ about $p$ and are unique up to a rotation of the form $\bar x^i = A^i_j\,x^j$ where $A = (A^i_j)$ is orthogonal. The $k$-jet of $g$ in these coordinates is determined by the $k$-jets of the $g_{ij}(x)$ satisfying the above relation. (Of course, the $0$-jet is fixed, since that is $g_{ij}(0)=\delta_{ij}$.) The group $\mathrm{O}(n)$ acts on the rest of the polynomial solutions of the above equation of degree $k$ or less in the obvious linear fashion.

Now one can apply standard invariant theory to that sum of representations, as Weyl does, to write down a generating set of all of the $\mathrm{O}(n)$-invariant polynomials on this vector space. What Weyl does is show that these invariants are all generated by taking the Riemann curvature tensor and its first $k{-}2$ covariant derivatives with respect to the Levi-Civita connection at $p$, and then forming all possible contractions on the direct sum using the metric tensor. Consequently, all of the pointwise invariants can be expressed in terms of the Riemann curvature tensor and its covariant derivatives.