Functional moduli of jets of Riemannian metrics (Q1392331)

Consider the jet (i.e., the Taylor series) of a Riemannian metric defined in a neighborhood of 0 in \(\mathbb{R}^n\). Formal diffeomorphisms preserving the origin act in an obvious way on the set of such jets. The main result of the present paper is a normal form for these jets up to the action of the formal diffeomorphisms. For example, in two dimensions the normal form is \[ ds^2 = dx^2 + xy\varphi(x,y)dx dy + dy^2, \] where \(\varphi\) is a formal power series in \(x\) and \(y\). Let \(a_k\) denote the dimension of the orbit space of \(k\)-jets of Riemannian metrics modulo formal diffeomorphisms. As an application of the above theorem, the author can show that the Poincaré series \[ p(t) = a_0 + \sum_{k=1}^\infty (a_k-a_{k-1})t^k \] is a rational function.