Moduli spaces for jets of Riemannian metrics at a point

We construct the moduli space of r-jets of Riemannian metrics at a point on a smooth manifold. The construction is closely related to the problem of classification of metric jets via scalar differential invariants.The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratification of moduli spaces is carried out for metrics in dimension n=2n=2.