Moduli spaces for finite-order jets of Riemannian metrics
  We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via 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=2. 
A. Gordillo, J. Navarro, and J. B. Sancho