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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 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 .1 Preliminaries Quotient spacesThroughout this paper, we are going to handle geometric objects of a more general nature than smooth manifolds, which appear when one considers the quotient of a smooth manifold by the action of a Lie group.Definition 1.1. Let X be a topological space. A sheaf of continuous functions on X is a map O X which assigns a subalgebra O X (U ) ⊆ C(U, R) to every open subset U ⊆ X , with the following condition:

We describe the ringed-space structure of moduli spaces of jets of linear connections (at a point) as orbit spaces of certain linear representations of the general linear group.Then, we use this fact to prove that the only (scalar) differential invariants associated to linear connections are constant functions, as well as to recover various expressions appearing in the literature regarding the Poincaré series of these moduli spaces.

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