We compute all 2-covariant tensors naturally constructed from a semiriemannian metric g which are divergence-free and have weight greater than −2.As a consequence, it follows a characterization of the Einstein tensor as the only, up to a constant factor, 2-covariant tensor naturally constructed from a semiriemannian metric which is divergence-free and has weight 0 (i.e., is independent of the unit of scale). Since these two conditions are also satisfied by the energy-momentum tensor of a relativistic space-time, we discuss in detail how these theorems lead to the field equation of General Relativity.

Let (X, g) be an arbitrary pseudo-riemannian manifold. A celebrated result by Lovelock ([4], [5], [6]) gives an explicit description of all second-order natural (0,2)-tensors on X, that satisfy the conditions of being symmetric and divergence-free. Apart from the dual metric, the Einstein tensor of g is the simplest example.In this paper, we give a short and self-contained proof of this theorem, simplifying the existing one by formalizing the notion of derivative of a natural tensor.

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:

MSC: 58Axx 58A32 53C05 Keywords: Natural operations Chern-Weil formsWe prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior differential. Our result generalises work by Palais [8] and Freed-Hopkins [2].As an application, we also deduce a theorem, originally due to Kolář [3], that determines those natural differential forms that can be associated to a connection on a principal bundle.

Let X be a smooth manifold of dimension 1 + n endowed with a Lorentzian metric g. The energy tensor of a 2-form F is locally defined asIn this paper we characterize this tensor as the only 2-covariant natural tensor associated to a Lorentzian metric and a 2-form that is independent of the unit of scale and satisfies certain condition on its divergence. This characterization is motivated on physical grounds, and can be used to justify the Einstein-Maxwell field equations.More generally, we characterize in a similar manner the energy tensor associated to a differential form of arbitrary order k.Finally, we develop a generalized theory of electromagnetism where charged particles are not punctual, but of an arbitrary fixed dimension p. In this theory, the electromagnetic field F is a differential form of order 2 + p and its electromagnetic energy tensor is precisely the energy tensor associated to F .