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In this paper we show how to translate into tensorial language the Chern-Weiltheorem for the Lorentz symmetry, which equates the difference of the Eulerdensities of two manifolds to the exterior derivative of a transgression form.For doing so we need to introduce an auxiliary, hybrid, manifold whose geometrywe construct explicitely. This allows us to find the vector density,constructed out of spacetime quantities only, whose divergence is the exteriorderivative of the transgression form. As a consequence we can show how theEinstein-Hilbert, Gauss-Bonnet and, in general, the Euler scalar densities canbe written as the divergences of genuine vector densities in the criticaldimensions $D=2,4$, etc. As Lovelock gravity is a dimensional continuation ofEuler densities, these results are of relevance for Gauss-Bonnet and, ingeneral, Lovelock gravity. Indeed, these vectors which can be calledgeneralized Katz vectors ensure, in particular, a well-posed Dirichletvariational principle.

Chern-Weil theorem, Lovelock Lagrangians in critical dimensions and boundary terms in gravity actions