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