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

Understand

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.