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Résumé |
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. |
