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Let pC : C = CN → N be the bundle of linear connections on a smooth manifold N and let pM : M ! N be the bundle of pseudo-Riemannian metrics of a given signature (n₊ ; n-) , n₊ n- = n = dimN on N. The structure of the rst-order Lagrangians dened on the bundle M x N C → N that are invariant under the natural action of the diffeomorphisms of N, is determined.

Structure of diffeomorphism-invariant Lagrangians on the product bundle of metrics and linear connections