On vector-tensor minimally coupled field theories

We consider vector-tensor minimally coupled Lagrangians, i.e., scalar densities of the form = g ^1/2 R +L(g _ij ; _i ; _i,j ). We prove that the gauge invariance of any of the sets of Euler-Lagrange expressions implies the gauge invariance of the Lagrangian itself forn even, and an almost gauge invariance forn odd. We also find those for whichE ⁱ () = 0 orE ^ij (L) = 0, generalizing well-known results by Lovelock and a result by the authors.