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We classify invariant Lagrangians of the form$L(g_{ij},g_{ij,k},g_{ij,kl},D_I,D_{I,j})$ depending at most quadratically onthe variables $g_{ij,k},g_{ij,kl}$ and $D_I,D_{I,j}$, where $g$ is a Lorentzmetric and $D$ is a tensor field of arbitrary rank on a smooth manifold. As acorollary, we prove a conjecture of Bray's regarding the classification ofcertain variational principles with variables a Lorentz metric and an affineconnection.

Low-order geometric actions with fields a metric and a matter field of arbitrary rank (undergraduate honors thesis)