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The Einstein theory of general relativity provides a peculiar example ofclassical field theory ruled by non-linear partial differential equations. Anumber of supplementary conditions (more frequently called gauge conditions)have also been considered in the literature. In the present paper, startingfrom the de Donder gauge, which is not conformally invariant but is thegravitational counterpart of the Lorenz gauge, we consider, led by geometricstructures on vector bundles, a new family of gauges in general relativity,which involve fifth-order covariant derivatives of metric perturbations. Areview of recent results by the authors is presented: restrictions on thegeneral form of the metric on the vector bundle of symmetric rank-two tensorfields over space-time; admissibility of such gauges in the case of linearizedtheory about flat Euclidean space; generalization to a suitable class of curvedRiemannian backgrounds, by solving an integral equation. Eventually, theapplications to Euclidean quantum gravity are discussed.Comment: 8 pages, plain Tex. Talk prepared for the Conference: "Nonlinearity,  integrability and all that (twenty years after needs '79)". Gallipoli, July 1  - July 10, 199

NEW SUPPLEMENTARY CONDITIONS FOR A NON-LINEAR FIELD THEORY: GENERAL RELATIVITY