  The Einstein theory of general relativity provides a peculiar example of
classical field theory ruled by non-linear partial differential equations. A
number of supplementary conditions (more frequently called gauge conditions)
have also been considered in the literature. In the present paper, starting
from the de Donder gauge, which is not conformally invariant but is the
gravitational counterpart of the Lorenz gauge, we consider, led by geometric
structures on vector bundles, a new family of gauges in general relativity,
which involve fifth-order covariant derivatives of metric perturbations. A
review of recent results by the authors is presented: restrictions on the
general form of the metric on the vector bundle of symmetric rank-two tensor
fields over space-time; admissibility of such gauges in the case of linearized
theory about flat Euclidean space; generalization to a suitable class of curved
Riemannian backgrounds, by solving an integral equation. Eventually, the
applications to Euclidean quantum gravity are discussed.
