  Finding diffeomorphism-invariant observables to characterize the properties
of gravity and spacetime at the Planck scale is essential for making progress
in quantum gravity. The holonomy and Wilson loop of the Levi-Civita connection
are potentially interesting ingredients in the construction of quantum
curvature observables. Motivated by recent developments in nonperturbative
quantum gravity, we establish new relations in three and four dimensions
between the holonomy of a finite loop and certain curvature integrals over the
surface spanned by the loop. They are much simpler than a gravitational version
of the nonabelian Stokes' theorem, but require the presence of totally geodesic
surfaces in the manifold, which follows from the existence of suitable Killing
vectors. We show that the relations are invariant under smooth surface
deformations, due to the presence of a conserved geometric flux.
