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Finding suitable diffeomorphism-invariant observables to probe gravity at
the Planck scale is essential in quantum gravity. The Wilson loop of the
4-dimensional Christoffel connection is a potentially interesting
ingredient for the construction of such an observable. We have
investigated to what extent and what form of curvature information of the
underlying spacetime may be extracted from Wilson loops through a Stokes’
theorem-like relation. We present an expression for the conservation of
geometric flux as the quantity related to the gravitational Wilson loop.
This expression is surface-independent and it holds for a certain class of manifolds with global symmetries.