  The Heisenberg Uncertainty Principle (HUP) limits the accuracy in the
simultaneous measurements of the position and momentum variables of any quantum
system. This is known to be true in the context of non-relativistic quantum
mechanics. Based on a semiclassical geometric approach, here we propose an
effective generalization of this principle, which is well-suited to be extended
to general relativity scenarios as well. We apply our formalism to
Schwarzschild and de Sitter space-time, showing that the ensuing uncertainty
relations can be mapped into well-known deformations of the HUP. We also infer
the form of the perturbed metric that mimics the emergence of a discrete
spacetime structure at Planck scale, consistently with the predictions of the
Generalized Uncertainty Principle. Finally, we discuss our results in
connection with other approaches recently appeared in the literature.
