  A good hundred years after the necessity for a quantum theory of gravity was
acknowledged by Albert Einstein, the search for it continues to be an ongoing
endeavour. Nevertheless, the field still evolves rapidly as manifested by the
recent rise of quantum gravity phenomenology supported by an enormous surge in
experimental precision. In particular, the minimum length paradigm ingrained in
the program of generalized uncertainty principles (GUPs) is steadily growing in
importance. The present thesis is aimed at establishing a link between modified
uncertainty relations, derived from deformed canonical commutators, and curved
spaces - specifically, GUPs and nontrivial momentum space as well as the
related extended uncertainty principles (EUPs) and curved position space. In
that vein, we derive a new kind of EUP relating the radius of geodesic balls,
assumed to constrain the wave functions in the underlying Hilbert space, with
the standard deviation of the momentum operator. This result is gradually
generalized to relativistic particles in curved spacetime in accordance with
the 3+1 decomposition. In a sense pursuing the inverse route, we find an
explicit correspondence between theories yielding a GUP and quantum dynamics
set on non-Euclidean momentum space. Quantitatively, the coordinate
noncommutativity translates to momentum space curvature in the dual
description, allowing for an analogous transfer of constraints from the
literature. Finally, we find a formulation of quantum mechanics which proves
consistent on the arbitrarily curved cotangent bundle. Along these lines, we
show that the harmonic oscillator can not be used as a means to distinguish
between curvature in position and momentum space, thereby providing an explicit
instantiation of Born reciprocity in the context of curved spaces.
