  The minimal-length paradigm is a cornerstone of quantum gravity
phenomenology. Recently, it has been demonstrated that minimal-length quantum
mechanics can alternatively be described as an undeformed theory set on a
nontrivial momentum space. However, there is no fully consistent formulation of
these theories beyond Cartesian coordinates in flat space and, in particular,
no position representation. This paper is intended to take the first steps in
bridging this gap. We find a natural position representation of the position
and momentum operators on general curved cotangent bundles. In an expansion
akin to Riemann normal coordinates with curvature in both position and momentum
space, we apply the formalism perturbatively to the isotropic harmonic
oscillator and the hydrogenic atom. Due to the symmetry of the harmonic
oscillator under exchange of positions and momenta, we show that it is
impossible to distinguish position- from momentum- space curvature with
oscillators alone. Thus, we obtain an instantiation of Born reciprocity on the
curved cotangent bundle, i. e. in precisely the way Born originally envisioned.
It manifests itself as a symmetry mixing UV and IR physics, reminiscent of
T-duality in string theory.
