  The concept of minimum length, widely accepted as a low-energy effect of
quantum gravity, manifests itself in quantum mechanics through generalized
uncertainty principles. Curved momentum space, on the other hand, is at the
heart of similar applications such as doubly special relativity. We introduce a
duality between theories yielding generalized uncertainty principles and
quantum mechanics on nontrivial momentum space. In particular, we find
canonically conjugate variables which map the former into the latter. In that
vein, we explicitly derive the vielbein corresponding to a generic generalized
uncertainty principle in $d$ dimensions. Assuming the predominantly used
quadratic form of the modification, the curvature tensor in momentum space is
proportional to the noncommutativity of the coordinates in the modified
Heisenberg algebra. Yet, the metric is non-Euclidean even in the flat case
corresponding to commutative space, because the resulting momentum basis is
noncanonical. These insides are used to constrain the curvature and the
deviation from the canonical basis.
