  Minimal and maximal uncertainties of position measurements are widely
considered possible hallmarks of low-energy quantum as well as classical
gravity. While General Relativity describes interactions in terms of spatial
curvature, its quantum analogue may also extend to the realm of curved momentum
space as suggested, e. g. in the context of Relative Locality in Deformed
Special Relativity. Drawing on earlier work, we show in an entirely Born
reciprocal, i. e. position and momentum space covariant, way that the quadratic
Generalized Extended Uncertainty principle can alternatively be described in
terms of quantum dynamics on a general curved cotangent manifold. In the case
of the Extended Uncertainty Principle the curvature tensor in position space is
proportional to the noncommutativity of the momenta, while an analogous
relation applies to the curvature tensor in momentum space and the
noncommutativity of the coordinates for the Generalized Uncertainty Principle.
In the process of deriving this map, the covariance of the approach constrains
the admissible models to an interesting subclass of noncommutative geometries
which has not been studied before. Furthermore, we reverse the approach to
derive general anisotropically deformed uncertainty relations from general
background geometries. As an example, this formalism is applied to (anti)-de
Sitter spacetime.
