  In this work we investigate the relation between curved momentum space and
momentum-dependent gauge fields. While the former is a classic idea that has
been shown to be tied to minimal-length models, the latter constitutes a
relatively recent development in quantum gravity phenomenology. In particular,
the gauge principle in momentum space amounts to a modification of the position
operator of the form $\hat{X}^\mu\rightarrow\hat{X}^\mu-g A^\mu (\hat{P})$ akin
to a gauge-covariant derivative in momentum space according to the minimal
coupling prescription. Here, we derive both effects from a Kaluza-Klein
reduction of a higher-dimensional geometry exhibiting curvature in momentum
space. The interplay of the emerging gauge fields as well as the remaining
curved momentum space lead to modifications of the Heisenberg algebra. While
the gauge fields imply Moyal-type noncommutativity dependent on the analogue
field strength tensor, the dimensionally reduced curved momentum space geometry
translates to a Snyder-type noncommutative geometry.
