  We observe that in type I twistless torsional Newton--Cartan (TTNC) geometry,
one can always find (at least locally) a gauge transformation that transforms a
specific locally Galilei-invariant function -- that we dub the `locally
Galilei-invariant potential' -- to zero, due to the corresponding equation for
the gauge parameter taking the form of a Hamilton--Jacobi equation. In the case
of type II TTNC geometry, the same gauge fixing may locally be performed by
subleading spatial diffeomorphisms. We show (a) how this generalises a
classical result in standard Newton--Cartan geometry, and (b) how it allows to
parametrise the metric structure of a Galilei manifold as well as the gauge
equivalence class of the Bargmann form of TTNC geometry in terms of just the
space metric and a unit timelike vector field.
