  The "metric" structure of nonrelativistic spacetimes consists of a one-form
(the absolute clock) whose kernel is endowed with a positive-definite metric.
Contrarily to the relativistic case, the metric structure and the torsion do
not determine a unique Galilean (i.e. compatible) connection. This subtlety is
intimately related to the fact that the timelike part of the torsion is
proportional to the exterior derivative of the absolute clock. When the latter
is not closed, torsionfreeness and metric-compatibility are thus mutually
exclusive. We will explore generalisations of Galilean connections along the
two corresponding alternative roads in a series of papers. In the present one,
we focus on compatible connections and investigate the equivalence problem
(i.e. the search for the necessary data allowing to uniquely determine
connections) in the torsionfree and torsional cases. More precisely, we
characterise the affine structure of the spaces of such connections and display
the associated model vector spaces. In contrast with the relativistic case, the
metric structure does not single out a privileged origin for the space of
metric-compatible connections. In our construction, the role of the Levi-Civita
connection is played by a whole class of privileged origins, the so-called
torsional Newton-Cartan (TNC) geometries recently investigated in the
literature. Finally, we discuss a generalisation of Newtonian connections to
the torsional case.
