  There is significant recent work on coupling matter to Newton-Cartan
spacetimes with the aim of investigating certain condensed matter phenomena. To
this end, one needs to have a completely general spacetime consistent with
local non-relativisitic symmetries which supports massive matter fields. In
particular, one can not impose a priori restrictions on the geometric data if
one wants to analyze matter response to a perturbed geometry. In this paper we
construct such a Bargmann spacetime in complete generality without any prior
restrictions on the fields specifying the geometry. The resulting spacetime
structure includes the familiar Newton-Cartan structure with an additional
gauge field which couples to mass. We illustrate the matter coupling with a few
examples. The general spacetime we construct also includes as a special case
the covariant description of Newtonian gravity, which has been thoroughly
investigated in previous works. We also show how our Bargmann spacetimes arise
from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion
paper [arXiv:1503.02680] we use this Bargmann spacetime structure to
investigate the details of matter couplings, including the Noether-Ward
identities, and transport phenomena and thermodynamics of non-relativistic
fluids.
