  We consider non-relativistic curved geometries and argue that the background
structure should be generalized from that considered in previous works. In this
approach the derivative operator is defined by a Galilean spin connection
valued in the Lie algebra of the Galilean group. This includes the usual spin
connection plus an additional "boost connection" which parameterizes the
freedom in the derivative operator not fixed by torsion or metric
compatibility. As an example we write down the most general theory of
dissipative fluids consistent with the second law in curved non-relativistic
geometries and find significant differences in the allowed transport
coefficients from those found previously. Kubo formulas for all response
coefficients are presented. Our approach also immediately generalizes to
systems with independent mass and charge currents as would arise in
multicomponent fluids. Along the way we also discuss how to write general
locally Galilean invariant non-relativistic actions for multiple particle
species at any order in derivatives. A detailed review of the geometry and its
relation to non-relativistic limits may be found in a companion paper
[arXiv:1503.02682].
