  We develop the geometric description of submanifolds in Newton--Cartan
spacetime. This provides the necessary starting point for a covariant spacetime
formulation of Galilean-invariant hydrodynamics on curved surfaces. We argue
that this is the natural geometrical framework to study fluid membranes in
thermal equilibrium and their dynamics out of equilibrium. A simple model of
fluid membranes that only depends on the surface tension is presented and,
extracting the resulting stresses, we show that perturbations away from
equilibrium yield the standard result for the dispersion of elastic waves. We
also find a generalisation of the Canham--Helfrich bending energy for lipid
vesicles that takes into account the requirements of thermal equilibrium.
