  We show that the Lagrangian for interacting nonrelativistic particles can be
coupled to an external gauge field and metric tensor in a way that exhibits a
nonrelativistic version of general coordinate invariance. We explore the
consequences of this invariance on the example of the degenerate Fermi gas at
infinite scattering length, where conformal invariance also plays an important
role. We find the most general effective Lagrangian consistent with both
general coordinate and conformal invariance to leading and next-to-leading
orders in the momentum expansion. At the leading order the Lagrangian contains
one phenomenological constant and reproduces the results of the Thomas-Fermi
theory and superfluid hydrodynamics. At the next-to-leading order there are two
additional constants. We express various physical quantities through these
constants.
