  We present a new approach to the problem of the thermodynamical equilibrium
of a quantum relativistic fluid in a curved spacetime in the limit of small
curvature. We calculate the mean value of local operators by expanding the
four-temperature Killing vector field in Riemann normal coordinates about the
same spacetime point and we derive corrections with respect to the flat
spacetime expressions. Thereby, we clarify the origin of the terms proportional
to Riemann and Ricci tensors introduced in general hydrodynamic expansion of
the stress-energy tensor.
