  Quantum inequalities bound the extent to which weighted time averages of the
renormalized energy density of a quantum field can be negative. They have
mostly been proved in flat spacetime, but we need curved-spacetime inequalities
to disprove the existence of exotic phenomena, such as closed timelike curves.
In this work we derive such an inequality for a minimally-coupled scalar field
on a geodesic in a spacetime with small curvature, working to first order in
the Ricci tensor and its derivatives. Since only the Ricci tensor enters, there
are no first-order corrections to the flat-space quantum inequalities on paths
which do not encounter any matter or energy.
