  Superfluids under an intense gravitational field are typically found in
compact stellar cores. Most treatments of these superfluids, however, are done
using a flat spacetime background. In this paper, the effect of spacetime
curvature on relativistic superfluids is investigated. The scalar-field
superfluid in a background metric of a typical star core is considered first.
It is found that the superfluid formed inside a compact object can not be of
spherical shape. Explicit numerical calculation of the gravitational correction
to the superfluid order parameter is performed for two specific examples with
different boundary conditions. It is found that even in the weak-gravity limit,
gravity can have a significant effect on superfluidity. The relativistic
superfluids formed by various fermion pairings are also considered. Two
possible cases are considered in the mean-field treatment: antifermion-fermion
and fermion-fermion pairings. The effective actions and the self-consistent
equations are derived for both cases. An analytical expression for the
first-order gravitational correction to the superfluid order parameter is given
for the antifermion-fermion pairing; while, the analytical expressions for the
matrix elements of the heat kernel operator, which is useful in
curved-spacetime QFT calculations, is derived for the fermion-fermion pairing.
