  The goal of this paper is to present a formalism that allows to handle
four-fermion effective theories at finite temperature and density in curved
space. The formalism is based on the use of the effective action and zeta
function regularization, supports the inclusion of inhomogeneous and
anisotropic phases. One of the key points of the method is the use of a
non-perturbative ansatz for the heat-kernel that returns the effective action
in partially resummed form, providing a way to go beyond the approximations
based on the Ginzburg-Landau expansion for the partition function. The
effective action for the case of ultra-static Riemannian spacetimes with
compact spatial section is discussed in general and a series representation,
valid when the chemical potential satisfies a certain constraint, is derived.
To see the formalism at work, we consider the case of static Einstein spaces at
zero chemical potential. Although in this case we expect inhomogeneous phases
to occur only as meta-stable states, the problem is complex enough and allows
to illustrate how to implement numerical studies of inhomogeneous phases in
curved space. Finally, we extend the formalism to include arbitrary chemical
potentials and obtain the analytical continuation of the effective action in
curved space.
